Higher Inductive Types as Homotopy-Initial Algebras

Authors

Kristina Sojakova

Type: Article

Publication Date: 2014-12-19

Citations: 45

DOI: https://doi.org/10.1145/2676726.2676983

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Abstract

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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Higher Inductive Types as Homotopy-Initial Algebras

2014-02-04

Kristina Sojakova

Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs constructive type theory and abstract homotopy theory. We have a powerful interplay … Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs 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. A crucial ingredient in this new system are higher inductive types, which allow us to represent objects such as spheres, tori, pushouts, and quotients. We investigate a variant of higher inductive types whose computational behavior is determined up to a higher path. We show that in this setting, higher inductive types are characterized by the universal property of being a homotopy-initial algebra.

Higher Inductive Types as Homotopy-Initial Algebras

2014-01-01

Kristina Sojakova

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Higher inductive types in Homotopy Type Theory: applications and theory

2016-04-01

Fredrik Nordvall Forsberg

Higher Inductive Types as Homotopy-Initial Algebras

2015-01-14

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. Higher inductive types form a crucial part of this new system since they allow us to represent mathematical objects, such as spheres, tori, pushouts, and quotients, in the type theory. We investigate a class of higher inductive types called W-suspensions which generalize Martin-Löf's well-founded trees. We show that a propositional variant of W-suspensions, whose computational behavior is determined up to a higher path, is characterized by the universal property of being a homotopy-initial algebra. As a corollary we get that W-suspensions in the strict form are homotopy-initial.

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Type theory and homotopy

2010-01-01

Steve Awodey

The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an … The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-Löf into homotopy theory, resulting in new examples of higher-dimensional categories.

Homotopy-initial algebras in type theory

2015-04-21

Steve Awodey , Nicola Gambino , Kristina Sojakova

We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a … We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition which replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterises the types that are equivalent to W-types as homotopy-initial algebras.

Homotopy-initial algebras in type theory

2015-01-01

Steve Awodey , Nicola Gambino , Kristina Sojakova

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Homotopy-Initial Algebras in Type Theory

2017-01-30

Steve Awodey , Nicola Gambino , Kristina Sojakova

We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a … We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition that replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterizes the types that are equivalent to W-types as homotopy-initial algebras.

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π n (S n ) in Homotopy Type Theory

2013-01-01

Daniel R. Licata , Guillaume Brunerie

Homotopy type theory and the formalization of mathematics

2016-01-01

Egbert Rijke , Bas Spitters

Computational Higher Type Theory IV: Inductive Types

2018-01-01

Evan Cavallo , Robert Harper

This is the fourth in a series of papers extending Martin-Löf's meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type … This is the fourth in a series of papers extending Martin-Löf's meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of indexed cubical inductive types whose constructors may take dimension parameters and have a specified boundary. Using this schema, we are able to specify and implement many of the higher inductive types which have been postulated in homotopy type theory, including homotopy pushouts, the torus, $W$-quotients, truncations, arbitrary localizations. By including indexed inductive types, we enable the definition of identity types. The addition of higher inductive types makes computational higher type theory a model of homotopy type theory, capable of interpreting almost all of the constructions in the HoTT Book (with the exception of inductive-inductive types). This is the first such model with an explicit canonicity theorem, which specifies the canonical values of higher inductive types and confirms that every term in an inductive type evaluates to such a value.

Inductive types in homotopy type theory

2012-01-18

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.

Inductive Types in Homotopy Type Theory

2012-06-01

Steve Awodey , Nicola Gambino , Kristina Sojakova

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Inductive types in homotopy type theory

2012-01-01

Steve Awodey , Nicola Gambino , Kristina Sojakova

Homotopy type theory is an interpretation of Martin-Löf'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-Löf'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.

Homotopy Theoretic Aspects of Constructive Type Theory

2008-01-01

Michael A. Warren

On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory

2018-01-01

Floris van Doorn

The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction … The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.

