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Homotopy Type Theory: Univalent Foundations of Mathematics

2013-01-01
Peter Aczel , Benedikt Ahrens , Thorsten Altenkirch +7 more
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of … Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
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Quotient Inductive-Inductive Types

2018-01-01
Thorsten Altenkirch , Paolo Capriotti , Gabe Dijkstra +2 more
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Two-Level Type Theory and Applications.

2017-05-09
Danil Annenkov , Paolo Capriotti , Nicolai Kraus
We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

Constructions with Non-Recursive Higher Inductive Types

2016-07-05
Nicolai Kraus
Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to … Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs.
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Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type

2016-10-28
Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus
Capretta's delay monad can be used to model partial computations, but it has the wrong notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by … Capretta's delay monad can be used to model partial computations, but it has the wrong notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the right notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice. Using an idea from homotopy type theory - a higher inductive-inductive type - we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.
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Partiality, Revisited

2017-01-01
Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus
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Univalent higher categories via complete Semi-Segal types

2017-12-27
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first ( n +3) levels of a semisimplicial type S , we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n -type . This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n -category (more precisely, ( n ,1)-category) for n ∈ {0,1,2}, and we show that the type of complete semi-Segal n -types is equivalent to the type of univalent n -categories in these cases. Thus, we believe that the notion of a complete semi-Segal n -type can be taken as the definition of a univalent n -category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
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Notions of Anonymous Existence in Martin-Löf Type Theory

2016-01-01
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we … As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.

Two-level type theory and applications

2023-05-30
Danil Annenkov , Paolo Capriotti , Nicolai Kraus +1 more
We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.
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Extending Homotopy Type Theory with Strict Equality

2016-04-13
Thorsten Altenkirch , Paolo Capriotti , Nicolai Kraus
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of … In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an one, containing a strict equality type former, and an one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of infinite structures which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with schematic definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

Generalizations of Hedberg’s Theorem

2013-01-01
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
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Higher Homotopies in a Hierarchy of Univalent Universes

2015-03-21
Nicolai Kraus , Christian Sattler
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding … For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.
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Free Higher Groups in Homotopy Type Theory

2018-06-27
Nicolai Kraus , Thorsten Altenkirch
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this … Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit: F(A), cons: A~F(A)~F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [22, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ||F(A)||1 is a set.
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Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-06-01
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
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The General Universal Property of the Propositional Truncation

2015-01-01
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega^op. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of coherently constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in standard syntactical homotopy type theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a propositional type Q with A -> Q and Q -> B.

Space-Valued Diagrams, Type-Theoretically (Extended Abstract).

2017-04-14
Nicolai Kraus , Christian Sattler
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion … Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.

Two-Level Type Theory and Applications

2017-01-01
Danil Annenkov , Paolo Capriotti , Nicolai Kraus +1 more
We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

Coherence via Well-Foundedness

2020-05-26
Nicolai Kraus , Jakob von Raumer
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of … Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
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A Lambda Term Representation Inspired by Linear Ordered Logic

2011-10-30
Andreas Abel , Nicolai Kraus
We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, … We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework. Keywords: representation of binders, explicit substitutions, ordered contexts, space leaks, Logical Framework.
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Notions of Anonymous Existence in Martin-L\"of Type Theory

2017-03-24
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we … As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda. Comment: 36 pages, to appear in the special issue of TLCA'13 (LMCS)
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Functions out of Higher Truncations

2015-07-04
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of … In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B that are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.

Extending Homotopy Type Theory with Strict Equality

2016-01-01
Thorsten Altenkirch , Paolo Capriotti , Nicolai Kraus
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of … In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an "outer" one, containing a strict equality type former, and an "inner" one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of "infinite structures" which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.
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Univalent Higher Categories via Complete Semi-Segal Types

2018-01-07
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (Ahrens … Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (Ahrens et al. 2015) solves this by considering only truncated types, and it roughly corresponds to an ordinary category. The drawback is that univalent categories fail to capture many naturally occurring structures, such as type universes or the type of univalent categories themselves. This stems from the fact that the natural notion of a category in homotopy type theory is not that of an ordinary, but rather a higher category.

