DOCSLIB.ORG

Lecture Notes on Motivic Cohomology Motivic on Notes Lecture

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Clay Mathematics Monographs 2 Volume 2 Lecture Lecture Notes on Motivic Cohomology The notion of a motive is an elusive one, like its namesake “the motif” of Cezanne’s impressionist method of painting. Its existence was fi rst suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffi ces for the development Lecture Notes on of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. This book provides an account of the triangulated theory of motives. Its purpose Motivic Cohomology is to introduce Motivic Cohomology, to develop its main properties, and fi nally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, étale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The fi rst part presents the defi nition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (étale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part fi ve. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related fi eld. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999–2000. In addition, many of Carlo Mazza the original proofs have been simplifi ed and improved so that this book will also Vladimir Voevodsky be a useful tool for research mathematicians. Charles Weibel Mazza, Voevodsky Mazza, and Voevodsky Weibel For additional information and updates on this book, visit www.ams.org/bookpages/cmim-2 www.ams.org American Mathematical Society CMIM/2 AMS www.claymath.org CMI Clay Mathematics Institute 4 color process 232 pages pages on 50 lb stock • 1 3/16 spine Lecture Notes on Motivic Cohomology Clay Mathematics Monographs Volume 2 Lecture Notes on Motivic Cohomology Carlo Mazza Vladimir Voevodsky Charles Weibel American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 14F42; Secondary 19E15, 14C25, 14–01, 14F20. For additional information and updates on this book, visit www.ams.org/bookpages/cmim-2 Library of Congress Cataloging-in-Publication Data Mazza, Carlo, 1974– Lecture notes on motivic cohomology / Carlo Mazza, Vladimir Voevodsky, Charles A. Weibel. p. cm. — (Clay mathematics monographs, ISSN 1539-6061 ; v. 2) Includes bibliographical references and index. ISBN-10: 0-8218-3847-4 (alk. paper) ISBN-13: 978-0-8218-3847-1 (alk. paper) 1. Homology theory. I. Voevodsky, Vladimir. II. Weibel, Charles A., 1950– III. Title. IV. Series. QA612.3.M39 2006 514.23—dc22 2006045973 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the Clay Mathematics Institute. Requests for such permission should be addressed to the Clay Mathematics Institute, One Bow Street, Cambridge, MA 02138, USA. Requests can also be made by e-mail to [email protected]. c 2006 by the authors. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 111009080706 Contents Preface vii Introduction xi Part 1. Presheaves with Transfers 1 Lecture 1. The category of finite correspondences 3 Appendix 1A. The category CorS 7 Lecture 2. Presheaves with transfers 13 Homotopy invariant presheaves 17 Lecture 3. Motivic cohomology 21 Lecture 4. Weight one motivic cohomology 25 Lecture 5. Relation to Milnor K-Theory 29 Part 2. Etale´ Motivic Theory 35 Lecture 6. Etale´ sheaves with transfers 37 Lecture 7. The relative Picard group and Suslin’s Rigidity Theorem 47 Lecture 8. Derived tensor products 55 Appendix 8A. Tensor triangulated categories 63 Lecture 9. A1-weak equivalence 67 Etale´ A1-local complexes 71 Lecture 10. Etale´ motivic cohomology and algebraic singular homology 75 Part 3. Nisnevich Sheaves with Transfers 81 Lecture 11. Standard triples 83 Lecture 12. Nisnevich sheaves 89 The cdh topology 94 v vi CONTENTS Lecture 13. Nisnevich sheaves with transfers 99 Cdh sheaves with transfers 105 Part 4. The Triangulated Category of Motives 107 Lecture 14. The category of motives 109 Nisnevich A1-local complexes 111 Motives with Q-coefficients 116 Lecture 15. The complex Z(n) and Pn 119 Lecture 16. Equidimensional cycles 125 Motives with compact support 128 Part 5. Higher Chow Groups 133 Lecture 17. Higher Chow groups 135 Appendix 