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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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Homotopy type theory

2012-09-28

Egbert Rijke

Introduction 3 1 A short guide to constructive type theory 7 1.1 A dependent type over a type . . . . . . . . . . . . … Introduction 3 1 A short guide to constructive type theory 7 1.1 A dependent type over a type . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 Dependent products . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Dependent sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Defining types inductively . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Type theory with identity types 14 2.1 The inductive definition of identity types . . . . . . . . . . . . . . . . . . 14 2.2 More properties of paths . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Preservation of composition . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Preservation of inversion . . . . . . . . . . . . . . . . . . . . . . 23 2.2.3 The dependent type Y(a) . . . . . . . . . . . . . . . . . . . . . . 23

Homotopy Type Theory: A synthetic approach to higher equalities

2016-01-01

Michael Shulman

This is an introduction to Homotopy Type Theory and Univalent Foundations for philosophers, written as a chapter for the book "Categories for the Working Philosopher" (ed. Elaine Landry) This is an introduction to Homotopy Type Theory and Univalent Foundations for philosophers, written as a chapter for the book "Categories for the Working Philosopher" (ed. Elaine Landry)

On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory.

2018-08-31

Floris van Doorn

Higher Inductive Types as Homotopy-Initial Algebras

2014-12-19

Kristina Sojakova

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Finite sets in homotopy type theory

2018-01-08

Dan Frumin , Herman Geuvers , Léon Gondelman +1 more

We study different formalizations of finite sets in homotopy type theory to obtain a general definition that exhibits both the computational facilities and the proof principles expected from finite sets. … We study different formalizations of finite sets in homotopy type theory to obtain a general definition that exhibits both the computational facilities and the proof principles expected from finite sets. We use higher inductive types to define the type K(A) of "finite sets over type A" à la Kuratowski without assuming that K(A) has decidable equality. We show how to define basic functions and prove basic properties after which we give two applications of our definition.

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Computational Higher Type Theory IV: Inductive Types

2018-01-01

Evan Cavallo , Robert Harper

Impredicative Encodings of (Higher) Inductive Types

2018-06-27

Steve Awodey , Jonas Frey , Sam Speight

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit … Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit dependent eliminators. To recover η and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1.

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What inductive explanations could not be

2017-06-05

John Dougherty

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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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Quotients, inductive types, and quotient inductive types

2022-06-07

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories … This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

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Quotient Inductive-Inductive Types

2018-01-01

Thorsten Altenkirch , Paolo Capriotti , Gabe Dijkstra +2 more

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Homotopy-Initial Algebras in Type Theory

2017-01-30

Steve Awodey , Nicola Gambino , Kristina Sojakova

We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a … We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition that replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterizes the types that are equivalent to W-types as homotopy-initial algebras.

View PDF Chat

Construction of the circle in UniMath

2021-01-13

Marc Bezem , Ulrik Buchholtz , Daniel R. Grayson +1 more

We show that the type TZ of Z-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence … We show that the type TZ of Z-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.

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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.

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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Finitary Higher Inductive Types in the Groupoid Model

2018-04-01

Peter Dybjer , Hugo Moeneclaey

A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces … A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces too. We restrict attention to finitary higher inductive types and present general schemata for the types of their point, path, and surface constructors. We also derive the elimination and equality rules from the types of constructors for 1-hits and 2-hits. Moreover, we construct a groupoid model for dependent type theory with 2-hits and point out that we obtain a setoid model for dependent type theory with 1-hits by truncating the groupoid model.

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Constructing Higher Inductive Types as Groupoid Quotients

2021-04-22

Niccolò Veltri , Niels van der Weide

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. … In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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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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Higher Inductive Types as Homotopy-Initial Algebras

2015-01-14

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. Higher inductive types form a crucial part of this new system since they allow us to represent mathematical objects, such as spheres, tori, pushouts, and quotients, in the type theory. We investigate a class of higher inductive types called W-suspensions which generalize Martin-Löf's well-founded trees. We show that a propositional variant of W-suspensions, whose computational behavior is determined up to a higher path, is characterized by the universal property of being a homotopy-initial algebra. As a corollary we get that W-suspensions in the strict form are homotopy-initial.