Shallow Embedding of Type Theory is Morally Correct

2019-01-01
Ambrus Kaposi , András Kovács , Nicolai Kraus
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Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

2021-06-29
Nicolai Kraus
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or … Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type Con of contexts, a type family Ty indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP. In this paper, we develop a notion of ∞-categories with families (∞-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised (set-level) syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT). Apart from introducing ∞-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.
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Space-Valued Diagrams, Type-Theoretically (Extended Abstract)

2017-01-01
Nicolai Kraus , Christian Sattler
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion … Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.
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Type-theoretic approaches to ordinals

2023-03-26
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
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Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-01-17
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.

A rewriting coherence theorem with applications in homotopy type theory

2022-08-01
Nicolai Kraus , Jakob von Raumer
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. … Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
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The General Universal Property of the Propositional Truncation

2014-01-01
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B.
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Set-Theoretic and Type-Theoretic Ordinals Coincide

2023-06-26
Tom de Jong , Nicolai Kraus , Fredrik Nordvall Forsberg +1 more
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We … In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.
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Functions out of Higher Truncations

2015-07-04
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and … In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B which are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.

The General Universal Property of the Propositional Truncation

2014-11-11
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B.

From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2019-02-27
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher groupoids. Bezem, Coquand, and Huber have presented a constructive model of univalence using a specific cube category, which we call the BCH category. The higher categories encoded with the BCH category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).

From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2020-01-01
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category. The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).

Connecting Constructive Notions of Ordinals in Homotopy Type Theory

2021-01-01
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We … In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.
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On symmetries of spheres in univalent foundations

2024-06-21
Pierre Cagne , Ulrik Buchholtz , Nicolai Kraus +1 more
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Univalent Higher Categories via Complete Semi-Segal Types

2017-07-12
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated to homotopy, and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, … Category theory in homotopy type theory is intricate as categorical laws can only be stated to homotopy, and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first (n+3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n,1)-category) for n in {0,1,2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent $n$-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.

Shallow Embedding of Type Theory is Morally Correct

2019-07-17
Ambrus Kaposi , András Kovács , Nicolai Kraus
There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary … There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-Lof type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.

Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory

2020-01-21
Nicolai Kraus , Jakob von Raumer
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of … Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence. Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty. We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.
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Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT

2020-01-01
Nicolai Kraus
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or … Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type CON of contexts, a type family TY indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP. In this paper, we develop a notion of $\infty$-categories with families ($\infty$-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT). Apart from introducing $\infty$-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.

Shallow Embedding of Type Theory is Morally Correct

2019-01-01
Ambrus Kaposi , András Kovács , Nicolai Kraus
There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary … There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-L\"of type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.
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From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2019-01-01
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher groupoids. Bezem, Coquand, and Huber have presented a constructive model of univalence using a specific cube category, which we call the BCH category. The higher categories encoded with the BCH category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).
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Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-01-01
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.
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Functions out of Higher Truncations

2015-01-01
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and … In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B which are constant on all (n+1)-st loop spaces. We give one "elementary" proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct "set-based" representations of 1-types, as long as they have "braided" loop spaces. The main result with one of its proofs and the application have been formalised in Agda.
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Type-Theoretic Approaches to Ordinals

2022-01-01
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences … In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensional equality. Using homotopy type theory as the foundational setting, we develop an abstract framework for ordinal theory and establish a collection of desirable properties and constructions. We then study and compare three concrete implementations of ordinals in homotopy type theory: first, a notation system based on Cantor normal forms (binary trees); second, a refined version of Brouwer trees (infinitely-branching trees); and third, extensional well-founded orders. Each of our three formulations has the central properties expected of ordinals, such as being equipped with an extensional and well-founded ordering as well as allowing basic arithmetic operations, but they differ with respect to what they make possible in addition. For example, for finite collections of ordinals, Cantor normal forms have decidable properties, but suprema of infinite collections cannot be computed. In contrast, extensional well-founded orders work well with infinite collections, but almost all properties are undecidable. Brouwer trees take the sweet spot in the middle by combining a restricted form of decidability with the ability to work with infinite increasing sequences. Our three approaches are connected by canonical order-preserving functions from the "more decidable" to the "less decidable" notions. We have formalised the results on Cantor normal forms and Brouwer trees in cubical Agda, while extensional well-founded orders have been studied and formalised thoroughly by Escardo and his collaborators. Finally, we compare the computational efficiency of our implementations with the results reported by Berger.
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Univalent Higher Categories via Complete Semi-Segal Types

