Publications

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Author Title Type Year(Desc)
1989
Voevodsky, V.. A., and G.. B. Shabat, "Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields", Dokl. Akad. Nauk SSSR, vol. 304, pp. 265–268, 1989. [PDF] (439.49 KB)
Voevodsky, V.. A., "Galois Group over Q and Teihmuller Modular Groups", Proc. Conf. Constr. Methods and Alg. Number Theory, Minsk, 1989.
1990
Shabat, G.. B., and V.. A. Voevodsky, "Drawing curves over number fields", The Grothendieck Festschrift, Vol. III, vol. 88: Birkhäuser Boston, Boston, MA, pp. 199–227, 1990. [PDF] (2.02 MB)
Voevodsky, V.. A., "Etale topologies of schemes over fields of finite type over Q", Izv. Akad. Nauk SSSR Ser. Mat., vol. 54, pp. 1155–1167, 1990. [PDF] (713.56 KB)
Voevodsky, V.. A., "Flags and Grothendieck Cartographical Group in Higher Dimensions", CSTARCI Math. Preprints, 1990. [PDF] (583.06 KB)
Voevodsky, V.. A., and M.. M. Kapranov, "infty-groupoids as a model for a homotopy category", Uspekhi Mat. Nauk, vol. 45, pp. 183–184, 1990.
Warning: the main theorem of this paper was shown by Carlos Simpson to be false.
[PDF] (202.35 KB)
Voevodsky, V.. A., and M.. M. Kapranov, "Multidimensional Categories (in Russian)", Proc. Conf. of Young Scientists: Moscow Univ. Press, 1990.
Voevodsky, V.. A., and G.. B. Shabat, "Piece-Wise Euclidean Approximation of Jacobians of Algebraic Curves", CSTARCI Math. Preprints, 1990. [PDF] (1.23 MB)
Voevodsky, V.. A., "Triangulations of Oriented Manifolds and Ramified Coverings of Sphere (in Russian)", Proc. Conf. of Young Scientists: Moscow Univ. Press, 1990.
1991
Kapranov, M.. M., and V.. A. Voevodsky, "Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results)", Cahiers Topologie Géom. Différentielle Catég., vol. 32, pp. 11–27, 1991.
International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990)
[PDF] (1.28 MB)
Voevodsky, V.. A., and M.. M. Kapranov, "The free n-category generated by a cube, oriented matroids and higher Bruhat orders", Funktsional. Anal. i Prilozhen., vol. 25, pp. 62–65, 1991. [PDF] (235.89 KB)
Voevodsky, V.. A., "Galois groups of function fields over fields of finite type over Q", Uspekhi Mat. Nauk, vol. 46, pp. 163–164, 1991. [PDF] (148.48 KB)
Voevodsky, V.. A., "Galois representations connected with hyperbolic curves", Izv. Akad. Nauk SSSR Ser. Mat., vol. 55, pp. 1331–1342, 1991. [PDF] (667.54 KB)
Kapranov, M.. M., and V.. A. Voevodsky, "infty-groupoids and homotopy types", Cahiers Topologie Géom. Différentielle Catég., vol. 32, pp. 29–46, 1991.
International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990)

Warning: the main theorem of this paper was shown by Carlos Simpson to be false.   
[PDF] (1.34 MB)
1992
Voevodsky, V., Homology of schemes and covariant motives, : ProQuest LLC, Ann Arbor, MI, pp. 64, 1992.
Thesis (Ph.D.)–Harvard University
[PDF] (3.48 MB)
1994
Kapranov, M.. M., and V.. A. Voevodsky, "2-categories and Zamolodchikov tetrahedra equations", Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991), vol. 56: Amer. Math. Soc., Providence, RI, pp. 177–259, 1994. [PDF] (7.42 MB)
Kapranov, M.., and V.. Voevodsky, "Braided monoidal 2-categories and Manin-Schechtman higher braid groups", J. Pure Appl. Algebra, vol. 92, pp. 241–267, 1994. [PDF] (1.51 MB)
1995
Voevodsky, V.., "A nilpotence theorem for cycles algebraically equivalent to zero", Internat. Math. Res. Notices, pp. 187–198 (electronic), 1995. [PDF] (98.87 KB)
1996
Voevodsky, V.., "Homology of schemes", Selecta Math. (N.S.), vol. 2, pp. 111–153, 1996. [PDF] (2.44 MB)
Suslin, A., and V. Voevodsky, "Singular homology of abstract algebraic varieties", Invent. Math., vol. 123, pp. 61–94, 1996. [PDF] (1.56 MB)
1998
Voevodsky, V., "A^1-homotopy theory", Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998. [PDF] (355.89 KB)
1999
Morel, F., and V. Voevodsky, "A^1-homotopy theory of schemes", Inst. Hautes Études Sci. Publ. Math., pp. 45–143 (2001), 1999. [PDF] (9.25 MB)
Voevodsky, V., "Categories and functors in mathematics", Unpublished, pp. 1–18, 1999.