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Construction of the Circle in UniMath

2019-10-04

Marc Bezem , Ulrik Buchholtz , Daniel R. Grayson +1 more

We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence … We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.

Non-wellfounded trees in Homotopy Type Theory

2015-04-12

Benedikt Ahrens , Paolo Capriotti , Régis Spadotti

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence … We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-Lof type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.

Truncation levels in homotopy type theory

2015-07-15

Nicolai Kraus

Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type … Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky. To begin, we give a few simple criteria for determining whether a type is 0-truncated (a set), inspired by a well-known theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs. A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other non-trivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type. A further result is the construction of strict n-types in Martin-Lof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not n-truncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton). The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite non-empty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an n-type for some given finite n, (non-trivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions. To obtain these results, we develop some theory on equality diagrams, especially equality semi-simplicial types. In particular, we show that the semi-simplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omega-groupoids. Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semi-simplicial types work.

Finite sets in homotopy type theory

2017-12-22

Dan Frumin , Herman Geuvers , Léon Gondelman +1 more

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Sequential Colimits in Homotopy Type Theory

2020-05-25

Kristina Sojakova , Floris van Doorn , Egbert Rijke

Sequential colimits are an important class of higher inductive types. We present a self-contained and fully formalized proof of the conjecture that in homotopy type theory sequential colimits appropriately commute … Sequential colimits are an important class of higher inductive types. We present a self-contained and fully formalized proof of the conjecture that in homotopy type theory sequential colimits appropriately commute with Σ-types. This result allows us to give short proofs of a number of useful corollaries, some of which were conjectured in other works: the commutativity of sequential colimits with identity types, with homotopy fibers, loop spaces, and truncations, and the preservation of the properties of truncatedness and connectedness under sequential colimits. Our entire development carries over to (∞, 1)-toposes using Shulman's recent interpretation of homotopy type theory into these structures.

Impredicative Encodings of (Higher) Inductive Types

2018-01-01

Steve Awodey , Jonas Frey , Sam Speight

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit … Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1.

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A class of higher inductive types in Zermelo-Fraenkel set theory

2020-05-28

Andrew Swan

We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or … We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

The HoTT reals coincide with the Escardó-Simpson reals.

2017-06-19

Auke Bart Booij

Escardo and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, … Escardo and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy this universal property. We show that this is indeed the case in the category of sets of any universe. We also show that the type of HoTT reals is the least Cauchy complete subset of the Dedekind reals containing the rationals.

Constructing the Propositional Truncation using Non-recursive HITs

2015-12-07

Floris van Doorn

Constructing quotient inductive-inductive types

2019-01-02

Ambrus Kaposi , András Kovács , Thorsten Altenkirch

Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In … Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting with uniqueness of identity proofs, hence we use the term QIIT instead of higher inductive-inductive type. An example of a QIIT is the well-typed (intrinsic) syntax of type theory quotiented by conversion. In this paper first we specify finitary QIITs using a domain-specific type theory which we call the theory of signatures. The syntax of the theory of signatures is given by a QIIT as well. Then, using this syntax we show that all specified QIITs exist and they have a dependent elimination principle. We also show that algebras of a signature form a category with families (CwF) and use the internal language of this CwF to show that dependent elimination is equivalent to initiality.

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Higher inductive types in cubical computational type theory

2019-01-02

Evan Cavallo , Robert Harper

Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory … Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory does not specify the computational behavior of HITs. Computational interpretations have now been provided for univalence and specific HITs by way of cubical type theories, which use a judgmental infrastructure of dimension variables. We extend the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher inductive types to the cubical setting. In doing so, we isolate the canonical values of a cubical inductive type and prove a canonicity theorem with respect to these values.

