Univalent categories and the Rezk completion

Authors

Benedikt Ahrens, Krzysztof Kapulkin, Michael Shulman

Type: Article

Publication Date: 2015-01-19

Citations: 75

DOI: https://doi.org/10.1017/s0960129514000486

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Abstract

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

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Mathematical Structures in Computer Science
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arXiv (Cornell University)
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Univalent Categories and the Rezk Completion

2015-01-01

Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman

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The Univalence Principle

2025-01-14

Benedikt Ahrens , Paige Randall North , Michael Shulman +1 more

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined … The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk’s completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai’s First-Order Logic with Dependent Sorts, but is expressed in Voevodsky’s Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

The Univalence Principle

2021-01-01

Benedikt Ahrens , Paige Randall North , Michael Shulman +1 more

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined … The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

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Univalent Higher Categories via Complete Semi-Segal Types

2018-01-07

Paolo Capriotti , Nicolai Kraus

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

The Univalence Principle.

2021-02-11

Benedikt Ahrens , Paige Randall North , Michael Shulman +1 more

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined … The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

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Categorical structures for type theory in univalent foundations

2017-01-01

Benedikt Ahrens , Peter LeFanu Lumsdaine , Vladimir Voevodsky

In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps … In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in univalent type theory, where the comparisons between them can be given more elementarily than in set-theoretic foundations. Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences. We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.

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Univalence and completeness of Segal objects

2022-11-11

Raffael Stenzel

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Univalence and completeness of Segal objects

2019-01-01

Raffael Stenzel

Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps … Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in $(\infty,1)$-categories. Completeness is a property of Segal spaces introduced by Rezk that characterizes those Segal spaces which are $(\infty,1)$-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study its ramifications. The core aspect of this analogy can be understood as a translation between internal and external notions, motivated by model categorical considerations of Joyal and Tierney. As a result, we characterize the internal notion of univalence in logical model categories by the external notion of completeness defined as the right Quillen condition of suitably indexed Set-weighted limit functors. Furthermore, we extend the analogy and show that univalent completion in the sense of van den Berg and Moerdijk translates to Rezk-completion of associated Segal objects as well. Motivated by these correspondences, we exhibit univalence as a homotopical locality condition whenever univalent completion exists.

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Univalent Enriched Categories and the Enriched Rezk Completion

2024-01-01

Niels van der Weide

Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics … Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk completion for enriched categories to construct univalent enriched Kleisli categories.

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Homotopy Type Theory: A Synthetic Approach to Higher Equalities

2018-01-18

Michael Shulman

Homotopy type theory and univalent foundations (HoTT/UF) is a new foundation of mathematics, based not on set theory but on “infinity-groupoids”, which consist of collections of objects, ways in which … Homotopy type theory and univalent foundations (HoTT/UF) is a new foundation of mathematics, based not on set theory but on “infinity-groupoids”, which consist of collections of objects, ways in which two objects can be equal, ways in which those ways-to-be-equal can be equal, ad infinitum. Though apparently complicated, such structures are increasingly important in mathematics. Philosophically, they are an inevitable result of the notion that whenever we form a collection of things, we must simultaneously consider when two of those things are the same. The “synthetic” nature of HoTT/UF enables a much simpler description of infinity groupoids than is available in set theory, thereby aligning with modern mathematics while placing “equality” back in the foundations of logic. This chapter will introduce the basic ideas of HoTT/UF for a philosophical audience, including Voevodsky’s univalence axiom and higher inductive types.

