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Vladimir Voevodsky

Publications and source records attributed to Vladimir Voevodsky.

At least 19 recordsLinked to original sources
arXiv2018

Categorical structures for type theory in univalent foundations

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.

math.LO↗
arXiv2017

Products of families of types and (Pi,lambda)-structures on C-systems

In this paper we continue the study of the most important structures on C-systems, the structures that correspond, in the case of the syntactic C-systems, to the $(Pi,lambda,app,beta,eta)$-system of inference rules. One such structure was introduced by J. Cartmell and later studied by T. Streicher under the name of the products of families of types. We introduce the notion of a (Pi,lambda)-structure and construct a bijection, for a given C-system, between the set of (Pi,lambda)-structures and the set of Cartmell-Streicher structures. In the following paper we will show how to construct, and in some cases fully classify, the (Pi,λ)-structures on the C-systems that correspond to universe categories. The first section of the paper provides careful proofs of many of the properties of general C-systems. Methods of the paper are fully constructive, that is, neither the axiom of excluded middle nor the axiom of choice are used.

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arXiv2017

The (Pi,lambda)-structures on the C-systems defined by universe categories

We define the notion of a (P,P-tilde)-structure on a universe p in a locally cartesian closed category category C with a binary product structure and construct a (Pi,lambda)-structure on the C-systems CC(C,p) from a (P,P-tilde)-structure on p. We then define homomorphisms of C-systems with (Pi,lambda)-structures and functors of universe categories with (P,P-tilde)-structures and show that our construction is functorial relative to these definitions.

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arXiv2017

C-systems defined by universe categories: presheaves

The main result of this paper may be stated as a construction of "almost representations" for the canonical presheaves of object extensions of length n on the C-systems defined by locally cartesian closed universe categories with binary product structures and the study of the behavior of these "almost representations" with respect to the universe category functors. In addition, we study a number of constructions on presheaves on C-systems and on universe categories that are used in the proofs of our main results, but are expected to have other applications as well.

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arXiv2016

Subsystems and regular quotients of C-systems

C-systems were introduced by J. Cartmell under the name "contextual categories". In this note we study sub-objects and quotient-objects of C-systems. In the case of the sub-objects we consider all sub-objects while in the case of the quotient-objects only {\em regular} quotients that in particular have the property that the corresponding projection morphism is surjective both on objects and on morphisms. It is one of several short papers based on the material of the "Notes on Type Systems" by the same author. This version is essentially identical with the version published in Contemporary Mathematics n.658.

math.LO↗
arXiv2016

C-system of a module over a $Jf$-relative monad

Let $F$ be the category with the set of objects $\bf N$ and morphisms being the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:F\rightarrow Sets$ be the obvious functor from this category to the category of sets. In this paper we construct, for any relative monad $\bf RR$ on $Jf$ and a left module $\bf LM$ over $\bf RR$, a C-system $C({\bf RR},{\bf LM})$ and explicitly compute the action of the B-system operations on its B-sets. In the following paper it is used to provide a rigorous mathematical approach to the construction of the C-systems underlying the term models of a wide class of dependent type theories. This paper is a result of evolution of arXiv:1407.3394. However this paper is much more detailed and contains a lot of material that is not contained in arXiv:1407.3394. It also does not cover some material that is covered in arXiv:1407.3394.

math.LO↗
arXiv2016

Lawvere theories and Jf-relative monads

In this paper we provide a detailed construction of an equivalence between the category of Lawvere theories and the category of relative monads on the obvious functor $Jf:F\rightarrow Sets$ where $F$ is the category with the set of objects ${\bf N}$ and morphisms being the functions between the standard finite sets of the corresponding cardinalities. The methods of this paper are fully constructive and it should be formalizable in the Zermelo-Fraenkel theory without the axiom of choice and the excluded middle. It is also easily formalizable in the UniMath.