2017-01-01
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first (n+3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n,1)-category) for n in {0,1,2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent $n$-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
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A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

2021-01-01
Nicolai Kraus , Jakob von Raumer
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. … Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
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Set-Theoretic and Type-Theoretic Ordinals Coincide

2023-01-01
Tom de Jong , Nicolai Kraus , Fredrik Nordvall Forsberg +1 more
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We … In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.
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Type-Theoretic Approaches to Ordinals

2023-02-20
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
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Ordinal Exponentiation in Homotopy Type Theory

2025-01-24
Tom de Jong , Nicolai Kraus , Fredrik Nordvall Forsberg +1 more
While ordinals have traditionally been studied mostly in classical frameworks, constructive ordinal theory has seen significant progress in recent years. However, a general constructive treatment of ordinal exponentiation has thus … While ordinals have traditionally been studied mostly in classical frameworks, constructive ordinal theory has seen significant progress in recent years. However, a general constructive treatment of ordinal exponentiation has thus far been missing. We present two seemingly different definitions of constructive ordinal exponentiation in the setting of homotopy type theory. The first is abstract, uses suprema of ordinals, and is solely motivated by the expected equations. The second is more concrete, based on decreasing lists, and can be seen as a constructive version of a classical construction by Sierpi\'{n}ski based on functions with finite support. We show that our two approaches are equivalent (whenever it makes sense to ask the question), and use this equivalence to prove algebraic laws and decidability properties of the exponential. All our results are formalized in the proof assistant Agda.
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On symmetries of spheres in univalent foundations

2024-06-21
Pierre Cagne , Ulrik Buchholtz , Nicolai Kraus +1 more
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On symmetries of spheres in univalent foundations

2024-01-26
Pierre Cagne , Ulrik Buchholtz , Nicolai Kraus +1 more
Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries … Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle. For higher-dimensional spheres, the type of symmetries has again two connected components, namely the components of the maps of degree plus or minus one. Each of the two components has $\mathbb{Z}/2\mathbb{Z}$ as fundamental group. For the latter result, we develop an EHP long exact sequence.
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Set-Theoretic and Type-Theoretic Ordinals Coincide

2023-06-26
Tom de Jong , Nicolai Kraus , Fredrik Nordvall Forsberg +1 more
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We … In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.
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Two-level type theory and applications

2023-05-30
Danil Annenkov , Paolo Capriotti , Nicolai Kraus +1 more
We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.
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Type-theoretic approaches to ordinals

2023-03-26
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
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Type-Theoretic Approaches to Ordinals

2023-02-20
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
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Set-Theoretic and Type-Theoretic Ordinals Coincide

2023-01-01
Tom de Jong , Nicolai Kraus , Fredrik Nordvall Forsberg +1 more
In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We … In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.
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A rewriting coherence theorem with applications in homotopy type theory

2022-08-01
Nicolai Kraus , Jakob von Raumer
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. … Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
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Type-Theoretic Approaches to Ordinals

2022-01-01
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences … In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensional equality. Using homotopy type theory as the foundational setting, we develop an abstract framework for ordinal theory and establish a collection of desirable properties and constructions. We then study and compare three concrete implementations of ordinals in homotopy type theory: first, a notation system based on Cantor normal forms (binary trees); second, a refined version of Brouwer trees (infinitely-branching trees); and third, extensional well-founded orders. Each of our three formulations has the central properties expected of ordinals, such as being equipped with an extensional and well-founded ordering as well as allowing basic arithmetic operations, but they differ with respect to what they make possible in addition. For example, for finite collections of ordinals, Cantor normal forms have decidable properties, but suprema of infinite collections cannot be computed. In contrast, extensional well-founded orders work well with infinite collections, but almost all properties are undecidable. Brouwer trees take the sweet spot in the middle by combining a restricted form of decidability with the ability to work with infinite increasing sequences. Our three approaches are connected by canonical order-preserving functions from the "more decidable" to the "less decidable" notions. We have formalised the results on Cantor normal forms and Brouwer trees in cubical Agda, while extensional well-founded orders have been studied and formalised thoroughly by Escardo and his collaborators. Finally, we compare the computational efficiency of our implementations with the results reported by Berger.
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Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

