This paper was written to present the constructions that were originally introduced for the proof of functoriality of $(\Pi,\lambda)$ structures obtained from the $(P,\tilde{P})$ structures. In the process of writing up this proof it became clear that the main objects required for it -- the "almost representations" $\mu_n$ and $\tilde{\mu}_n$ of the presheaves\ $Ob_n$ and $\tilde{Ob}_n$ on the C-systems defined by locally cartesian closed universe categories with binary product structures and the lemmas describing the behavior of these "almost representations" with respect to the universe category functors, play important role in the general theory that is quite independent from the analysis of the $\Pi,\lambda,\beta,\eta$ system of inference rules. This led to the decision to present these constructions together with the important intermediate ones such as $Sig$ and $D_p$ in a separate paper. Keywords: Type theory, Contextual category, Universe category

}, url = {http://www.tac.mta.ca/tac/volumes/32/3/32-03abs.html}, author = {Voevodsky, Vladimir} } @article {114, title = {The $(\Pi,\lambda)$-structures on the C-systems defined by universe categories}, journal = {Theory and Applications of Categories}, volume = {32}, year = {2017}, month = {01/2017}, pages = {113-121}, abstract = {We define the notion of a $(P,\tilde{P})$-structure on a universe $p$ in a locally cartesian closed category category with a binary product structure and construct a $(\Pi,\lambda)$-structure on the C-systems $CC(C,p)$ from a $(P,\tilde{P})$-structure on $p$. We then define homomorphisms of C-systems with $(\Pi,\lambda)$-structures and functors of universe categories with $(P,\tilde{P})$-structures and show that our construction is functorial relative to these definitions.The proofs in this paper rely on the general constructions of the previous paper "C-systems defined by universe categories: presheaves". Keywords: Type theory, Contextual category, Universe category, dependent product, product of families of types

}, url = {http://www.tac.mta.ca/tac/volumes/32/4/32-04abs.html}, author = {Voevodsky, Vladimir} } @article {110, title = {Lawvere theories and Jf-relative monads}, journal = {arXiv 1601.02158}, year = {2016}, pages = {1--21}, url = {http://arxiv.org/abs/1601.02158}, author = {Voevodsky, Vladimir} } @article {112, title = {Products of families of types and $(\Pi,\lambda)$-structures on C-systems}, journal = {Theory Appl. Categ.}, volume = {31}, year = {2016}, month = {11/2016}, pages = {No. 36, 1044-1094}, url = {http://www.tac.mta.ca/tac/volumes/31/36/31-36.pdf}, author = {Voevodsky, Vladimir} } @article {109, title = {A C-system defined by a universe category}, journal = {Theory Appl. Categ.}, volume = {30}, year = {2015}, note = {Prepublication versions in arXiv.

}, pages = {No. 37, 1181{\textendash}1215}, issn = {1201-561X}, url = {http://www.tac.mta.ca/tac/volumes/30/37/30-37abs.html}, author = {Voevodsky, Vladimir} } @article {38, title = {An experimental library of formalized Mathematics based on the univalent foundations}, journal = {Mathematical Structures in Computer Science}, volume = {FirstView}, year = {2015}, note = {Pre-publication version is here.

}, month = {2}, pages = {1{\textendash}17}, issn = {1469-8072}, doi = {10.1017/S0960129514000577}, url = {http://journals.cambridge.org/article_S0960129514000577}, author = {Voevodsky, Vladimir} } @article {115, title = {Lawvere theories and C-systems}, journal = {arXiv:1512.08104}, year = {2015}, month = {12/2015}, pages = {15pp}, abstract = {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.}, url = {https://arxiv.org/abs/1512.08104}, author = {Voevodsky, Vladimir} } @article {103, title = {Martin-Lof identity types in the C-systems defined by a universe category}, journal = {arXiv 1505.06446, submitted}, year = {2015}, pages = {1{\textendash}51}, url = {http://arxiv.org/abs/1505.06446}, author = {Voevodsky, Vladimir} } @article {104, title = {Products of families of types in the C-systems defined by a universe category}, journal = {arXiv 1503.07072, submitted}, year = {2015}, pages = {1{\textendash}30}, url = {http://arxiv.org/abs/1503.07072}, author = {Voevodsky, Vladimir} } @article {37, title = {A univalent formalization of the p-adic numbers}, journal = {Mathematical Structures in Computer Science}, volume = {FirstView}, year = {2015}, note = {The arXiv version is here.