17A. Cycle maps 143 Lecture 18. Higher Chow groups and equidimensional cycles 149 Appendix 18A. Generic equidimensionality 155 Lecture 19. Motivic cohomology and higher Chow groups 159 Lecture 20. Geometric motives 167 Part 6. Zariski Sheaves with Transfers 173 Lecture 21. Covering morphisms of triples 175 Lecture 22. Zariski sheaves with transfers 183 Lecture 23. Contractions 191 Lecture 24. Homotopy invariance of cohomology 197 Bibliography 203 Glossary 207 Index 211 Preface This book was written by Carlo Mazza and Charles Weibel on the basis of the lectures on motivic cohomology which I gave at the Institute for Advanced Study in Princeton in 1999/2000. From the point of view taken in these lectures, motivic cohomology with coef- ficients in an abelian group A is a family of contravariant functors H p,q(−,A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by integers p and q. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum. Most of the known and expected properties of motivic cohomology (predicted in [ABS87]and[Lic84]) can be divided into two families. The first family concerns properties of motivic cohomology itself – there are theorems about homotopy in- variance, Mayer-Vietoris and Gysin long exact sequences, projective bundles, etc. This family also contains conjectures such as the Beilinson-Soule´ vanishing con- jecture (H p,q = 0forp < 0) and the Beilinson-Lichtenbaum conjecture, which can be interpreted as a partial etale´ descent property for motivic cohomology. The sec- ond family of properties relates motivic cohomology to other known invariants of algebraic varieties and rings. The power of motivic cohomology as a tool for prov- ing results in algebra and algebraic geometry lies in the interaction of the results in these two families; applying general theorems of motivic cohomology to the specific cases of classical invariants, one gets new results about these invariants. The idea of these lectures was to define motivic cohomology and to give care- ful proofs for the elementary results in the second family. In this sense they are complementary to the study of [VSF00], where the emphasis is on the properties of motivic cohomology itself. The structure of the proofs forces us to deal with the main properties of motivic cohomology as well (such as homotopy invariance). As a result, these lectures cover a considerable portion of the material of [VSF00], but from a different point of view. One can distinguish the following “elementary” comparison results for motivic cohomology. Unless otherwise specified, all schemes below are assumed to be smooth or (in the case of local or semilocal schemes) limits of smooth schemes. vii viii PREFACE (1) H p,q(X,A)=0forq < 0, and for a connected X one has A for p = 0 H p,0(X,A)= 0forp = 0; (2) one has O∗(X) for p = 1 p,1 Z H (X, )= Pic(X) for p = 2 0forp = 1,2; p,p M ⊗ M (3) for a field k, one has H (Spec(k),A)=Kp (k) A where Kp (k) is the p-th Milnor K-group of k (see [Mil70]); (4) for a strictly Hensel local scheme S over k and an integer n prime to char(k), one has ⊗ µ q(S) for p = 0 H p,q(S,Z/n)= n 0forp = 0 where µn(S) is the groups of n-th roots of unity in S; (5) one has H p,q(X,A)=CHq(X,2q − p;A).HereCHi(X, j;A) denotes the higher Chow groups of X introduced by S. Bloch in [Blo86], [Blo94]. In particular, H2q,q(X,A)=CHq(X) ⊗ A, where CHq(X) is the classical Chow group of cycles of codimension q modulo rational equivalence. The isomorphism between motivic cohomology and higher Chow groups leads to connections between motivic cohomology and algebraic K-theory, but we do not discuss these connections in the present lectures. See [Blo94], [BL94], [FS02], [Lev98]and[SV00]. Deeper comparison results include the theorem of M. Levine comparing CHi(X, j;Q) with the graded pieces of the gamma filtration in K∗(X)⊗Q [Lev94], and the construction of the spectral sequence relating motivic cohomology and al- gebraic K-theory for arbitrary coefficients in [BL94]and[FS02].
Recommended publications
  • What Is the Motivation Behind the Theory of Motives?