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On the homotopy groups of spheres in homotopy type theory

2016-06-15

Guillaume Brunerie

The goal of this thesis is to prove that π4(S3) ≃ Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the … The goal of this thesis is to prove that π4(S3) ≃ Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(Sn) with k < n, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number n such that π4(S3) ≃ Z/nZ. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the n to either 1 or 2. The Hopf invariant also allows us to prove that all the groups of the form π4n−1(S2n) are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of CP2 and to prove that π4(S3) ≃ Z/2Z and that more generally πn+1(Sn) ≃ Z/2Z for every n ≥ 3

On the homotopy groups of spheres in homotopy type theory

2016-01-01

Guillaume Brunerie

The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the … The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form $\pi_k(S^n)$ with $k < n$, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number $n$ such that $\pi_4(S^3) \simeq \mathbb{Z}/n\mathbb{Z}$. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the $n$ to either $1$ or $2$. The Hopf invariant also allows us to prove that all the groups of the form $\pi_{4n-1}(S^{2n})$ are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of $\mathbb{C}P^2$ and to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ and that more generally $\pi_{n+1}(S^n) \simeq \mathbb{Z}/2\mathbb{Z}$ for every $n \ge 3$.

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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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Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks

2022-08-02

Magnus Baunsgaard Kristensen , Rasmus Ejlers Møgelberg , Andrea Vezzosi

We present Clocked Cubical Type Theory, the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory. Guarded recursion is an abstract form of step-indexing, which … We present Clocked Cubical Type Theory, the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory. Guarded recursion is an abstract form of step-indexing, which can be used for construction of advanced programming language models. In its multi-clocked version, it can also be used for coinductive programming and reasoning, encoding productivity in types. Combining this with Higher Inductive Types (HITs) the encoding extends to coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems.

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A class of higher inductive types in Zermelo‐Fraenkel set theory

2022-01-21

Andrew Swan

Abstract We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo‐Fraenkel set theory without the axiom of choice … Abstract We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo‐Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

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Constructing Higher Inductive Types as Groupoid Quotients

2020-05-26

Niels van der Weide

In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for … In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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Constructing Infinitary Quotient-Inductive Types

2020-01-01

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

Abstract This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types … Abstract This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel’s size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem prover.

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A model of Clocked Cubical Type Theory.

2021-02-03

Magnus Baunsgaard Kristensen , Rasmus Ejlers Møgelberg , Andrea Vezzosi

Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming … Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain equations also with negative occurrences. In its multi-clocked version, guarded recursion can also be used to program with and reason about coinductive types, encoding the productivity condition required for recursive definitions in types. This paper presents the first denotational model of a type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory. Using the combination of Higher Inductive Types (HITs) and guarded recursion allows for simple programming and reasoning about coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. For example, our results imply that bisimilarity for these imply path equality, and so proofs can be transported along bisimilarity proofs.

Large and Infinitary Quotient Inductive-Inductive Types

2020-05-26

András Kovács , Ambrus Kaposi

Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic … Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

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Partiality, Revisited

2017-01-01

Thorsten Altenkirch , Nils Anders Danielsson , Nicolai Kraus

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A HoTT Quantum Equational Theory (Extended Version)

2019-01-01

Jennifer Paykin , Steve Zdancewic

This paper presents an equational theory for the QRAM model of quantum computation, formulated as an embedded language inside of homotopy type theory. The embedded language approach is highly expressive, … This paper presents an equational theory for the QRAM model of quantum computation, formulated as an embedded language inside of homotopy type theory. The embedded language approach is highly expressive, and reflects the style of state-of-the art quantum languages like Quipper and QWIRE. The embedding takes advantage of features of homotopy type theory to encode unitary transformations as higher inductive paths, simplifying the presentation of an equational theory. We prove that this equational theory is sound and complete with respect to established models of quantum computation.

