Homotopical Algebra Yuri Berest, Sasha Patotski Fall 2015 These are lecture notes of a graduate course MATH 7400 currently taught by Yuri Berest at Cornell University. The notes are taken by Sasha Patotski; the updates will appear every week. If you are reading the notes, please send us corrections; your questions and comments are also very much welcome. Emails: [email protected] and [email protected]. ii Contents 1 Homotopy theory of categories 1 1 Basics of Simplicial sets . .1 1.1 Definitions . .1 1.2 Motivation . .3 1.3 Examples of (co-)simplicial sets . .5 1.4 Remarks on basepoints and reduced simplicial sets . .7 1.5 The nerve of a category . .7 1.6 Examples of nerves . .9 1.7 One example: the nerve of a groupoid . 10 2 Geometric realization . 12 2.1 General remarks on spaces . 12 2.2 Definition of geometric realization . 13 2.3 Two generalizations of geometric realization . 15 2.4 Kan extensions . 17 2.5 Comma categories . 19 2.6 Kan extensions using comma categories . 21 3 Homotopy theory of categories . 23 3.1 The classifying space of a small category . 23 3.2 Homotopy-theoretic properties of the classifying spaces . 25 3.3 Connected components . 27 3.4 Coverings of categories . 28 3.5 Explicit presentations of fundamental groups of categories . 29 3.6 Homology of small categories . 30 3.7 Quillen's Theorem A . 33 3.8 Fibred and cofibred functors . 36 3.9 Quillen's Theorem B . 38 2 Application: algebraic K-theory 41 1 Introduction and overview . 41 2 Classical K-theory . 42 2.1 The group K0(A)............................. 42 2.2 The group K1(A)............................. 42 iii 2.3 The group K2(A)............................. 43 2.4 Universal central extensions . 44 3 Higher K-theory via \plus"-construction . 46 3.1 Acyclic spaces and maps . 46 3.2 Plus construction . 50 3.3 Higher K-groups via plus construction . 50 3.4 Milnor K-theory of fields . 54 3.5 Loday product . 56 3.6 Bloch groups . 57 3.7 Homology of Lie groups \made discrete" . 58 3.8 Relation to polylogarithms, and some conjectures . 62 4 Higher K-theory via Q-construction . 63 4.1 Exact categories . 63 4.2 K-group of an exact category . 64 4.3 Q-construction . 65 4.4 Some remarks on the Q-construction . 66 4.5 The K0-group via Q-construction . 67 4.6 Higher K-theory . 69 4.7 Elementary properties . 69 4.8 Quillen{Gersten theorem . 70 5 The \plus = Q" theorem . 72 5.1 The category S−1S ............................ 72 5.2 K-groups of a symmetric monoidal groupoid . 73 5.3 Some facts about H-spaces . 74 5.4 Actions on categories . 76 5.5 Application to \plus"-construction . 79 5.6 Proving the \plus=Q" theorem . 82 6 Final remarks on Algebraic K-theory . 88 6.1 Historic remarks on algebraic and topological K-theory . 88 6.2 Remarks on delooping . 91 3 Model categories 93 1 Introduction to Model categories . 93 1.1 Axioms of model categories . 93 1.2 Examples of Model categories . 94 1.3 Natural contructions . 96 A Topological and geometric background 99 1 Connections on principal bundles . 99 2 Coverings of spaces . 101 3 Homotopy fibration sequences and homotopy fibers . 104 iv Chapter 1 Homotopy theory of categories 1 Basics of Simplicial sets 1.1 Definitions Definition 1.1.1. The simplicial category ∆ is a category having finite totally ordered sets [n] := f0; 1; : : : ; ng, n 2 N as objects, and order-preserving functions f : [n] ! [m] as morphisms (i.e. functions f such that i 6 j ) f(i) 6 f(j)). Definition 1.1.2. A simplicial set is a contravariant functor from ∆ to the category of sets Sets. Morphism of simplicial sets is just a natural transformation of functors. Notation 1.1.3. We denote the category Fun(∆op; Sets) of simplicial sets by sSets. Definition 1.1.4. Dually, we define cosimplicial sets as covariant functors ∆ ! Sets. We denote the category of cosimplicial sets by csSets. There are two distinguished classes of maps in ∆: i d :[n − 1] ,! [n] n > 1; 0 6 i 6 n j s :[n + 1] [n] n > 0; 0 6 j 6 n called the face maps and degeneracy maps respectively. Informally, di skips value i in its image and sj repeats the value j twice. More precisely, they are defined by ( ( k if k < i k if k j di(k) = sj(k) = 6 k + 1 if k > i k − 1 if k > j Theorem 1.1.5. Every morphism f 2 Hom∆ ([n]; [m]) can be decomposed in a unique way as f = di1 di2 ··· dir sj1 ··· sjs such that m = n − s + r and i1 < ··· < ir and j1 < ··· < js. 1 The proof of this theorem is a little technical, but a few examples make it clear what is going on. For the proof, see for example Lemma 2.2 in [GZ67]. Example 1.1.6. Let f : [3] ! [1] be f0; 1 7! 