# Publications

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Author Title Type Year
2
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)
A
Voevodsky, V., "A^1-homotopy theory", Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998. [PDF] (355.89 KB)
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)
B
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)
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)
Voevodsky, V., "B-systems", arXiv 1410.5389, submitted, pp. 1–17, 2014.
C
Voevodsky, V., "Cancellation theorem", Doc. Math., pp. 671–685, 2010. [PDF] (155.85 KB)
Voevodsky, V., "Categories and functors in mathematics", Unpublished, pp. 1–18, 1999.
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)
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 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., "C-system of a module over a monad on sets", arXiv 1407.3394, submitted, pp. 1–20, 2014.
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)
D
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)
E
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., "The equivalence axiom and univalent models of type theory", arXiv 1402.5556, pp. 1–11, 2010.
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)
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., "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.
F
Voevodsky, V.. A., "Flags and Grothendieck Cartographical Group in Higher Dimensions", CSTARCI Math. Preprints, 1990. [PDF] (583.06 KB)
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)
Voevodsky, V., "Framed Correspondences", Unpublished, 2001. [PDF] (116.24 KB)
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)
G
Voevodsky, V.. A., "Galois Group over Q and Teihmuller Modular Groups", Proc. Conf. Constr. Methods and Alg. Number Theory, Minsk, 1989.
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)
H
Voevodsky, V.., "Homology of schemes", Selecta Math. (N.S.), vol. 2, pp. 111–153, 1996. [PDF] (2.44 MB)
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)
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)
I
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)
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)
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)
L
Voevodsky, V., "Lawvere theories and C-systems", arXiv:1512.08104, pp. 15pp, 12/2015. [PDF] (250.57 KB)
Voevodsky, V., "Lawvere theories and Jf-relative monads", arXiv 1601.02158, pp. 1--21, 2016.
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)
M
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., "Motives over simplicial schemes", J. K-Theory, vol. 5, pp. 1–38, 2010. [PDF] (325.11 KB)
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., "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., "On motivic cohomology with Z/l-coefficients", Ann. of Math. (2), vol. 174, pp. 401–438, 2011. [PDF] (505.13 KB)
Voevodsky, V., "Motivic Eilenberg-Maclane spaces", Publ. Math. Inst. Hautes Études Sci., pp. 1–99, 2010. [PDF] (915.29 KB)
Voevodsky, V.. A., and M.. M. Kapranov, "Multidimensional Categories (in Russian)", Proc. Conf. of Young Scientists: Moscow Univ. Press, 1990.
N
Voevodsky, V.., "A nilpotence theorem for cycles algebraically equivalent to zero", Internat. Math. Res. Notices, pp. 187–198 (electronic), 1995. [PDF] (98.87 KB)
O
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)
P
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., "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., "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)
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., "Products of families of types in the C-systems defined by a universe category", arXiv 1503.07072, submitted, pp. 1–30, 2015.
R
Voevodsky, V., "Reduced power operations in motivic cohomology", Publ. Math. Inst. Hautes Études Sci., pp. 1–57, 2003. [PDF] (542.94 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)
S
Voevodsky, V., "Simplicial radditive functors", J. K-Theory, vol. 5, pp. 201–244, 2010. [PDF] (389.16 KB)
Suslin, A., and V. Voevodsky, "Singular homology of abstract algebraic varieties", Invent. Math., vol. 123, pp. 61–94, 1996. [PDF] (1.56 MB)
Voevodsky, V., "Subsystems and regular quotients of C-systems", arXiv 1406.5389, submitted, pp. 1–11, 2014.
T
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)
Voevodsky, V.. A., "Triangulations of Oriented Manifolds and Ramified Coverings of Sphere (in Russian)", Proc. Conf. of Young Scientists: Moscow Univ. Press, 1990.
U
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., "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)
V
Voevodsky, V., "A very short note on homotopy $\lambda$-calculus", Unpublished, pp. 1–7, September, 2006. [PDF] (113.85 KB)
Deligne, P., "Voevodsky's lectures on cross functors, Fall 2001", Unpublished, pp. 1–41, 2001. [PDF] (369.71 KB)
Deligne, P., "Voevodsky's lectures on motivic cohomology 2000/2001", Algebraic topology, vol. 4: Springer, Berlin, pp. 355–409, 2009. [PDF] (735.89 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)
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)
Z
Voevodsky, V.., "On the zero slice of the sphere spectrum", Tr. Mat. Inst. Steklova, vol. 246, pp. 106–115, 2004. [PDF] (207.32 KB)