2021-06-29
Nicolai Kraus
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or … Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type Con of contexts, a type family Ty indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP. In this paper, we develop a notion of ∞-categories with families (∞-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised (set-level) syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT). Apart from introducing ∞-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.
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Connecting Constructive Notions of Ordinals in Homotopy Type Theory

2021-01-01
Nicolai Kraus , Fredrik Nordvall Forsberg , Chuangjie Xu
In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We … In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.
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A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

2021-01-01
Nicolai Kraus , Jakob von Raumer
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. … Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
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Coherence via Well-Foundedness

2020-05-26
Nicolai Kraus , Jakob von Raumer
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of … Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
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Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory

2020-01-21
Nicolai Kraus , Jakob von Raumer
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of … Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence. Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty. We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.
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Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT

2020-01-01
Nicolai Kraus
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or … Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type CON of contexts, a type family TY indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP. In this paper, we develop a notion of $\infty$-categories with families ($\infty$-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT). Apart from introducing $\infty$-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.

From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2020-01-01
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category. The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).

Shallow Embedding of Type Theory is Morally Correct

2019-07-17
Ambrus Kaposi , András Kovács , Nicolai Kraus
There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary … There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-Lof type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.

Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-06-01
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
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From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2019-02-27
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher groupoids. Bezem, Coquand, and Huber have presented a constructive model of univalence using a specific cube category, which we call the BCH category. The higher categories encoded with the BCH category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).

Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-01-17
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.

Shallow Embedding of Type Theory is Morally Correct

2019-01-01
Ambrus Kaposi , András Kovács , Nicolai Kraus
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Shallow Embedding of Type Theory is Morally Correct

2019-01-01
Ambrus Kaposi , András Kovács , Nicolai Kraus
There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary … There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-L\"of type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.
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From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

2019-01-01
Gun Pinyo , Nicolai Kraus
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure … Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher groupoids. Bezem, Coquand, and Huber have presented a constructive model of univalence using a specific cube category, which we call the BCH category. The higher categories encoded with the BCH category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).
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Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-01-01
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.
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Free Higher Groups in Homotopy Type Theory

2018-06-27
Nicolai Kraus , Thorsten Altenkirch
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this … Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit: F(A), cons: A~F(A)~F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [22, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ||F(A)||1 is a set.
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Univalent Higher Categories via Complete Semi-Segal Types

2018-01-07
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (Ahrens … Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (Ahrens et al. 2015) solves this by considering only truncated types, and it roughly corresponds to an ordinary category. The drawback is that univalent categories fail to capture many naturally occurring structures, such as type universes or the type of univalent categories themselves. This stems from the fact that the natural notion of a category in homotopy type theory is not that of an ordinary, but rather a higher category.

Quotient Inductive-Inductive Types

2018-01-01
Thorsten Altenkirch , Paolo Capriotti , Gabe Dijkstra +2 more
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Univalent higher categories via complete Semi-Segal types

2017-12-27
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first ( n +3) levels of a semisimplicial type S , we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n -type . This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n -category (more precisely, ( n ,1)-category) for n ∈ {0,1,2}, and we show that the type of complete semi-Segal n -types is equivalent to the type of univalent n -categories in these cases. Thus, we believe that the notion of a complete semi-Segal n -type can be taken as the definition of a univalent n -category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
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Univalent Higher Categories via Complete Semi-Segal Types

2017-07-12
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated to homotopy, and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, … Category theory in homotopy type theory is intricate as categorical laws can only be stated to homotopy, and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first (n+3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n,1)-category) for n in {0,1,2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent $n$-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.

Two-Level Type Theory and Applications.

2017-05-09
Danil Annenkov , Paolo Capriotti , Nicolai Kraus
We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

Space-Valued Diagrams, Type-Theoretically (Extended Abstract).