    What Is the Motivation Behind the Theory of Motives?

    What is the motivation behind the Theory of Motives? Barry Mazur How much of the algebraic topology of a connected simplicial complex X is captured by its one-dimensional cohomology? Specifically, how much do you know about X when you know H1(X, Z) alone? For a (nearly tautological) answer put GX := the compact, connected abelian Lie group (i.e., product of circles) which is the Pontrjagin dual of the free abelian group H1(X, Z). Now H1(GX, Z) is canonically isomorphic to H1(X, Z) = Hom(GX, R/Z) and there is a canonical homotopy class of mappings X −→ GX which induces the identity mapping on H1. The answer: we know whatever information can be read off from GX; and are ignorant of anything that gets lost in the projection X → GX. The theory of Eilenberg-Maclane spaces offers us a somewhat analogous analysis of what we know and don’t know about X, when we equip ourselves with n-dimensional cohomology, for any specific n, with specific coefficients. If we repeat our rhetorical question in the context of algebraic geometry, where the structure is somewhat richer, can we hope for a similar discussion? In algebraic topology, the standard cohomology functor is uniquely characterized by the basic Eilenberg-Steenrod axioms in terms of a simple normalization (the value of the functor on a single point). In contrast, in algebraic geometry we have a more intricate set- up to deal with: for one thing, we don’t even have a cohomology theory with coefficients in Z for varieties over a field k unless we provide a homomorphism k → C, so that we can form the topological space of complex points on our variety, and compute the cohomology groups of that topological space.
  • Cohomological Descent on the Overconvergent Site

    Cohomological Descent on the Overconvergent Site

    COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable.
  • Homological Algebra

    Homological Algebra

    Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of
  • FIELDS MEDAL for Mathematical Efforts R

    FIELDS MEDAL for Mathematical Efforts R

    Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study,
  • The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P

    The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P

    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all &#x3E;&#x3E; homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different.
  • Lecture 15. De Rham Cohomology

    Lecture 15. De Rham Cohomology

    Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial.
  • Affine Grassmannians in A1-Algebraic Topology

    Affine Grassmannians in A1-Algebraic Topology

    AFFINE GRASSMANNIANS IN A1-ALGEBRAIC TOPOLOGY TOM BACHMANN Abstract. Let k be a field. Denote by Spc(k)∗ the unstable, pointed motivic homotopy category and A1 1 by R ΩGm : Spc(k)∗ →Spc(k)∗ the (A -derived) Gm-loops functor. For a k-group G, denote by GrG the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence A1 A1 R ΩGm G ≃ GrG. We use this to compute the motive M(R ΩGm G) ∈ DM(k, Z[1/e]). 1. Introduction This note deals with the subject of A1-algebraic topology. In other words it deals with with the ∞- category Spc(k) of motivic spaces over a base field k, together with the canonical functor Smk → Spc(k) and, importantly, convenient models for Spc(k). Since our results depend crucially on the seminal papers [1, 2], we shall use their definition of Spc(k) (which is of course equivalent to the other definitions in the literature): start with the category Smk of smooth (separated, finite type) k-schemes, form the universal homotopy theory on Smk (i.e. pass to the ∞-category P(Smk) of space-valued presheaves on k), and A1 then impose the relations of Nisnevich descent and contractibility of the affine line k (i.e. localise P(Smk) at an appropriate family of maps). The ∞-category Spc(k) is presentable, so in particular has finite products, and the functor Smk → Spc(k) preserves finite products. Let ∗ ∈ Spc(k) denote the final object (corresponding to the final k-scheme k); then we can form the pointed unstable motivic homotopy category Spc(k)∗ := Spc(k)/∗.
  • Derived Functors of /-Adic Completion and Local Homology