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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Type-Theoretic Signatures for Algebraic Theories and Inductive Types

2022-01-01

András Kovács

Summary of Thesis This thesis develops the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type … Summary of Thesis This thesis develops the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. In the thesis, we describe three type theories where each typing context can be viewed as an algebraic signature, specifying sorts, operations and equations. These signatures are useful in broader mathematical contexts, but we are also concerned with potential implementation in proof assistants. In Chapter 3, we describe a way to use two-level type theory as a metalanguage for developing semantics of algebraic signatures. This makes it possible to work in a concise internal notation of a type theory, and at the same time build semantics internally to arbitrary structured categories. For example, the signature for natural number objects can be interpreted in any category with finite products. In Chapter 4, we describe finitary quotient inductive-inductive (FQII) signatures. Most type theories themselves can be specified with FQII signatures. We build a structured category of algebras for each signature, where equivalence of initiality and induction can be shown. We additionally present term algebra constructions, constructions of left adjoint functors of signature morphisms, and we describe a way to use self-describing signatures to minimize necessary metatheoretic assumptions. In Chapter 5, we describe infinitary quotient inductive-inductive signatures. These allow specification of infinitely branching trees as initial algebras. We adapt the semantics from the previous chapter. We also revisit term models, left adjoints of signature morphisms and self-description of signatures. We also describe how to build semantics of signatures internally to the theory of signatures itself, which yields numerous ways to build new signatures from existing ones. In Chapter 6, we describe higher inductive-inductive signatures. These differ from previous semantics mostly in that their intended semantics is in homotopy type theory, and allows higher-dimensional equalities. In this more general setting we only consider enough semantics to compute notions of initiality and induction for each signature.

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Homotopy type theory

2012-09-28

Egbert Rijke

Univalent Foundations of Mathematics

2011-01-01

Vladimir Voevodsky

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Topological and Simplicial Models of Identity Types

2012-01-01

Benno van den Berg , Richard Garner

In this paper we construct new categorical models for the identity types of Martin-Löf type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do … In this paper we construct new categorical models for the identity types of Martin-Löf type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure, which we call a homotopy-theoretic model of identity types , and to prove that this is sufficient for a sound interpretation. Now, although both Top and SSet are categories endowed with a weak factorization system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting these leads us to introduce the notion of a path object category . This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet , we endow those categories with the structure of a homotopy-theoretic model and, in this way, obtain the desired topological and simplicial models of identity types.

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Higher Inductive Types as Homotopy-Initial Algebras

2014-12-19

Kristina Sojakova

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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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Homotopy Theoretic Aspects of Constructive Type Theory

2008-01-01

Michael A. Warren

A model of type theory in cubical sets

2014-01-01

M Bezem , Thierry Coquand , Simon Huber

We present a model of type theory with dependent product, sum, and identity, in cubical sets. We describe a universe and explain how to transform an equivalence between two types … We present a model of type theory with dependent product, sum, and identity, in cubical sets. We describe a universe and explain how to transform an equivalence between two types into an equality. We also explain how to model propositional truncation and the circle. While not expressed internally in type theory, the model is expressed in a constructive metalogic. Thus it is a step towards a computational interpretation of Voevodsky's Univalence Axiom.

THE SIMPLICIAL MODEL OF UNIVALENT FOUNDATIONS

2014-01-01

Krzysztof Kapulkin , Peter LeFanu Lumsdaine , Vladimir Voevodsky

In this paper, we construct and investigate a model of the Univalent Foundations of Mathematics in the category of simplicial sets. To this end, we rst give a new technique … In this paper, we construct and investigate a model of the Univalent Foundations of Mathematics in the category of simplicial sets. To this end, we rst give a new technique for constructing models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan bration, 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 Univalent Foundations are at least as consistent as ZFC with two inaccessible cardinals.

The identity type weak factorisation system

2008-09-03

Nicola Gambino , Richard Garner

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Two-dimensional models of type theory

2009-07-02

Richard Garner

We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories. We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.

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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.

Weak omega-categories from intensional type theory

2010-09-17

Peter LeFanu Lumsdaine

We show that for any type in Martin-L\"of Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. … We show that for any type in Martin-L\"of Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable composition laws, and give an action of this operad on the terms of any type and its identity types.

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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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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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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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π n (S n ) in Homotopy Type Theory

2013-01-01

Daniel R. Licata , Guillaume Brunerie

Inductive Types in Homotopy Type Theory

2012-06-01

Steve Awodey , Nicola Gambino , Kristina Sojakova

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