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Insights From Univalent Foundations: A Case Study Using Double Categories

2024-02-07

Nima Rasekh , Niels van der Weide , Benedikt Ahrens +1 more

Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional … Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional programming, and semantics. Certain objects naturally exhibit two classes of morphisms, leading to the concept of a double category, which has found applications in computing science (e.g., ornaments, profunctor optics, denotational semantics). The emergence of diverse categorical structures motivated a unified framework for category theory. However, unlike other mathematical objects, classification of categorical structures faces challenges due to various relevant equivalences. This poses significant challenges when pursuing the formalization of categories and restricts the applicability of powerful techniques, such as transport along equivalences. This work contends that univalent foundations offers a suitable framework for classifying different categorical structures based on desired notions of equivalences, and remedy the challenges when formalizing categories. The richer notion of equality in univalent foundations makes the equivalence of a categorical structure an inherent part of its structure. We concretely apply this analysis to double categorical structures. We characterize and formalize various definitions in Coq UniMath, including (pseudo) double categories and double bicategories, up to chosen equivalences. We also establish univalence principles, making chosen equivalences part of the double categorical structure, analyzing strict double setcategories (invariant under isomorphisms), pseudo double setcategories (invariant under isomorphisms), univalent pseudo double categories (invariant under vertical equivalences) and univalent double bicategories (invariant under gregarious equivalences).

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The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations

2024-02-07

Daniel Gratzer , Håkon Robbestad Gylterud , Anders Mörtberg +1 more

When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with … When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of Set hold for hSet ((co)completeness, exactness, local cartesian closure, etc.). Notably, however, the univalence axiom implies that Ob(hSet) is not itself an h-set, but an h-groupoid. This is expected in univalent foundations, but it is sometimes useful to also have a stricter universe of sets, for example when constructing internal models of type theory. In this work, we equip the type of iterative sets V0, due to Gylterud (2018) as a refinement of the pioneering work of Aczel (1978) on universes of sets in type theory, with the structure of a Tarski universe and show that it satisfies many of the good properties of h-sets. In particular, we organize V0 into a (non-univalent strict) category and prove that it is locally cartesian closed. This enables us to organize it into a category with families with the structure necessary to model extensional type theory internally in HoTT/UF. We do this in a rather minimal univalent type theory with W-types, in particular we do not rely on any HITs, or other complex extensions of type theory. Furthermore, the construction of V0 and the model is fully constructive and predicative, while still being very convenient to work with as the decoding from V0 into h-sets commutes definitionally for all type constructors. Almost all of the paper has been formalized in Agda using the agda-unimath library of univalent mathematics.

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Univalence and completeness of Segal objects

2019-11-15

Raffael Stenzel

Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property which characterizes descent and hence classifying maps … Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property which characterizes descent and hence classifying maps in $(\infty,1)$-categories. Completeness is a property of Segal spaces introduced by Rezk which characterizes those Segal spaces that are $(\infty,1)$-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study ramifications of this analogy. The core aspect of this analogy can be understood as a translation between internal and external notions, motivated by model categorical considerations of Joyal and Tierney. As a result, we characterize the internal notion of univalence in model categories by the external notion of completeness defined as the right Quillen condition of suitably indexed Set-weighted limit functors. Furthermore, we extend this analogy and show that univalent completion in the sense of van den Berg and Moerdijk translates to Rezk-completion of associated Segal objects as well. Motivated by these correspondences, we exhibit univalence as a locality condition whenever univalent completion exists.

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Univalent Monoidal Categories

2022-01-01

Kobe Wullaert , Ralph Matthes , Benedikt Ahrens

Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. … Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we zoom in on monoidal categories and study them in a univalent setting. Specifically, we show that the bicategory of univalent monoidal categories is univalent. Furthermore, we construct a Rezk completion for monoidal categories: we show how any monoidal category is weakly equivalent to a univalent monoidal category, universally. We have fully formalized these results in UniMath, a library of univalent mathematics in the Coq proof assistant.

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Univalent Monoidal Categories

2023-07-28

Kobe Wullaert , Ralph Matthes , Benedikt Ahrens

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Univalence in Higher Category Theory

2021-01-01

Nima Rasekh

Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently … Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $\infty$-categories by Gepner and Kock. These definitions were used to characterize various $\infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $\infty$-categories that generalizes the aforementioned definitions and completely focuses on the $\infty$-categorical aspects, characterizing it via representability of certain functors, which should remind the reader of concepts such as adjunctions or limits. We then prove that in a locally Cartesian closed $\infty$-category (that is not necessarily presentable) univalence of a morphism is equivalent to the completeness of a certain Segal object we construct out of the morphism, characterizing univalence via internal $\infty$-categories, which had been considered in a strict setting by Stenzel. We use these results to study the connection between univalence and elementary topos theory. We also study univalent morphisms in the category of groups, the $\infty$-category of $\infty$-categories, and pointed $\infty$-categories.