    Derived Functors of /-Adic Completion and Local Homology

    JOURNAL OF ALGEBRA 149, 438453 (1992) Derived Functors of /-adic Completion and Local Homology J. P. C. GREENLEES AND J. P. MAY Department q/ Mathematics, Unil;ersi/y of Chicago, Chicago, Illinois 60637 Communicated by Richard G. Swan Received June 1. 1990 In recent topological work [2], we were forced to consider the left derived functors of the I-adic completion functor, where I is a finitely generated ideal in a commutative ring A. While our concern in [2] was with a particular class of rings, namely the Burnside rings A(G) of compact Lie groups G, much of the foundational work we needed was not restricted to this special case. The essential point is that the modules we consider in [2] need not be finitely generated and, unless G is finite, say, the ring ,4(G) is not Noetherian. There seems to be remarkably little information in the literature about the behavior of I-adic completion in this generality. We presume that interesting non-Noetherian commutative rings and interesting non-finitely generated modules arise in subjects other than topology. We have therefore chosen to present our algebraic work separately, in the hope that it may be of value to mathematicians working in other fields. One consequence of our study, explained in Section 1, is that I-adic com- pletion is exact on a much larger class of modules than might be expected from the key role played by the Artin-Rees lemma and that the deviations from exactness can be computed in terms of torsion products. However, the most interesting consequence, discussed in Section 2, is that the left derived functors of I-adic completion usually can be computed in terms of certain local homology groups, which are defined in a fashion dual to the definition of the classical local cohomology groups of Grothendieck.
  • Algebraic Topology

    Algebraic Topology

    Algebraic Topology John W. Morgan P. J. Lamberson August 21, 2003 Contents 1 Homology 5 1.1 The Simplest Homological Invariants . 5 1.1.1 Zeroth Singular Homology . 5 1.1.2 Zeroth deRham Cohomology . 6 1.1.3 Zeroth Cecˇ h Cohomology . 7 1.1.4 Zeroth Group Cohomology . 9 1.2 First Elements of Homological Algebra . 9 1.2.1 The Homology of a Chain Complex . 10 1.2.2 Variants . 11 1.2.3 The Cohomology of a Chain Complex . 11 1.2.4 The Universal Coefficient Theorem . 11 1.3 Basics of Singular Homology . 13 1.3.1 The Standard n-simplex . 13 1.3.2 First Computations . 16 1.3.3 The Homology of a Point . 17 1.3.4 The Homology of a Contractible Space . 17 1.3.5 Nice Representative One-cycles . 18 1.3.6 The First Homology of S1 . 20 1.4 An Application: The Brouwer Fixed Point Theorem . 23 2 The Axioms for Singular Homology and Some Consequences 24 2.1 The Homotopy Axiom for Singular Homology . 24 2.2 The Mayer-Vietoris Theorem for Singular Homology . 29 2.3 Relative Homology and the Long Exact Sequence of a Pair . 36 2.4 The Excision Axiom for Singular Homology . 37 2.5 The Dimension Axiom . 38 2.6 Reduced Homology . 39 1 3 Applications of Singular Homology 39 3.1 Invariance of Domain . 39 3.2 The Jordan Curve Theorem and its Generalizations . 40 3.3 Cellular (CW) Homology . 43 4 Other Homologies and Cohomologies 44 4.1 Singular Cohomology .
  • An Intuitive Introduction to Motivic Homotopy Theory Vladimir Voevodsky