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arXiv2015

Lawvere theories and C-systems

In this paper we consider the class of l-bijective C-systems, i.e., C-systems for which the length function is a bijection. The main result of the paper is a construction of an isomorphism between two categories - the category of l-bijective C-systems and the category of Lawvere theories.

math.LO↗
arXiv2015

A C-system defined by a universe category

This is a major update of the previous version. The methods of the paper are now fully constructive and the style is "formalization ready" with the emphasis on the possibility of formalization both in type theory and in constructive set theory without the axiom of choice. This is the third paper in a series started in 1406.7413. In it we construct a C-system $CC({\cal C},p)$ starting from a category $\cal C$ together with a morphism $p:\widetilde{U}\rightarrow U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of $\cal C$. Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems $CC({\cal C},p)$ defined by universe category functors. As a corollary of this construction and its properties we show that the C-systems corresponding to different choices of pull-backs and final objects are constructively isomorphic. In the second part of the paper we provide for any C-system CC three constructions of pairs $(({\cal C},p),H)$ where $({\cal C},p)$ is a universe category and $H:CC\rightarrow CC({\cal C},p)$ is an isomorphism. In the third part we define, using the constructions of the previous parts, for any category $C$ with a final object and fiber products a C-system $CC(C)$ and an equivalence $(J^*,J_*):C \rightarrow CC$.

math.LO↗
arXiv2015

Products of families of types in the C-systems defined by a universe category

We introduce the notion of a $(Π,λ)$-structure on a C-system and show that C-systems with $(Π,λ)$-structures are constructively equivalent to contextual categories with products of families of types. We then show how to construct $(Π,λ)$-structures on C-systems of the form $CC({\cal C},p)$ defined by a universe $p$ in a locally cartesian closed category $\cal C$ from a simple pull-back square based on $p$. In the last section we prove a theorem that asserts that our construction is functorial. This version introduces some changes compared to the previous one to ensure rigorous compatibility with arXiv:1409.7925v3.

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arXiv2015

Martin-Lof identity types in the C-systems defined by a universe category

This paper continues the series of papers that develop a new approach to syntax and semantics of dependent type theories. Here we study the interpretation of the rules of the identity types in the intensional Martin-Lof type theories on the C-systems that arise from universe categories. In the first part of the paper we develop constructions that produce interpretations of these rules from certain structures on universe categories while in the second we study the functoriality of these constructions with respect to functors of universe categories. The results of the first part of the paper play a crucial role in the construction of the univalent model of type theory in simplicial sets.

math.CT↗
arXiv2014

B-systems

B-systems are algebras (models) of an essentially algebraic theory that is expected to be constructively equivalent to the essentially algebraic theory of C-systems which is, in turn, constructively equivalent to the theory of contextual categories. The theory of B-systems is closer in its form to the structures directly modeled by contexts and typing judgements of (dependent) type theories and further away from categories than contextual categories and C-systems.

math.LO↗
arXiv2014

C-system of a module over a monad on sets

This is the second paper in a series that aims to provide mathematical descriptions of objects and constructions related to the first few steps of the semantical theory of dependent type systems. We construct for any pair $(R,LM)$, where $R$ is a monad on sets and $LM$ is a left module over $R$, a C-system (contextual category) $CC(R,LM)$ and describe a class of sub-quotients of $CC(R,LM)$ in terms of objects directly constructed from $R$ and $LM$. In the special case of the monads of expressions associated with nominal signatures this construction gives the C-systems of general dependent type theories when they are specified by collections of judgements of the four standard kinds.

math.LO↗
arXiv2013

A preliminary univalent formalization of the p-adic numbers

In this paper we give a preliminary formalization of the p-adic numbers, in the context of the second author's univalent foundations program. We also provide the corresponding code verifying the construction in the proof assistant Coq. Because work in the univalent setting is ongoing, the structure and organization of the construction of the p-adic numbers we give in this paper is expected to change as Coq libraries are more suitably rearranged, and optimized, by the authors and other researchers in the future. So our construction here should be deemed as a first approximation which is subject to improvements.

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arXiv2010

On motivic cohomology with Z/l coefficients

In this paper we give a proof of the Bloch-Kato conjecture relating motivic cohomology and etale cohomology. It is a corrected version of the paper with the same title which posted earlier.

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