2017-04-14
Nicolai Kraus , Christian Sattler
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion … Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.

Notions of Anonymous Existence in Martin-L\"of Type Theory

2017-03-24
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we … As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda. Comment: 36 pages, to appear in the special issue of TLCA'13 (LMCS)
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Partiality, Revisited

2017-01-01
Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus
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Space-Valued Diagrams, Type-Theoretically (Extended Abstract)

2017-01-01
Nicolai Kraus , Christian Sattler
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion … Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.
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Univalent Higher Categories via Complete Semi-Segal Types

2017-01-01
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first (n+3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n,1)-category) for n in {0,1,2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent $n$-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
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Two-Level Type Theory and Applications

2017-01-01
Danil Annenkov , Paolo Capriotti , Nicolai Kraus +1 more
We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type

2016-10-28
Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus
Capretta's delay monad can be used to model partial computations, but it has the wrong notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by … Capretta's delay monad can be used to model partial computations, but it has the wrong notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the right notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice. Using an idea from homotopy type theory - a higher inductive-inductive type - we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.
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Constructions with Non-Recursive Higher Inductive Types

2016-07-05
Nicolai Kraus
Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to … Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs.
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Extending Homotopy Type Theory with Strict Equality

2016-04-13
Thorsten Altenkirch , Paolo Capriotti , Nicolai Kraus
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of … In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an one, containing a strict equality type former, and an one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of infinite structures which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with schematic definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

Notions of Anonymous Existence in Martin-Löf Type Theory

2016-01-01
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we … As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.

Extending Homotopy Type Theory with Strict Equality

2016-01-01
Thorsten Altenkirch , Paolo Capriotti , Nicolai Kraus
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of … In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an "outer" one, containing a strict equality type former, and an "inner" one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of "infinite structures" which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.
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Functions out of Higher Truncations

2015-07-04
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of … In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B that are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.

Functions out of Higher Truncations

2015-07-04
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and … In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B which are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.

Higher Homotopies in a Hierarchy of Univalent Universes

2015-03-21
Nicolai Kraus , Christian Sattler
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding … For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.
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The General Universal Property of the Propositional Truncation

2015-01-01
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega^op. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of coherently constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in standard syntactical homotopy type theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a propositional type Q with A -> Q and Q -> B.

Functions out of Higher Truncations

2015-01-01
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and … In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B which are constant on all (n+1)-st loop spaces. We give one "elementary" proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct "set-based" representations of 1-types, as long as they have "braided" loop spaces. The main result with one of its proofs and the application have been formalised in Agda.
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The General Universal Property of the Propositional Truncation

2014-11-11
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B.

The General Universal Property of the Propositional Truncation

2014-01-01
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B.
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Generalizations of Hedberg’s Theorem

2013-01-01
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
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Coauthor Papers Together
Paolo Capriotti 13
Thorsten Altenkirch 10
Fredrik Nordvall Forsberg 8
Jakob von Raumer 7
Christian Sattler 5
Chuangjie Xu 5
Martı́n Hötzel Escardó 4
Thierry Coquand 4
Marc Bezem 3
Ambrus Kaposi 3
Danil Annenkov 3
Gun Pinyo 3
Andrea Vezzosi 3
Tom de Jong 3
Ulrik Buchholtz 2
Nils Anders Danielsson 2
András Kovács 2
Chuangjie Xu 2
Pierre Cagne 2
Benno van den Berg 1
Bas Spitters 1
Erik Palmgren 1
Richard Garner 1
Steve Awodey 1
Nicola Gambino 1
Peter Aczel 1
Peter Dybjer 1
Michael Nahas 1
Vladimir Voevodsky 1
Hugo Herbelin 1
Philip Scott 1
Kuen-Bang Hou 1
Matthieu Sozeau 1
Michael Shulman 1
András Kovács 1
Álvaro Pelayo 1
Andrej Bauer 1
André Joyal 1
Georges Gonthier 1
Guillaume Brunerie 1
Carlo Angiuli 1
Kristina Sojakova 1
Egbert Rijke 1
Zhaohui Luo 1
Nuo Li 1
Martin Hofmann 1
Michael A. Warren 1
Robert L. Constable 1
Cyril Cohen 1
Dana Scott 1