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On the $\infty$-topos semantics of homotopy type theory

2022-01-01

Emily Riehl

Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes … Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course delivered at the Logic and Higher Structures workshop at CIRM-Luminy, attempts to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any $\infty$-topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.

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First-Order Logic with Isomorphism

2016-01-01

Dimitris Tsementzis

The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\text{FOL}_=$) allows us to define structures on sets. … The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\text{FOL}_=$) allows us to define structures on sets. We develop the syntax, semantics and deductive system for such a logic, which we call first-order logic with isomorphism ($\text{FOL}_{\cong}$). The syntax of $\text{FOL}_{\cong}$ extends $\text{FOL}_{=}$ in two ways. First, by incorporating into its signatures a notion of dependent sorts along the lines of Makkai's FOLDS as well as a notion of an $h$-level of each sort. Second, by specifying three new logical sorts within these signatures: isomorphism sorts, reflexivity predicates and transport structure. The semantics for $\text{FOL}_{\cong}$ are then defined in homotopy type theory with the isomorphism sorts interpreted as identity types, reflexivity predicates as relations picking out the trivial path, and transport structure as transport along a path. We then define a deductive system $\mathcal{D}_{\cong}$ for $\text{FOL}_{\cong}$ that encodes the sense in which the inhabitants of isomorphism sorts really do behave like isomorphisms and prove soundness of the rules of $\mathcal{D}_{\cong}$ with respect to its homotopy semantics. Finally, as an application, we prove that precategories, strict categories and univalent categories are axiomatizable in $\text{FOL}_{\cong}$.

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The category of iterative sets in homotopy type theory and univalent foundations

2024-11-19

Daniel Gratzer , Håkon Robbestad Gylterud , Anders Mörtberg +1 more

Abstract When working in homotopy type theory and univalent foundations, the traditional role of the category of sets, $\mathcal{Set}$ , is replaced by the category $\mathcal{hSet}$ of homotopy sets (h-sets); … Abstract When working in homotopy type theory and univalent foundations, the traditional role of the category of sets, $\mathcal{Set}$ , is replaced by the category $\mathcal{hSet}$ of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of $\mathcal{Set}$ hold for $\mathcal{hSet}$ ((co)completeness, exactness, local cartesian closure, etc.). Notably, however, the univalence axiom implies that $\mathsf{Ob}\,\mathcal{hSet}$ is not itself an h-set, but an h-groupoid. This is expected in univalent foundations, but it is sometimes useful to also have a stricter universe of sets, for example, when constructing internal models of type theory. In this work, we equip the type of iterative sets $\mathsf{V}^0$ , due to Gylterud ((2018). The Journal of Symbolic Logic 83 (3) 1132–1146) as a refinement of the pioneering work of Aczel ((1978). Logic Colloquium’77 , Studies in Logic and the Foundations of Mathematics, vol. 96, Elsevier, 55–66.) on universes of sets in type theory, with the structure of a Tarski universe and show that it satisfies many of the good properties of h-sets. In particular, we organize $\mathsf{V}^0$ into a (non-univalent strict) category and prove that it is locally cartesian closed. This enables us to organize it into a category with families with the structure necessary to model extensional type theory internally in HoTT/UF. We do this in a rather minimal univalent type theory with W-types, in particular we do not rely on any HITs, or other complex extensions of type theory. Furthermore, the construction of $\mathsf{V}^0$ and the model is fully constructive and predicative, while still being very convenient to work with as the decoding from $\mathsf{V}^0$ into h-sets commutes definitionally for all type constructors. Almost all of the paper has been formalized in $\texttt{Agda}$ using the $\texttt{agda}$ - $\texttt{unimath}$ library of univalent mathematics.