}, month = {2}, pages = {1{\textendash}25}, issn = {1469-8072}, doi = {10.1017/S0960129514000541}, url = {http://journals.cambridge.org/article_S0960129514000541}, author = {Pelayo, {\'A}lvaro and Voevodsky, Vladimir and Warren, Michael A.} } @article {40, title = {B-systems}, journal = {arXiv 1410.5389, submitted}, year = {2014}, pages = {1{\textendash}17}, url = {http://arxiv.org/abs/1410.5389}, author = {Voevodsky, Vladimir} } @article {42, title = {C-system of a module over a monad on sets}, journal = {arXiv 1407.3394, submitted}, year = {2014}, pages = {1{\textendash}20}, url = {http://arxiv.org/abs/1407.3394}, author = {Voevodsky, Vladimir} } @article {39, title = {Subsystems and regular quotients of C-systems}, journal = {arXiv 1406.5389, submitted}, year = {2014}, pages = {1{\textendash}11}, url = {http://arxiv.org/abs/1406.7413}, author = {Voevodsky, Vladimir} } @article {23, title = {On motivic cohomology with Z/l-coefficients}, journal = {Ann. of Math. (2)}, volume = {174}, year = {2011}, pages = {401{\textendash}438}, issn = {0003-486X}, doi = {10.4007/annals.2011.174.1.11}, url = {http://dx.doi.org/10.4007/annals.2011.174.1.11}, author = {Voevodsky, Vladimir} } @article {26, title = {Cancellation theorem}, journal = {Doc. Math.}, year = {2010}, pages = {671{\textendash}685}, issn = {1431-0635}, url = {http://www.math.uiuc.edu/documenta/vol-suslin/voevodsky.html}, author = {Voevodsky, Vladimir} } @article {107, title = {The equivalence axiom and univalent models of type theory}, journal = {arXiv 1402.5556}, year = {2010}, pages = {1{\textendash}11}, url = {http://arxiv.org/abs/1402.5556}, author = {Vladimir Voevodsky} } @article {49, title = {Homotopy theory of simplicial sheaves in completely decomposable topologies}, journal = {J. Pure Appl. Algebra}, volume = {214}, year = {2010}, pages = {1384{\textendash}1398}, issn = {0022-4049}, doi = {10.1016/j.jpaa.2009.11.004}, url = {http://dx.doi.org/10.1016/j.jpaa.2009.11.004}, author = {Voevodsky, Vladimir} } @article {47, title = {Motives over simplicial schemes}, journal = {J. K-Theory}, volume = {5}, year = {2010}, pages = {1{\textendash}38}, issn = {1865-2433}, doi = {10.1017/is010001030jkt107}, url = {http://dx.doi.org/10.1017/is010001030jkt107}, author = {Voevodsky, Vladimir} } @article {45, title = {Motivic Eilenberg-Maclane spaces}, journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.}, year = {2010}, pages = {1{\textendash}99}, issn = {0073-8301}, doi = {10.1007/s10240-010-0024-9}, url = {http://dx.doi.org/10.1007/s10240-010-0024-9}, author = {Voevodsky, Vladimir} } @article {46, title = {Simplicial radditive functors}, journal = {J. K-Theory}, volume = {5}, year = {2010}, pages = {201{\textendash}244}, issn = {1865-2433}, doi = {10.1017/is010003026jkt097}, url = {http://dx.doi.org/10.1017/is010003026jkt097}, author = {Voevodsky, Vladimir} } @article {97, title = {Univalent Foundations Project}, journal = {a modified version of an NSF grant application}, year = {2010}, month = {October}, pages = {1{\textendash}12}, url = {http://www.math.ias.edu/vladimir/files/univalent_foundations_project.pdf}, author = {Voevodsky, Vladimir} } @article {48, title = {Unstable motivic homotopy categories in Nisnevich and cdh-topologies}, journal = {J. Pure Appl. Algebra}, volume = {214}, year = {2010}, pages = {1399{\textendash}1406}, issn = {0022-4049}, doi = {10.1016/j.jpaa.2009.11.005}, url = {http://dx.doi.org/10.1016/j.jpaa.2009.11.005}, author = {Voevodsky, Vladimir} } @inbook {92, title = {Voevodsky{\textquoteright}s lectures on motivic cohomology 2000/2001}, booktitle = {Algebraic topology}, series = {Abel Symp.}, volume = {4}, year = {2009}, pages = {355{\textendash}409}, publisher = {Springer, Berlin}, organization = {Springer, Berlin}, doi = {10.1007/978-3-642-01200-6_12}, url = {http://dx.doi.org/10.1007/978-3-642-01200-6_12}, author = {Deligne, Pierre} } @article {51, title = {An exact sequence for K^M/2 with applications to quadratic forms}, journal = {Ann. of Math. (2)}, volume = {165}, year = {2007}, pages = {1{\textendash}13}, issn = {0003-486X}, doi = {10.4007/annals.2007.165.1}, url = {http://dx.doi.org/10.4007/annals.2007.165.1}, author = {Orlov, D. and Vishik, A. and Voevodsky, V.} } @inbook {50, title = {Voevodsky{\textquoteright}s Nordfjordeid lectures: motivic homotopy theory}, booktitle = {Motivic homotopy theory}, series = {Universitext}, year = {2007}, pages = {147{\textendash}221}, publisher = {Springer, Berlin}, organization = {Springer, Berlin}, doi = {10.1007/978-3-540-45897-5_7}, url = {http://dx.doi.org/10.1007/978-3-540-45897-5_7}, author = {Voevodsky, Vladimir and R{\"o}ndigs, Oliver and {\O}stv{\ae}r, Paul Arne} } @book {52, title = {Lecture notes on motivic cohomology}, series = {Clay Mathematics Monographs}, volume = {2}, year = {2006}, pages = {xiv+216}, publisher = {American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA}, organization = {American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA}, isbn = {978-0-8218-3847-1; 0-8218-3847-4}, url = {http://www2.maths.ox.ac.uk/cmi/library/monographs/cmim02.pdf}, author = {Mazza, Carlo and Voevodsky, Vladimir and Weibel, Charles} } @article {96, title = {A very short note on homotopy $\lambda$-calculus}, journal = {Unpublished}, year = {2006}, month = {September}, pages = {1{\textendash}7}, url = {http://www.math.ias.edu/vladimir/files/2006_09_Hlambda.pdf}, author = {Voevodsky, Vladimir} } @article {53, title = {On the zero slice of the sphere spectrum}, journal = {Tr. Mat. Inst. Steklova}, volume = {246}, year = {2004}, pages = {106{\textendash}115}, issn = {0371-9685}, url = {http://www.mathnet.ru/php/archive.phtml?wshow=paper\&jrnid=tm\&paperid=148\&option_lang=eng}, author = {Voevodsky, V.} } @article {54, title = {Motivic cohomology with Z/2-coefficients}, journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.}, year = {2003}, note = {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:

- Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories (June\ 1995).\
- The Milnor Conjecture\ (December\ 1996)
- On 2-torsion in motivic cohomology\ (July 2001)
- Motivic cohomology with Z/2-coefficients (published version, 2003).

\

}, pages = {59{\textendash}104}, issn = {0073-8301}, doi = {10.1007/s10240-003-0010-6}, url = {http://dx.doi.org/10.1007/s10240-003-0010-6}, author = {Voevodsky, Vladimir} } @article {55, title = {Reduced power operations in motivic cohomology}, journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.}, year = {2003}, pages = {1{\textendash}57}, issn = {0073-8301}, doi = {10.1007/s10240-003-0009-z}, url = {http://dx.doi.org/10.1007/s10240-003-0009-z}, author = {Voevodsky, Vladimir} } @article {58, title = {Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic}, journal = {Int. Math. Res. Not.}, year = {2002}, pages = {351{\textendash}355}, issn = {1073-7928}, doi = {10.1155/S107379280210403X}, url = {http://dx.doi.org/10.1155/S107379280210403X}, author = {Voevodsky, Vladimir} } @inbook {56, title = {Open problems in the motivic stable homotopy theory. I}, booktitle = {Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998)}, series = {Int. Press Lect. Ser.}, volume = {3}, year = {2002}, pages = {3{\textendash}34}, publisher = {Int. Press, Somerville, MA}, organization = {Int. Press, Somerville, MA}, author = {Voevodsky, Vladimir} } @inbook {57, title = {A possible new approach to the motivic spectral sequence for algebraic K-theory}, booktitle = {Recent progress in homotopy theory Baltimore, MD, 2000)}, series = {Contemp. Math.}, volume = {293}, year = {2002}, pages = {371{\textendash}379}, publisher = {Amer. Math. Soc., Providence, RI}, organization = {Amer. Math. Soc., Providence, RI}, doi = {10.1090/conm/293/04956}, url = {http://dx.doi.org/10.1090/conm/293/04956}, author = {Voevodsky, Vladimir} } @article {98, title = {Framed Correspondences}, journal = {Unpublished}, year = {2001}, url = {http://www.math.ias.edu/vladimir/files/framed.pdf}, author = {Vladimir Voevodsky} } @article {94, title = {Voevodsky{\textquoteright}s lectures on cross functors, Fall 2001}, journal = {Unpublished}, year = {2001}, pages = {1{\textendash}41}, url = {http://www.math.ias.edu/vladimir/files/2015_transfer_from_ps_delnotes01.pdf}, author = {Deligne, Pierre} } @inbook {60, title = {Bivariant cycle cohomology}, booktitle = {Cycles, transfers, and motivic homology theories}, series = {Ann. of Math. Stud.}, volume = {143}, year = {2000}, pages = {138{\textendash}187}, publisher = {Princeton Univ. Press, Princeton, NJ}, organization = {Princeton Univ. Press, Princeton, NJ}, url = {http://www.jstor.org/stable/j.ctt7tcnh}, author = {Friedlander, Eric M. and Voevodsky, Vladimir} } @inbook {64, title = {Bloch-Kato conjecture and motivic cohomology with finite coefficients}, booktitle = {The arithmetic and geometry of algebraic cycles (Banff, AB, 1998)}, series = {NATO Sci. Ser. C Math. Phys. Sci.}, volume = {548}, year = {2000}, pages = {117{\textendash}189}, publisher = {Kluwer Acad. Publ., Dordrecht}, organization = {Kluwer Acad. Publ., Dordrecht}, url = {http://link.springer.com/chapter/10.1007\%2F978-94-011-4098-0_5}, author = {Suslin, Andrei and Voevodsky, Vladimir} } @inbook {61, title = {Cohomological theory of presheaves with transfers}, booktitle = {Cycles, transfers, and motivic homology theories}, series = {Ann. of Math. Stud.}, volume = {143}, year = {2000}, pages = {87{\textendash}137}, publisher = {Princeton Univ. Press, Princeton, NJ}, organization = {Princeton Univ. Press, Princeton, NJ}, url = {http://www.jstor.org/stable/j.ctt7tcnh}, author = {Voevodsky, Vladimir} } @inbook {63, title = {Introduction}, booktitle = {Cycles, transfers, and motivic homology theories}, series = {Ann. of Math. Stud.}, volume = {143}, year = {2000}, pages = {3{\textendash}9}, publisher = {Princeton Univ. Press, Princeton, NJ}, organization = {Princeton Univ. Press, Princeton, NJ}, url = {http://www.jstor.org/stable/j.ctt7tcnh}, author = {Friedlander, Eric M. and Suslin, A. and Voevodsky, V.} } @inbook {62, title = {Relative cycles and Chow sheaves}, booktitle = {Cycles, transfers, and motivic homology theories}, series = {Ann. of Math. Stud.}, volume = {143}, year = {2000}, pages = {10{\textendash}86}, publisher = {Princeton Univ. Press, Princeton, NJ}, organization = {Princeton Univ. Press, Princeton, NJ}, url = {http://www.jstor.org/stable/j.ctt7tcnh}, author = {Suslin, Andrei and Voevodsky, Vladimir} } @inbook {59, title = {Triangulated categories of motives over a field}, booktitle = {Cycles, transfers, and motivic homology theories}, series = {Ann. of Math. Stud.}, volume = {143}, year = {2000}, pages = {188{\textendash}238}, publisher = {Princeton Univ. Press, Princeton, NJ}, organization = {Princeton Univ. Press, Princeton, NJ}, url = {http://www.jstor.org/stable/j.ctt7tcnh}, author = {Voevodsky, Vladimir} } @article {65, title = {A^1-homotopy theory of schemes}, journal = {Inst. Hautes {\'E}tudes Sci. Publ. Math.}, year = {1999}, pages = {45{\textendash}143 (2001)}, issn = {0073-8301}, url = {http://www.numdam.org/item?id=PMIHES_1999__90__45_0}, author = {Morel, Fabien and Voevodsky, Vladimir} } @article {93, title = {Categories and functors in mathematics}, journal = {Unpublished}, year = {1999}, pages = {1{\textendash}18}, url = {http://www.math.ias.edu/vladimir/files/wolf.pdf}, author = {Voevodsky, Vladimir} } @article {95, title = {Four functors formalism}, journal = {Unpublished}, year = {1999}, note = {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