Voevodsky, V., "Four functors formalism", Unpublished, pp. 1–20, March, 1999.
One of the most important ideas of the four functor formalism was the proof that a projective morphism is lower transversal (satisfies a generalized analog of the proper base change theorem). Unfortunately no complete proof remained but all of the main ideas are contained in these notes:

1998-11-13-notes.pdf
1998-11-21-notes.pdf
2002-01-08-notes.pdf
[PDF] (77.92 KB)
Weibel, C., "Voevodsky's Seattle lectures: K-theory and motivic cohomology", Algebraic K-theory (Seattle, WA, 1997), vol. 67: Amer. Math. Soc., Providence, RI, pp. 283–303, 1999. [PDF] (2.35 MB)
2000
Friedlander, E. M., and V. Voevodsky, "Bivariant cycle cohomology", Cycles, transfers, and motivic homology theories, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 138–187, 2000. [PDF] (348.23 KB)
Suslin, A., and V. Voevodsky, "Bloch-Kato conjecture and motivic cohomology with finite coefficients", The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), vol. 548: Kluwer Acad. Publ., Dordrecht, pp. 117–189, 2000. [PDF] (618.57 KB)
Voevodsky, V., "Cohomological theory of presheaves with transfers", Cycles, transfers, and motivic homology theories, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 87–137, 2000. [PDF] (346.26 KB)
Friedlander, E. M., A.. Suslin, and V.. Voevodsky, "Introduction", Cycles, transfers, and motivic homology theories, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 3–9, 2000. [PDF] (76.77 KB)
Suslin, A., and V. Voevodsky, "Relative cycles and Chow sheaves", Cycles, transfers, and motivic homology theories, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 10–86, 2000. [PDF] (524.85 KB)
Voevodsky, V., "Triangulated categories of motives over a field", Cycles, transfers, and motivic homology theories, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 188–238, 2000. [PDF] (346.26 KB)
2001
Voevodsky, V., "Framed Correspondences", Unpublished, 2001. [PDF] (116.24 KB)
Deligne, P., "Voevodsky's lectures on cross functors, Fall 2001", Unpublished, pp. 1–41, 2001. [PDF] (369.71 KB)
2002
Voevodsky, V., "Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic", Int. Math. Res. Not., pp. 351–355, 2002. [PDF] (87.25 KB)
Voevodsky, V., "Open problems in the motivic stable homotopy theory. I", Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), vol. 3: Int. Press, Somerville, MA, pp. 3–34, 2002. [PDF] (3.04 MB)
Voevodsky, V., "A possible new approach to the motivic spectral sequence for algebraic K-theory", Recent progress in homotopy theory Baltimore, MD, 2000), vol. 293: Amer. Math. Soc., Providence, RI, pp. 371–379, 2002. [PDF] (810.81 KB)
2003
Voevodsky, V., "Motivic cohomology with Z/2-coefficients", Publ. Math. Inst. Hautes Études Sci., pp. 59–104, 2003.
There were several papers that preceded this paper that explored other proofs of the Milnor Conjecture based on the same general direction of approach. Ultimately it was the present paper that was published in a journal since the proof that it contained was most direct and required the least amount of preliminary work. The earlier versions of the proof are available here:  
Voevodsky, V., "Reduced power operations in motivic cohomology", Publ. Math. Inst. Hautes Études Sci., pp. 1–57, 2003. [PDF] (542.94 KB)
2004
Voevodsky, V.., "On the zero slice of the sphere spectrum", Tr. Mat. Inst. Steklova, vol. 246, pp. 106–115, 2004. [PDF] (207.32 KB)
2006
Mazza, C., V. Voevodsky, and C. Weibel, "Lecture notes on motivic cohomology", Clay Mathematics Monographs, vol. 2: American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, pp. xiv+216, 2006. [PDF] (1.34 MB)
Voevodsky, V., "A very short note on homotopy $\lambda$-calculus", Unpublished, pp. 1–7, September, 2006. [PDF] (113.85 KB)
2007
Orlov, D.., A.. Vishik, and V.. Voevodsky, "An exact sequence for K^M/2 with applications to quadratic forms", Ann. of Math. (2), vol. 165, pp. 1–13, 2007. [PDF] (336.38 KB)
Voevodsky, V., O. Röndigs, and P. Arne Ø. stvær, "Voevodsky's Nordfjordeid lectures: motivic homotopy theory", Motivic homotopy theory: Springer, Berlin, pp. 147–221, 2007. [PDF] (1.12 MB)
2009
Deligne, P., "Voevodsky's lectures on motivic cohomology 2000/2001", Algebraic topology, vol. 4: Springer, Berlin, pp. 355–409, 2009. [PDF] (735.89 KB)
2010
Voevodsky, V., "The equivalence axiom and univalent models of type theory", arXiv 1402.5556, pp. 1–11, 2010.