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Univalent higher categories via complete Semi-Segal types

2017-12-27

Paolo Capriotti , Nicolai Kraus

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

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Formalizing the ∞-Categorical Yoneda Lemma

2024-01-09

Nikolai Kudasov , Emily Riehl , Jonathan Weinberger

Formalized 1-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," … Formalized 1-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," rely on infinite-dimensional categories in place of 1-dimensional categories, and ∞-category theory has thusfar proved unamenable to computer formalization.

Experimental library of univalent formalization of mathematics

2014-01-01

Vladimir Voevodsky

This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13. This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.

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Homotopy Type Theory: The Logic of Space

2021-03-22

Michael Shulman

This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book New Spaces … This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book New Spaces for Mathematics and Physics (ed. Gabriel Catren and Mathieu Anel).

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An introduction to univalent foundations for mathematicians

2018-03-05

Daniel R. Grayson

We offer an introduction for mathematicians to the univalent foundations of Vladimir Voevodsky, aiming to explain how he chose to encode mathematics in type theory and how the encoding reveals … We offer an introduction for mathematicians to the univalent foundations of Vladimir Voevodsky, aiming to explain how he chose to encode mathematics in type theory and how the encoding reveals a potentially viable foundation for all of modern mathematics that can serve as an alternative to set theory.

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Univalence and completeness of Segal objects

2022-11-11

Raffael Stenzel

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A Higher Structure Identity Principle

2020-05-25

Benedikt Ahrens , Paige Randall North , Michael Shulman +1 more

The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of structures … The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of structures coincide with isomorphisms. We prove a version of this principle for a wide range of higher-categorical structures, adapting FOLDS-signatures to specify a general class of structures, and using two-level type theory to treat all categorical dimensions uniformly. As in the previously known case of 1-categories (which is an instance of our theory), the structures themselves must satisfy a local univalence principle, stating that identifications coincide with "isomorphisms" between elements of the structure. Our main technical achievement is a definition of such isomorphisms, which we call "indiscernibilities," using only the dependency structure rather than any notion of composition.

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Higher Structures in Homotopy Type Theory

2019-01-01

Ulrik Buchholtz

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Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT

2020-01-01

Nicolai Kraus

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

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Two-sided cartesian fibrations of synthetic $$(\infty ,1)$$-categories

2024-06-21

Jonathan Weinberger

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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Types Are Internal ∞-Groupoids

2021-06-29

Eric Finster , Antoine Allioux , Matthieu Sozeau

By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode … By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of ∞-groupoid internal to type theory and we prove that the type of such ∞-groupoids is equivalent to the universe of types. That is, every type admits the structure of an ∞-groupoid internally, and this structure is unique.

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Univalence in Higher Category Theory

2021-01-01

Nima Rasekh

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Abstraction, equality and univalence

2023-07-30

Gabriel Catren

We shall propose a conceptual-oriented discussion of the so-called Univalent Foundations Program, that is, of Martin-Löf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by … We shall propose a conceptual-oriented discussion of the so-called Univalent Foundations Program, that is, of Martin-Löf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by Voevodsky. We shall argue that the type-theoretic notion of propositional equality encodes the notion of indiscernibility, we shall address the homotopic interpretation of Martin-Löf type theory, and we shall analyse whether Leibniz's principle of the identity of indiscernibles holds or not in Univalent Foundations. We shall finally argue that univalence can be understood as a particular implementation of a constructive notion of abstraction that resolves Fregean abstraction. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.

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Reedy categories and their generalizations

2015-01-01

Michael Shulman

We observe that the Reedy model structure on a diagram category can be constructed by iterating an operation of "bigluing" model structures along a pair of functors and a natural … We observe that the Reedy model structure on a diagram category can be constructed by iterating an operation of "bigluing" model structures along a pair of functors and a natural transformation. This yields a new explanation of the definition of Reedy categories: they are almost exactly those small categories for which the category of diagrams and its model structure can be constructed by iterated bigluing. It also gives a consistent way to produce generalizations of Reedy categories, including the generalized Reedy categories of Cisinski and Berger-Moerdijk and the enriched Reedy categories of Angeltveit, but also new versions such as a combined notion of "enriched generalized Reedy category".

