Constructions with Non-Recursive Higher Inductive Types

Authors

Type: Article
Publication Date: 2016-07-05
Citations: 21
DOI: https://doi.org/10.1145/2933575.2933586

Abstract

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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  • Repository@Nottingham (University of Nottingham)
  • University of Birmingham Research Portal (University of Birmingham)

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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.
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.
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.
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.
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.
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.
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.
A further innovation is the introduction of an <italic>intensional</italic> type theory. Here one need not reduce equivalence to mere identity. <italic>How</italic> two entities are the same tells us more than … A further innovation is the introduction of an <italic>intensional</italic> type theory. Here one need not reduce equivalence to mere identity. <italic>How</italic> two entities are the same tells us more than <italic>whether</italic> they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s <italic>definite descriptions</italic>. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.
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.
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.
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.
We define a naturality construction for the operations of weak {\omega}-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product, … We define a naturality construction for the operations of weak {\omega}-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product, and we realise it as a globular analogue of Reynolds parametricity. Our construction operates as a "power tool" to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in {\omega}-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.
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
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.
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.
Finster and Mimram have defined a dependent type theory called CaTT, which describes the structure of omega-categories. Types in homotopy type theory with their higher identity types form weak omega-groupoids, … Finster and Mimram have defined a dependent type theory called CaTT, which describes the structure of omega-categories. Types in homotopy type theory with their higher identity types form weak omega-groupoids, so they are in particular weak omega-categories. In this article, we show that this principle makes homotopy type theory into a model of CaTT, by defining a translation principle that interprets an operation on the cell of an omega-category as an operation on higher identity types. We then illustrate how this translation allows to leverage several mechanisation principles that are available in CaTT, to reduce the proof effort required to derive results about the structure of identity types, such as the existence of an Eckmann-Hilton cell.
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.
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.
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.
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.
We combine homotopy type theory with axiomatic cohesion, expressing the latter internally with a version of ‘adjoint logic’ in which the discretization and codiscretization modalities are characterized using a judgemental … We combine homotopy type theory with axiomatic cohesion, expressing the latter internally with a version of ‘adjoint logic’ in which the discretization and codiscretization modalities are characterized using a judgemental formalism of ‘crisp variables.’ This yields type theories that we call ‘spatial’ and ‘cohesive,’ in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the ‘fundamental ∞-groupoid’ or ‘shape’), disentangling the ‘identifications’ of homotopy type theory from the ‘continuous paths’ of topology. In a further refinement called ‘real-cohesion,’ the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.
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.
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.
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.
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.
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.
Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about … Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> to be the empty type. In the inductive step, we take ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> to be the mapping cone of the projection map of the tautological bundle of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , and we use its universal property and the univalence axiom to define the tautological bundle on ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> . By showing that the total space of the tautological bundle of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is the n-sphere S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , we retrieve the classical description of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> as ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> with an (n + 1)-disk attached to it. The infinite dimensional real projective space ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> , defined as the sequential colimit of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(ℤ/2ℤ, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.
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.
We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent foundations. In their place we propose something more well-behaved: … We show that total, very well-behaved lenses are not very well-behaved when treated proof-relevantly in the setting of homotopy type theory/univalent foundations. In their place we propose something more well-behaved: higher lenses. Such a lens contains an equivalence between the lens's source type and the product of its view type and a remainder type, plus a function from the remainder type to the propositional truncation of the view type. It can equivalently be formulated as a getter function and a proof that its family of fibres is coherently constant, i.e. factors through propositional truncation.We explore the properties of higher lenses. For instance, we prove that higher lenses are equivalent to traditional ones for types that satisfy the principle of uniqueness of identity proofs. We also prove that higher lenses are n-truncated for n-truncated types, using a coinductive characterisation of coherently constant functions.
In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map … In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element. Moreover, we show that the idempotents of the join of maps are precisely the embeddings, and we prove the `join connectivity theorem', which states that the connectivity of the join of maps equals the join of the connectivities of the individual maps. We define the image of a map $f:A\to X$ in $U$ via the join construction, as the colimit of the finite join powers of $f$. The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity. A modified version of the join construction can be used to show that for any map $f:A\to X$ in which $X$ is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of set-quotients, the Rezk completion of a precategory, and we define the $n$-truncation for any $n:\mathbb{N}$. Thus we see that each of these are definable operations on a univalent universe for Martin-Lof type theory with a natural numbers object, that is moreover closed under homotopy coequalizers.
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.
Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about … Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RPn, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RPn by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP-1 to be the empty type. In the inductive step, we take RPn+1 to be the mapping cone of the projection map of the tautological bundle of RPn, and we use its universal property and the univalence axiom to define the tautological bundle on RPn+1.By showing that the total space of the tautological bundle of RPn is the n-sphere Sn, we retrieve the classical description of RPn+1 as RPn with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP∞, defined as the sequential colimit of RPn with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles.These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.
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.
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.
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.
The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many … The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes the study of the set theoretic semantics (models of homotopy type theory and univalence in a meta-theory of sets or categories), type theoretic semantics (internal models of homotopy type theory), and computational semantics, as well as the study of various questions in the internal language of homotopy type theory which are not necessarily motivated by homotopy theory, or questions related to the development of formalized libraries of mathematics based on homotopy type theory. This thesis concerns the development of synthetic homotopy theory.
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.
Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids … Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi W$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.
If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in … If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Sigma type via the Curry-Howard interpretation. But with an intuitionistic notion of anonymous existence, defined as the propositional truncation of Sigma, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N -> 2) -> N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn't show Agda code.
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.
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.
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.
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed … Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.
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.
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.