Homotopy theoretic models of identity types

2008-07-14
Steve Awodey , Michael A. Warren
This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing … This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.
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Homotopy Type Theory: Univalent Foundations of Mathematics

2013-01-01
Peter Aczel , Benedikt Ahrens , Thorsten Altenkirch +7 more
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of … Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
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Univalent categories and the Rezk completion

2015-01-19
Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman
We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition … We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.
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The Simplicial Model of Univalent Foundations (after Voevodsky)

2012-11-12
Chris Kapulkin , Peter LeFanu Lumsdaine
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models … We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-Lof type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

Univalence for inverse diagrams and homotopy canonicity

2014-11-24
Michael Shulman
We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory … We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky's univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.
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Calculating the Fundamental Group of the Circle in Homotopy Type Theory

2013-06-01
Daniel R. Licata , Michael Shulman
Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new … Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new principles to add to type theory, while the type theory can be used in novel ways to do computer-checked proofs in a proof assistant. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Our proof illustrates the new features of homotopy type theory, such as higher inductive types and Voevodsky's univalence axiom. It also introduces a new method for calculating the path space of a type, which has proved useful in many other examples.
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Models of Type Theory with Strict Equality

2017-01-01
Paolo Capriotti
This thesis introduces the idea of two-level type theory, an extension of Martin-Löf type theory that adds a notion of strict equality as an internal primitive. A type theory with … This thesis introduces the idea of two-level type theory, an extension of Martin-Löf type theory that adds a notion of strict equality as an internal primitive. A type theory with a strict equality alongside the more conventional form of equality, the latter being of fundamental importance for the recent innovation of homotopy type theory (HoTT), was first proposed by Voevodsky, and is usually referred to as HTS. Here, we generalise and expand this idea, by developing a semantic framework that gives a systematic account of type formers for two-level systems, and proving a conservativity result relating back to a conventional type theory like HoTT. Finally, we show how a two-level theory can be used to provide partial solutions to open problems in HoTT. In particular, we use it to construct semi-simplicial types, and lay out the foundations of an internal theory of $(\infty, 1)$-categories.

A model for the homotopy theory of homotopy theory

2000-06-20
Charles Rezk
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy … We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.

Free Higher Groups in Homotopy Type Theory

2018-06-27
Nicolai Kraus , Thorsten Altenkirch
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this … Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit: F(A), cons: A~F(A)~F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [22, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ||F(A)||1 is a set.
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Internal type theory

1996-01-01
Peter Dybjer

Quotient Inductive-Inductive Types

2018-01-01
Thorsten Altenkirch , Paolo Capriotti , Gabe Dijkstra +2 more
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Univalent higher categories via complete Semi-Segal types

2017-12-27
Paolo Capriotti , Nicolai Kraus
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first ( n +3) levels of a semisimplicial type S , we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n -type . This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n -category (more precisely, ( n ,1)-category) for n ∈ {0,1,2}, and we show that the type of complete semi-Segal n -types is equivalent to the type of univalent n -categories in these cases. Thus, we believe that the notion of a complete semi-Segal n -type can be taken as the definition of a univalent n -category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
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The General Universal Property of the Propositional Truncation

2015-01-01
Nicolai Kraus
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently … In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega^op. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of coherently constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in standard syntactical homotopy type theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a propositional type Q with A -> Q and Q -> B.

Higher Inductive Types as Homotopy-Initial Algebras

2014-12-19
Kristina Sojakova
Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these … Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can use geometric intuition to formulate new concepts in type theory and, conversely, use type-theoretic machinery to verify and often simplify existing mathematical proofs.
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Axiom of choice and complementation

1975-08-01
Radu Diaconescu
It is shown that an intuitionistic model of set theory with the axiom of choice has to be a classical one. It is shown that an intuitionistic model of set theory with the axiom of choice has to be a classical one.
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Normalisation by Evaluation for Type Theory, in Type Theory