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Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

2021-06-29

Nicolai Kraus

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

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Univalent higher categories via complete Semi-Segal types

2017-12-27

Paolo Capriotti , Nicolai Kraus

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Experience Implementing a Performant Category-Theory Library in Coq

2014-01-29

Jason Gross , Adam Chlipala , David I. Spivak

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering … We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.

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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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Experience Implementing a Performant Category-Theory Library in Coq

2014-01-01

Jason Gross , Adam Chlipala , David I. Spivak

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A type theory for synthetic ∞-categories

2017-12-22

Emily Riehl , Michael Shulman

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe … We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define \emph{Segal types}, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define \emph{Rezk types}, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities --- a ``local univalence'' condition. And we define \emph{covariant fibrations}, which are type families varying functorially over a Segal type, and prove a ``dependent Yoneda lemma'' that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using ``extension types'' that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.

Univalent categories of modules

2023-02-01

Jarl G. Taxerås Flaten

Abstract We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies … Abstract We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X . From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $\infty$ -topos $ {\mathscr{X}} $ . Given a ring R in $ {\tau_{\leq 0}({{\mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R -modules in $ {\mathscr{X}} $ represents the presheaf which sends an object $ X \in {\mathscr{X}} $ to the category of $ (X{\times}R) $ -modules in ${\mathscr{X}} / X$ . In general, our results yield a product-preserving left adjoint to base change of modules over X . When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.

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An experimental library of formalized Mathematics based on the univalent foundations

2015-01-20

Vladimir Voevodsky

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started … This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).

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Bicategories in Univalent Foundations

2019-03-04

Benedikt Ahrens , Dan Frumin , Marco Maggesi +2 more

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples … We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories, we develop the notion of `displayed bicategories', an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. Displayed bicategories allow us to construct univalent bicategories in a modular fashion. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-category is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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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.

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The category of iterative sets in homotopy type theory and univalent foundations

2024-11-19

Daniel Gratzer , Håkon Robbestad Gylterud , Anders Mörtberg +1 more

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Homotopy limits in type theory

2015-01-19

Jeremy Avigad , Krzysztof Kapulkin , Peter LeFanu Lumsdaine

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.

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Displayed Categories

2017-01-01

Benedikt Ahrens , Peter LeFanu Lumsdaine

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but … We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

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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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Types are Internal $\infty$-Groupoids

2021-01-01

Antoine Allioux , Eric Finster , Matthieu Sozeau

By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode … By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of $\infty$-groupoid internal to type theory and we prove that the type of such $\infty$-groupoids is equivalent to the universe of types. That is, every type admits the structure of an $\infty$-groupoid internally, and this structure is unique.

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Homotopy type theory

2014-01-01

Álvaro Pelayo , Michael A. Warren

We give a glimpse of an emerging field at the intersection of homotopy theory, logic, and theoretical computer science: homotopy type theory. One key ingredient of this approach is Vladimir … We give a glimpse of an emerging field at the intersection of homotopy theory, logic, and theoretical computer science: homotopy type theory. One key ingredient of this approach is Vladimir Voevodsky’s Univalence Axiom. It is the goal of this paper to provide a short introduction to some of the ideas of homotopy type theory and univalence. The approach taken here is to first develop some of the historical and mathematical context in which homotopy type theory arose and then to describe the Univalence Axiom and related technical machinery.

Univalent categories and the Rezk completion

2015-01-19

Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman

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

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A Coq Formalization of Finitely Presented Modules

2014-01-01

Cyril Cohen , Anders Mörtberg

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.