0; 2; 3 7! 1g. One can easily check that f = s0 ◦ s2. Corollary 1.1.7. For any f 2 Hom∆([n]; [m]), there is a unique factorization s d f :[n] / / [k] / [m] where s is surjective and d injective. Corollary 1.1.8. The category ∆ can be presented by fdig and fsjg as generators with the following relations: djdi = didj−1 i < j j i i j+1 s s = s s i 6 j (1.1) 8 disj−1 if i < j <> sjdi = id if i = j or i = j + 1 :>di−1sj if i > j + 1 Corollary 1.1.9. Giving a simplicial set X∗ = fXngn>0 is equivalent to giving a family of sets fXng equipped with morphisms di : Xn ! Xn−1 and si : Xn ! Xn+1 satisfying didj = dj−1di i < j sisj = sj+1si i 6 j (1.2) 8 s d if i < j <> j−1 i disj = id if i = j or i = j + 1 > :sjdi−1 if i > j + 1 i i The relation between (1.1) and (1.2) is given by di = X(d ) and si = X(s ). Remark 1.1.10. It is convenient (at least morally) to think of simplicial sets as a graded right \module" over the category ∆, and of cosimplicial set as a graded left \module". A standard way to write X 2 Ob(sSets) is o o / X0 / X1 o X2 ::: o o / with solid arrows denoting the maps di and dashed arrows denoting the maps sj. Definition 1.1.11. For a simplicial set X the elements of Xn := X[n] 2 Sets are called n-simplices. 2 Definition 1.1.12. For a simplicial set X, an n-simplex x 2 Xn is called degenerate if x 2 Im(sj : Xn−1 ! Xn) for some j. The set of degenerate n-simplices is given by n−1 [ fdegenerate n-simplicesg = sj(Xn−1) j=0 An alternative way to describe degenerate simplices is provided by the following S Lemma 1.1.13. The set of degenerate n-simplices is exactly the set X(f)(Xk). f :[n][k] Proof. Exercise. Definition 1.1.14. If X = fXng is a simplicial set, and Yn ⊆ Xn is a family of subsets, 8n > 0, then we call Y = fYng a simplicial subset of X if • Y forms a simplicial set, • the inclusion Y,! X is a morphism of simplicial sets. The definition of simplicial (and cosimplicial) sets can be easily generalized to simplicial and cosimplicial objects in any category. Definition 1.1.15. For any category C we define a simplicial object in C as a functor ∆op ! C. The simplicial objects in any category C form a category, which we will denote by sC. Definition 1.1.16. Dually, we define a cosimplicial object in C as a (covariant) functor ∆ ! C. The category Fun(∆; C) of cosimplicial objects in C will be denoted by csC. 1.2 Motivation In this section we provide some motivation for the definition of simplicial sets, which might seem rather counter-intuitive. We will discuss various examples later in section 1.3 which will hopefully help to develop some intuition. Definition 1.2.1. The geometric n-dimensional simplex is the topological space ( n ) n n+1 X ∆ = (x0; : : : ; xn) 2 R : xi = 1; xi > 0 i=0 Thus ∆0 is a point, ∆1 is an interval, ∆2 is an equilateral triangle, ∆3 is a filled n n tetrahedron, etc. We will label the vertices of ∆ as e0; : : : ; en. Then any point x 2 ∆ P can be written as a linear combination x = xiei, where xi 2 R are called the barycentric coordinates of x. n To any subset α ⊂ [n] = f0; 1; : : : ; ng we can associate a simplex ∆α ⊂ ∆ , which is the n convex hull of vertices ei with i 2 α. The sub-simplex ∆α is also called α-face of ∆ . Thus we have a bijection fsubsets of [n]g $ ffaces of ∆ng. 3 Definition 1.2.2. By a finite polyhedron we mean a topological spaces X homeomorphic to a union of faces of a simplex ∆n: n X ' S = ∆α1 [···[ ∆αr ⊂ ∆ . The choice of such a homeomorphism is called a triangulation. Infinite polyhedra arize from simplicial complexes, which we define next. Definition 1.2.3. A simplicial complex X on a set V (called the set of vertices) is a collection of non-empty finite subsets of V , closed under taking subsets, i.e. 8σ 2 X; ;= 6 τ ⊂ σ ) τ 2 X.