2016-01-01
Thorsten Altenkirch , Ambrus Kaposi
We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a … We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction, and every construction respects the conversion relation. NBE for simple types uses a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We prove normalisation, completeness, stability and decidability of definitional equality. Most of the constructions were formalized in Agda.
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Types are weak <i>ω</i> -groupoids

2010-10-12
Benno van den Berg , Richard Garner
We define a notion of weak ω-category internal to a model of Martin-Löf's type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower … We define a notion of weak ω-category internal to a model of Martin-Löf's type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower of iterated identity types over that type. We show that the ω-categories arising in this way are in fact ω-groupoids.
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The Seifert-van Kampen Theorem in Homotopy Type Theory

2016-08-01
Kuen-Bang Hou , Michael Shulman
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic … Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding homotopy theory. Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the encode-decode method, enforced homotopy-invariance, and lack of underlying sets.

On Higher Inductive Types in Cubical Type Theory

2018-06-27
Thierry Coquand , Simon Huber , Anders Mörtberg
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly … Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.
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Two-Level Type Theory and Applications.

2017-05-09
Danil Annenkov , Paolo Capriotti , Nicolai Kraus
We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer … We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

Partiality, Revisited

2017-01-01
Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus
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Signatures and Induction Principles for Higher Inductive-Inductive Types

2019-01-01
Ambrus Kaposi , András Kovács
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed … Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code.
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Space-Valued Diagrams, Type-Theoretically (Extended Abstract).

2017-04-14
Nicolai Kraus , Christian Sattler
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion … Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.

Eilenberg-MacLane spaces in homotopy type theory

2014-07-14
Daniel R. Licata , Eric Finster
Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs … Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose nth homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.
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Generalizations of Hedberg’s Theorem

2013-01-01
Nicolai Kraus , Martı́n Hötzel Escardó , Thierry Coquand +1 more
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Quasi-unital ∞–categories

2015-09-10
Yonatan Harpaz
Inspired by Lurie's theory of quasi-unital algebras we prove an analogous result for 1-categories.By constructing a suitable model category of non-unital complete Segal spaces, we show that the unital structure … Inspired by Lurie's theory of quasi-unital algebras we prove an analogous result for 1-categories.By constructing a suitable model category of non-unital complete Segal spaces, we show that the unital structure of an 1-category can be uniquely recovered from the underlying non-unital structure once suitable candidates for units have been identified.The main result of this paper can be used to produce a proof of the 1-dimensional cobordism hypothesis, as described in a forthcoming paper of the author.
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Inductive Types in Homotopy Type Theory

2012-06-01
Steve Awodey , Nicola Gambino , Kristina Sojakova
Homotopy type theory is an interpretation of Martin-Lof's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional … Homotopy type theory is an interpretation of Martin-Lof's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq. The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof scripts for this verification form an essential component of this research.
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The algebra of oriented simplexes

1987-12-01
Ross Street

A dependently-typed construction of semi-simplicial types

2014-11-20
Hugo Herbelin
This paper presents a dependently-typed construction of semi-simplicial sets in a type theory where sets are taken to be types. This addresses an open question raised on the wiki of … This paper presents a dependently-typed construction of semi-simplicial sets in a type theory where sets are taken to be types. This addresses an open question raised on the wiki of the special year on Univalent Foundations at the Institute of Advanced Study (2012–2013).
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Extending Homotopy Type Theory with Strict Equality

2016-04-13
Thorsten Altenkirch , Paolo Capriotti , Nicolai Kraus
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of … In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an one, containing a strict equality type former, and an one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of infinite structures which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with schematic definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

Constructions with Non-Recursive Higher Inductive Types

2016-07-05
Nicolai Kraus
Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to … Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs.
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Higher Groups in Homotopy Type Theory

2018-06-27
Ulrik Buchholtz , Floris van Doorn , Egbert Rijke
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, … We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an n-type can be delooped n + 2 times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.
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Constructing the propositional truncation using non-recursive HITs

2016-01-12
Floris van Doorn
In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive … In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. This construction gives a characterization of functions from the propositional truncation to an arbitrary type, extending the universal property of the propositional truncation. We have fully formalized all the results in a new proof assistant, Lean.
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Cubical Type Theory: a constructive interpretation of the univalence axiom