    An Intuitive Introduction to Motivic Homotopy Theory Vladimir Voevodsky

    What follows is Vladimir Voevodsky's snapshot of his Fields Medal work on motivic homotopy, plus a little philosophy and \from my point of view the main fun of doing mathematics" Voevodsky (2002). Voevodsky uses extremely simple leading ideas to extend topological methods of homotopy into number theory and abstract algebra. The talk takes the simple ideas right up to the edge of some extremely refined applications. The talk is on-line as streaming video at claymath.org/video/. In a nutshell: classical homotopy studies a topological space M by looking at maps to M from the unit interval on the real line: [0; 1] = fx 2 R j 0 ≤ x ≤ 1 g as well as maps to M from the unit square [0; 1]×[0; 1], the unit cube [0; 1]3, and so on. Algebraic geometry cannot copy these naively, even over the real numbers, since of course the interval [0; 1] on the real line is not defined by a polynomial equation. The only polynomial equal to 0 all over [0; 1] is the constant polynomial 0, thus equals 0 over the whole line. The problem is much worse when you look at algebraic geometry over other fields than the real or complex numbers. Voevodsky shows how category theory allows abstract definitions of the basic apparatus of homotopy, where the category of topological spaces is replaced by any category equipped with objects that share certain purely categorical properties with the unit interval, unit square, and so on. For example, the line in algebraic geometry (over any field) does not have \endpoints" in the usual way but it does come with two naturally distinguished points: 0 and 1.
  • Homotopy Theoretic Models of Type Theory

    Homotopy Theoretic Models of Type Theory

    Homotopy Theoretic Models of Type Theory Peter Arndt1 and Krzysztof Kapulkin2 1 University of Oslo, Oslo, Norway [email protected] 2 University of Pittsburgh, Pittsburgh, PA, USA [email protected] Abstract. We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power so that we can soundly inter- pret dependent products and sums in it while also have purely intensional iterpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of models that are examined in the paper. 1 Introduction Starting with the Hofmann-Streicher groupoid model [HS98] it has become clear that there exist deep connections between Intensional Martin-L¨ofType Theory ([ML72], [NPS90]) and homotopy theory. These connections have recently been studied very intensively. We start by summarizing this work | a more complete survey can be found in [Awo10]. It is well-known that Identity Types (or Equality Types) play an important role in type theory since they provide a relaxed and coarser notion of equality between terms of a given type. For example, assuming standard rules for type Nat one cannot prove that n: Nat ` n + 0 = n: Nat but there is still an inhabitant n: Nat ` p: IdNat(n + 0; n): Identity types can be axiomatized in a usual way (as inductive types) by form, intro, elim, and comp rules (see eg. [NPS90]). A type theory where we do not impose any further rules on the identity types is called intensional.
  • CATERADS and MOTIVIC COHOMOLOGY Contents Introduction 1 1. Weighted Multiplicative Presheaves 3 2. the Standard Motivic Cochain

    CATERADS and MOTIVIC COHOMOLOGY Contents Introduction 1 1. Weighted Multiplicative Presheaves 3 2. the Standard Motivic Cochain

    CATERADS AND MOTIVIC COHOMOLOGY B. GUILLOU AND J.P. MAY Abstract. We define “weighted multiplicative presheaves” and observe that there are several weighted multiplicative presheaves that give rise to motivic cohomology. By neglect of structure, weighted multiplicative presheaves give symmetric monoids of presheaves. We conjecture that a suitable stabilization of one of the symmetric monoids of motivic cochain presheaves has an action of a caterad of presheaves of acyclic cochain complexes, and we give some fragmentary evidence. This is a snapshot of work in progress. Contents Introduction 1 1. Weighted multiplicative presheaves 3 2. The standard motivic cochain complex 5 3. A variant of the standard motivic complex 7 4. The Suslin-Friedlander motivic cochain complex 8 5. Towards caterad actions 10 6. The stable linear inclusions caterad 11 7. Steenrod operations? 13 References 14 Introduction The first four sections set the stage for discussion of a conjecture about the multiplicative structure of motivic cochain complexes. In §1, we define the notion of a “weighted multiplicative presheaf”, abbreviated WMP. This notion codifies formal structures that appear naturally in motivic cohomology and presumably elsewhere in algebraic geometry. We observe that WMP’s determine symmetric monoids of presheaves, as defined in [11, 1.3], by neglect of structure. In §§2–4, we give an outline summary of some of the constructions of motivic co- homology developed in [13, 17] and state the basic comparison theorems that relate them to each other and to Bloch’s higher Chow complexes. There is a labyrinth of definitions and comparisons in this area, and we shall just give a brief summary overview with emphasis on product structures.