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Category Theory in Coq 8.5

2016-06-01

Amin Timany , Bart Jacobs

We report on our experience implementing category theory in Coq 8.5. Our work formalizes most of basic category theory, including concepts not covered by existing formalizations, in a library that … We report on our experience implementing category theory in Coq 8.5. Our work formalizes most of basic category theory, including concepts not covered by existing formalizations, in a library that is fit to be used as a general-purpose category-theoretical foundation. Our development particularly takes advantage of two features new to Coq 8.5: primitive projections for records and universe polymorphism. Primitive projections allow for well-behaved dualities while universe polymorphism provides a relative notion of largeness and smallness. The latter is one of the main contributions of this paper. It pushes the limits of the new universe polymorphism and constraint inference algorithm of Coq 8.5. In this paper we present in detail smallness and largeness in categories and the foundation they are built on top of. We furthermore explain how we have used the universe polymorphism of Coq 8.5 to represent smallness and largeness arguments by simply ignoring them and entrusting them to the universe inference algorithm of Coq 8.5. We also briefly discuss our experience throughout this implementation, discuss concepts formalized in this development and give a comparison with a few other developments of similar extent.

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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.

Bicategories in univalent foundations

2021-11-01

Benedikt Ahrens , Dan Frumin , Marco Maggesi +2 more

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples … We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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Semantics for two-dimensional type theory

2022-08-02

Benedikt Ahrens , Paige Randall North , Niels van der Weide

We propose a general notion of model for two-dimensional type theory, in the form of comprehension bicategories. Examples of comprehension bicategories are plentiful; they include interpretations of directed type theory … We propose a general notion of model for two-dimensional type theory, in the form of comprehension bicategories. Examples of comprehension bicategories are plentiful; they include interpretations of directed type theory previously studied in the literature.

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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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Sets in homotopy type theory

2015-01-30

Egbert Rijke , Bas Spitters

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.

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Univalent Categories and the Rezk Completion

2015-01-01

Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman

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Experimental library of univalent formalization of mathematics

2014-01-01

Vladimir Voevodsky

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Univalent Foundations of Mathematics

2011-01-01

Vladimir Voevodsky

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On the Unicity of the Homotopy Theory of Higher Categories

2011-01-01

Clark Barwick , Christopher Schommer‐Pries

We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $Θ_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, … We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $Θ_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.

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A model for the homotopy theory of homotopy theory

2000-06-20

Charles Rezk

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

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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Univalent categories and the Rezk completion

2015-01-19

Benedikt Ahrens , Krzysztof Kapulkin , Michael Shulman

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

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Homotopy Theoretic Aspects of Constructive Type Theory

2008-01-01

Michael A. Warren

Homotopy-Theoretic Models of Type Theory

2011-01-01

Peter Arndt , Krzysztof Kapulkin

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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.

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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.

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Stack completions and Morita equivalence for categories in a topos

1979-01-01

Marta Bunge

Homotopy type theory and Voevodsky’s univalent foundations

2014-05-09

Álvaro Pelayo , Michael A. Warren

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which … Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened <italic>homotopy type theory</italic>. In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the <italic>Univalence Axiom</italic>, which has a number of striking consequences. He has subsequently advocated a program, which he calls <italic>univalent foundations</italic>, of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model. Because type theory possesses good computational properties, this program can be carried out in a computer proof assistant. In this paper we give an introduction to homotopy type theory in Voevodsky’s setting, paying attention to both theoretical and practical issues. In particular, the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky’s work using the well-known proof assistant Coq. The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology; the paper does not assume any preliminary knowledge of type theory, logic, or computer science. Because a defining characteristic of Voevodsky’s program is that the Coq code has fundamental mathematical content, and many of the mathematical concepts which are efficiently captured in the code cannot be explained in standard mathematical English without a lengthy detour through type theory, the later sections of this paper (beginning with Section \ref{sec2}) make use of code; however, all notions are introduced from the beginning and in a self-contained fashion.

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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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Strong stacks and classifying spaces

1991-01-01

André Joyal , Myles Tierney