2016-01-01
Cyril Cohen , Thierry Coquand , Simon Huber +1 more
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in … This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory

2016-07-05
Kuen-Bang Hou , Eric Finster , Daniel R. Licata +1 more
This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of a result … This paper contributes to recent investigations of the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory. We present a mechanized proof of a result called the Blakers-Massey connectivity theorem, which relates the higher-dimensional loop structures of two spaces sharing a common part (represented by a pushout type, which is a generalization of a disjoint sum type) to those of the common part itself. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which was used in previous formalizations. The proof is more direct than existing ones that apply in general category-theoretic settings for homotopy theory, and its mechanization is concise and high-level, due to novel combinations of ideas from homotopy theory and from type theory.
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Pointers in Recursion: Exploring the Tropics

2019-12-16
Paulin Jacobé de Naurois
We translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are … We translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are proven equivalent to the usual partial/primitive recursive functions. Complexity-wise, this framework captures in a streamlined way most of the relevant sub-polynomial classes. Pointer recursion with the safe/normal tiering discipline of Bellantoni and Cook corresponds to polylogtime computation. We introduce a new, non-size increasing tiering discipline, called tropical tiering. Tropical tiering and pointer recursion, used with some of the most common recursion schemes, capture the classes logspace, logspace/polylogtime, ptime, and NC. Finally, in a fashion reminiscent of the safe recursive functions, tropical tier-ing is expressed directly in the syntax of the function algebras, yielding the tropical recursive function algebras.
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Homotopy canonicity for cubical type theory

2019-02-18
Thierry Coquand , Simon Huber , Christian Sattler

Homotopy Type Theory in Lean

2017-01-01
Floris van Doorn , Jakob von Raumer , Ulrik Buchholtz
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Homotopy Invariant Algebraic Structures on Topological Spaces

1973-01-01
J. M. Boardman , R. M. Vogt
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Functions out of Higher Truncations

2015-07-04
Paolo Capriotti , Nicolai Kraus , Andrea Vezzosi
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of … In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B that are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.

Quasi-categories and Kan complexes

2002-11-01
André Joyal

Univalent Categories and the Rezk Completion

2015-01-01
Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman
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Semantics of higher inductive types

2019-06-17
Peter LeFanu Lumsdaine , Michael Shulman
Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development … Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy theory within type theory, as well as in formalizing ordinary set-level mathematics in type theory. In this article, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has *weakly stable typal initial algebras* for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specializes to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction, and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed $(\infty,1)$-category is presented by some model category to which our results apply.
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Coherence via Well-Foundedness

2020-05-26
Nicolai Kraus , Jakob von Raumer
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of … Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
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On Theories with a Combinatorial Definition of "Equivalence"

1942-04-01
M. H. A. Newman

Path Spaces of Higher Inductive Types in Homotopy Type Theory

2019-06-01
Nicolai Kraus , Jakob von Raumer
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove … The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
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Weak identity arrows in higher categories

2006-04-09
J.L.F. Kock
Journal Article Weak identity arrows in higher categories Get access Joachim Kock Joachim Kock Departament de Matemàtiques, Universitat Autònoma de Barcelona08193 Bellaterra (Barcelona), Spain E-mail address: [email protected] Search for other … Journal Article Weak identity arrows in higher categories Get access Joachim Kock Joachim Kock Departament de Matemàtiques, Universitat Autònoma de Barcelona08193 Bellaterra (Barcelona), Spain E-mail address: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Papers, Volume 2006, 2006, 69163, https://doi.org/10.1155/IMRP/2006/69163 Published: 01 January 2006 Article history Received: 29 July 2005 Revision received: 28 November 2005 Accepted: 05 December 2005 Published: 01 January 2006
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A Survey of (∞, 1)-Categories

2009-09-16
Julia E. Bergner
In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible … In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n > 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.

Weak ω-Categories from Intensional Type Theory

2009-01-01
Peter LeFanu Lumsdaine

Higher Homotopies in a Hierarchy of Univalent Universes

2015-03-21
Nicolai Kraus , Christian Sattler
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding … For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.
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