Thesis (Ph.D.){\textendash}Harvard University

}, pages = {64}, publisher = {ProQuest LLC, Ann Arbor, MI}, organization = {ProQuest LLC, Ann Arbor, MI}, url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004\&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation\&res_dat=xri:pqdiss\&rft_dat=xri:pqdiss:9228294}, author = {Voevodsky, Vladimir} } @article {77, title = {Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results)}, journal = {Cahiers Topologie G{\'e}om. Diff{\'e}rentielle Cat{\'e}g.}, volume = {32}, year = {1991}, note = {International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990)

}, pages = {11{\textendash}27}, issn = {0008-0004}, url = {https://eudml.org/doc/91468}, author = {Kapranov, M. M. and Voevodsky, V. A.} } @article {78, title = {The free n-category generated by a cube, oriented matroids and higher Bruhat orders}, journal = {Funktsional. Anal. i Prilozhen.}, volume = {25}, year = {1991}, pages = {62{\textendash}65}, issn = {0374-1990}, doi = {10.1007/BF01090678}, url = {http://dx.doi.org/10.1007/BF01090678}, author = {Voevodsky, V. A. and Kapranov, M. M.} } @article {74, title = {Galois groups of function fields over fields of finite type over Q}, journal = {Uspekhi Mat. Nauk}, volume = {46}, year = {1991}, pages = {163{\textendash}164}, issn = {0042-1316}, doi = {10.1070/RM1991v046n05ABEH002841}, url = {http://dx.doi.org/10.1070/RM1991v046n05ABEH002841}, author = {Voevodsky, V. A.} } @article {75, title = {Galois representations connected with hyperbolic curves}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {55}, year = {1991}, pages = {1331{\textendash}1342}, issn = {0373-2436}, url = {http://iopscience.iop.org/0025-5726/39/3/A10/pdf/IZV_39_3_A10.pdf}, author = {Voevodsky, V. A.} } @article {76, title = {infty-groupoids and homotopy types}, journal = {Cahiers Topologie G{\'e}om. Diff{\'e}rentielle Cat{\'e}g.}, volume = {32}, year = {1991}, note = {International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990)

