Vladimir Voevodsky

• Professor (Full) at Institute for Advanced Study

This paper continues a 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-Löf type theories on the C-systems that arise from universe categories. In the first part of the paper we develop constructions that pro...

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...

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...

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...

We define the notion of a (P, P)-structure on a universe p in a locally cartesian closed category category with a binary product structure and construct a (Π, λ)-structure on the C-systems CC(C, p) from a (P, P)-structure on p. We then define homomorphisms of C-systems with (Π, λ)-structures and functors of universe categories with (P, P)-structure...

The main result of this paper may be stated as a construction of “almost representations” μn and eμn for the presheaves Obn and fObn on the C-systems dened 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...

In this paper we continue, following the pioneering works by J. Cartmell and T. Streicher, the study of the most important structures on C-systems, the structures that correspond, in the case of the syntactic C-systems, to the (Π, λ, app, β, η)-system of inference rules. One such structure was introduced by J. Cartmell and later studied by T. Strei...

Let F be the category with the set of objects N and morphisms given by the functions between the standard finite sets of the corresponding cardinalities. Let Jf:F→Sets(U) be the obvious functor from this category to the category of sets in a given Grothendieck universe U. In this paper we construct, for any Jf-relative monad RR and any left RR-modu...

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 → Sets where F is the category with the set of objects N and morphisms being the functions between the standard finite sets of the corresponding cardinalities. The methods of t...

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.

This version contains a number of changes which were required to preserve exact compatibility with the latest version of the "C-system from a universe category" paper.

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 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 p...

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 typ...

This is the second paper in a series started in [13] which 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 (M, LM), where M is a monad on sets and LM is a left module over M , a C-system ("contextual category") CC(M, L...

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 regular quotients which in particular have the property that the corresponding projection morp...

This is the text of my talk at CMU on Feb. 4, 2010 were I gave the second public presentation of the Univalence Axiom (called "equivalence axiom" in the text). The first presentation of the axiom was in a lecture at LMU Munich in November 2009.

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...

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...

We construct and investigate a model of the Univalent Foundations of Mathematics in the category of simplicial sets. To this end, we first give a new technique for constructing 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....

In this talk I will outline a new semantics for dependent polymorphic type theories with Martin-Lof identity types. It is based on a class of models which interpret types as simplicial sets or topological spaces defined up to homotopy equivalence. The intuition based on the univalent semantics leads to new answers to some long standing questions of...

Over the last two years deep and unexpected connections have been discovered between constructive type theories and classical homotopy theory. These connections open a way to construct new foundations of mathematics alternative to the ZFC. These foundations promise to resolve several seemingly unconnected problems-provide a support for categorical...

In this short paper we show that the motivic cohomology groups defined in [3],[4] are isomorphic to the motivic cohomology groups defined in [1, ] for smooth schemes over any field. In view of [1, Corollary 11.2] it implies that motivic cohomology of [3],[4] are isomorphic to higher Chow groups. Because of the homotopy invariance property of higher...

The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category built out of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is very de...

There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a N...

In this paper we define the triangulated category of motives over a simplicial scheme. The morphisms between the Tate objects in this category compute the motivic cohomology of the underlying scheme. In the last section we consider the special case of "embedded" simplicial schemes, which correspond to the subsheaves of the constant sheaf and natura...

These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic complexes. There is a description of the equivariant motivic homotopy category for a finite flat group scheme...

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.

In this paper we construct symmetric powers in the motivic homotopy categories of morphisms and finite correspondences associated with f-admissible subcategories in the categories of schemes of finite type over a field. Using this construction we provide a description of the motivic Eilenberg-MacLane spaces representing motivic cohomology on some f...

The simplicial extension of any functor from Sets to Sets which commutes with directed colimits takes weak equivalences to weak equivalences. The goal of the present paper is construct a framework which can be used to proof results of this kind for a wide class of closed model categories and functors between those categories. Comment: This is a sub...

0.0. This paper develops some of the ideas outlined by Alexander Grothendieck in his unpublished Esquisse d’un programme [0] in 1984.

We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjectu...

Motivic homotopy theory is a new and in vogue blend of algebra and topology. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology.

We prove the motivic analogue of the statement saying that the zero stable homotopy group of spheres is ℤ. In topology, this is equivalent to the fact that the fiber of the obvious map from the sphere S n to the Eilenberg-MacLane space K(ℤ,n) is (n+1)-connected. We prove our motivic analogue by an explicit geometric investigation of a similar map i...

In this paper we give a direct geometric proof of the fact that tensoring with the Tate motive in the triangulated category of effective motives DM is a full embedding. The main part of the proof is given in the context of schemes of finite type over a noetherian base scheme.

In this paper we prove that two definitions of motivic cohomologyfor smooth varieties over any field agree. The first definitionis the one used in the proof of the Milnor conjecture. The secondone was shown by Friedlander and Suslin to agree withBloch's higher Chow groups. A proof of the main theorem of thispaper was previously known for varieties...

In this paper we construct an analog of Steenrod operations in motivic cohomology and prove their basic properties including the Cartan formula, the Adem relations and the realtions to characteristic classes.

In this paper we prove the 2-local part of the Beilinson-Lichtenbaum conjectures on tosion in motivic cohomology. In particular we prove the Milnor conjecture relating Milnor's K-theory and the Galois cohomology with Z/2-coefficients. This paper is a new version of the previously distributed preprint "The Milnor Conjecture".

We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjectu...

this paper is to prove the following conjecture:

this paper we introduce a class of cycles on X which are called relative cycles on X over S. Their most important property is the existence of well dened base change homomorphisms for arbitrary morphisms S

this paper can be obtained from a glance at the table of contents. To conclude this introduction, we give a somewhat more detailed summary of the contents of the various sections of our paper. Section 2 recalls from [22] the functoriality of the presheaf z equi (X; r) which sends a smooth scheme U of nite type over k to the group of cycles on X U e...

Introduction The homotopic category Homotopic excision, homotopic purity and projective blow-ups Homotopic classification of vector bundles Appendix A: Review of homotopic algebra Appendix B: Ample families of invertible bundles on a scheme References.

this paper we study contravariant functors from the category Sm=k to additive 1

this paper we construct for any perfect eld k a triangulated category DM

Contents Introduction 0. Notations, terminology and general remarks. 1. Homotopy invariant presfeaves with transfers. 2. Tensor structure on the category DM Gamma (F ). 3. Motivic cohomology. 4. Fundamental distinguished triangles in the category DM Gamma (F ). 5. Motivic cohomology of non-smooth schemes and the cdhtopology. 6. Truncated etale coho...

this paper we show that the existence of algebro-geometrical analogs of the higher Morava K-theories satisfying some basic properties would imply the Bloch-Kato conjecture with Z=2-coefficients for fields which admit resolution of singularities (see [2] for a precise formulation of this condition).

These notes are based on a series of talks given by V. Voevodsky at the AMS Joint Summer Research Conference on algebraic K-theory, held in Seattle during July 1997. The purpose of these talks was to outline the proof of “Milnor’s conjecture” [V. Voevodsky, “The Milnor conjecture”, Preprint (1996)]: if F is a field of characteristic ≠2, then the Mi...

this paper.

In this paper we prove that a correspondence from a smooth projective variety over a eld to itself which is algebraically equivalent to zero is a nilpotent in the ring of correspondences modulo rational equivalence (with rational coecien ts). We also show that a little more general result holds, namely that for any algebraic cycle Z on a smooth pro...