Voevodsky, V., "Cancellation theorem", Doc. Math., pp. 671–685, 2010. [PDF] (155.85 KB)
Voevodsky, V., "Homotopy theory of simplicial sheaves in completely decomposable topologies", J. Pure Appl. Algebra, vol. 214, pp. 1384–1398, 2010. [PDF] (762.49 KB)
Voevodsky, V., "Motives over simplicial schemes", J. K-Theory, vol. 5, pp. 1–38, 2010. [PDF] (325.11 KB)
Voevodsky, V., "Motivic Eilenberg-Maclane spaces", Publ. Math. Inst. Hautes Études Sci., pp. 1–99, 2010. [PDF] (915.29 KB)
Voevodsky, V., "Simplicial radditive functors", J. K-Theory, vol. 5, pp. 201–244, 2010. [PDF] (389.16 KB)
Voevodsky, V., "Univalent Foundations Project", a modified version of an NSF grant application, pp. 1–12, October, 2010.
Voevodsky, V., "Unstable motivic homotopy categories in Nisnevich and cdh-topologies", J. Pure Appl. Algebra, vol. 214, pp. 1399–1406, 2010. [PDF] (506.91 KB)
2011
Voevodsky, V., "On motivic cohomology with Z/l-coefficients", Ann. of Math. (2), vol. 174, pp. 401–438, 2011. [PDF] (505.13 KB)
2014
Voevodsky, V., "B-systems", arXiv 1410.5389, submitted, pp. 1–17, 2014.
Voevodsky, V., "C-system of a module over a monad on sets", arXiv 1407.3394, submitted, pp. 1–20, 2014.
Voevodsky, V., "Subsystems and regular quotients of C-systems", arXiv 1406.5389, submitted, pp. 1–11, 2014.
2015
Voevodsky, V., "Lawvere theories and C-systems", arXiv:1512.08104, pp. 15pp, 12/2015. [PDF] (250.57 KB)
Voevodsky, V., "A C-system defined by a universe category", Theory Appl. Categ., vol. 30, pp. No. 37, 1181–1215, 2015.
Prepublication versions in arXiv.
[PDF] (449.89 KB)
Voevodsky, V., "An experimental library of formalized Mathematics based on the univalent foundations", Mathematical Structures in Computer Science, vol. FirstView, pp. 1–17, 2, 2015.
Pre-publication version is here.
Pelayo, Á., V. Voevodsky, and M. A. Warren, "A univalent formalization of the p-adic numbers", Mathematical Structures in Computer Science, vol. FirstView, pp. 1–25, 2, 2015.
The arXiv version is here.
Voevodsky, V., "Martin-Lof identity types in the C-systems defined by a universe category", arXiv 1505.06446, submitted, pp. 1–51, 2015.
Voevodsky, V., "Products of families of types in the C-systems defined by a universe category", arXiv 1503.07072, submitted, pp. 1–30, 2015.
2016
Voevodsky, V., "Products of families of types and $(\Pi,\lambda)$-structures on C-systems", Theory Appl. Categ., vol. 31, pp. No. 36, 1044-1094, 11/2016. [PDF] (565.01 KB)
Voevodsky, V., "Lawvere theories and Jf-relative monads", arXiv 1601.02158, pp. 1--21, 2016.
2017
Voevodsky, V., "The $(\Pi,\lambda)$-structures on the C-systems defined by universe categories", Theory and Applications of Categories, vol. 32, issue 4, pp. 113-121, 01/2017. [PDF] (296.95 KB)
Voevodsky, V., "The C-systems defined by universe categories: presheaves", Theory and Applications of Categories, vol. 32, issue 3, pp. 53-112, 01/2017. [PDF] (618.12 KB)