Warning: the main theorem of this paper was shown by Carlos Simpson to be false.\

\

}, pages = {29{\textendash}46}, issn = {0008-0004}, url = {https://eudml.org/doc/91469}, author = {Kapranov, M. M. and Voevodsky, V. A.} } @inbook {79, title = {Drawing curves over number fields}, booktitle = {The Grothendieck Festschrift, Vol. III}, series = {Progr. Math.}, volume = {88}, year = {1990}, pages = {199{\textendash}227}, publisher = {Birkh{\"a}user Boston, Boston, MA}, organization = {Birkh{\"a}user Boston, Boston, MA}, doi = {10.1007/978-0-8176-4576-2_8}, url = {http://dx.doi.org/10.1007/978-0-8176-4576-2_8}, author = {Shabat, G. B. and Voevodsky, V. A.} } @article {80, title = {Etale topologies of schemes over fields of finite type over Q}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {54}, year = {1990}, pages = {1155{\textendash}1167}, issn = {0373-2436}, url = {http://iopscience.iop.org/0025-5726/37/3/A03/pdf/IZV_37_3_A03.pdf}, author = {Voevodsky, V. A.} } @article {85, title = {Flags and Grothendieck Cartographical Group in Higher Dimensions}, journal = {CSTARCI Math. Preprints}, year = {1990}, author = {Voevodsky, V. A.} } @article {81, title = {infty-groupoids as a model for a homotopy category}, journal = {Uspekhi Mat. Nauk}, volume = {45}, year = {1990}, note = {Warning: the main theorem of this paper was shown by Carlos Simpson to be false.

}, pages = {183{\textendash}184}, issn = {0042-1316}, doi = {10.1070/RM1990v045n05ABEH002673}, url = {http://dx.doi.org/10.1070/RM1990v045n05ABEH002673}, author = {Voevodsky, V. A. and Kapranov, M. M.} } @conference {87, title = {Multidimensional Categories (in Russian)}, booktitle = {Proc. Conf. of Young Scientists}, year = {1990}, publisher = {Moscow Univ. Press}, organization = {Moscow Univ. Press}, author = {Voevodsky, V. A. and Kapranov, M. M.} } @article {86, title = {Piece-Wise Euclidean Approximation of Jacobians of Algebraic Curves}, journal = {CSTARCI Math. Preprints}, year = {1990}, author = {Voevodsky, V. A. and Shabat, G. B.} } @conference {88, title = {Triangulations of Oriented Manifolds and Ramified Coverings of Sphere (in Russian)}, booktitle = {Proc. Conf. of Young Scientists}, year = {1990}, publisher = {Moscow Univ. Press}, organization = {Moscow Univ. Press}, author = {Voevodsky, V. A.} } @article {82, title = {Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields}, journal = {Dokl. Akad. Nauk SSSR}, volume = {304}, year = {1989}, pages = {265{\textendash}268}, issn = {0002-3264}, author = {Voevodsky, V. A. and Shabat, G. B.} } @conference {84, title = {Galois Group over Q and Teihmuller Modular Groups}, booktitle = {Proc. Conf. Constr. Methods and Alg. Number Theory}, year = {1989}, address = {Minsk}, author = {Voevodsky, V. A.} }