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 . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Examples of nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 One example: the nerve of a groupoid . . . . . . . . . . . . . . . . . . 11 2 Geometric realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 General remarks on spaces . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Definition of geometric realization . . . . . . . . . . . . . . . . . . . 14 2.3 Two generalizations of geometric realization . . . . . . . . . . . . . . 16 2.4 Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Comma categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Kan extensions using comma categories . . . . . . . . . . . . . . . . 22 3 Homotopy theory of categories . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 The classifying space of a small category . . . . . . . . . . . . . . . . 24 3.2 Homotopy-theoretic properties of the classifying spaces . . . . . . . 26 3.3 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Coverings of categories . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Explicit presentations of fundamental groups of categories . . . . . . 30 3.6 Homology of small categories . . . . . . . . . . . . . . . . . . . . . . . 31 3.7 Quillen’s Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8 Fibred and cofibred functors . . . . . . . . . . . . . . . . . . . . . . 37 3.9 Quillen’s Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 Algebraic K-theory 41 1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 Classical K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.1 The group K (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 0 2.2 The group K (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1 iii 2.3 The group K (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Some remarks on the Q-construction . . . . . . . . . . . . . . . . . . 66 4.5 The K -group via Q-construction . . . . . . . . . . . . . . . . . . . . 67 0 4.6 Higher K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.7 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8 Quillen–Gersten Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 The “plus = Q” Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 The category S−1S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 K-groups of a symmetric monoidal groupoid . . . . . . . . . . . . . . 74 5.3 Some facts about H-spaces . . . . . . . . . . . . . . . . . . . . . . . 75 5.4 Actions on categories. . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5 Application to “plus”-construction . . . . . . . . . . . . . . . . . . . 80 5.6 Proving the “plus=Q” theorem . . . . . . . . . . . . . . . . . . . . . 83 6 Algebraic K-theory of finite fields . . . . . . . . . . . . . . . . . . . . . . . . 89 6.1 Basics of topological K-theory . . . . . . . . . . . . . . . . . . . . . . 90 6.2 λ-structures and Adams operations . . . . . . . . . . . . . . . . . . . . 91 6.3 Witt vectors and special λ-rings . . . . . . . . . . . . . . . . . . . . 93 6.4 Adams operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 Higher topological K-theory and Bott periodicity . . . . . . . . . . . 96 6.6 Quillen Theorem for Finite fields . . . . . . . . . . . . . . . . . . . . 97 6.7 λ-operations in Higher K-theory . . . . . . . . . . . . . . . . . . . . 98 7 Final remarks on Algebraic K-theory . . . . . . . . . . . . . . . . . . . . . . 104 7.1 Historic remarks on algebraic and topological K-theory . . . . . . . 104 7.2 Remarks on delooping . . . . . . . . . . . . . . . . . . . . . . . . . . 106 iv 3 Introduction to model categories 109 1 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 1.2 Examples of model categories . . . . . . . . . . . . . . . . . . . . . . . 111 1.3 Natural constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 Formal consequences of axioms . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.1 Lifting properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.2 Homotopy equivalence relations . . . . . . . . . . . . . . . . . . . . . 116 2.3 Whitehead Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3 Homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.1 Definition of a homotopy category . . . . . . . . . . . . . . . . . . . 123 3.2 Homotopy category as a localization . . . . . . . . . . . . . . . . . . 124 3.3 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4 Quillen’s Adjunction Theorem . . . . . . . . . . . . . . . . . . . . . 127 3.5 Quillen’s Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . 129 3.6 About proofs of of Quillen’s Adjunction and Equivalence Theorems. 130 3.7 Example: representation functor . . . . . . . . . . . . . . . . . . . . 132 4 Algebraic K-theory as a derived functor . . . . . . . . . . . . . . . . . . . . 134 4.1 Simplicial groups and the Kan loop group construction. . . . . . . . 135 4.2 The Bousfield–Kan completion of a space . . . . . . . . . . . . . . . . 141 4.3 The K-theory space as pronilpotent completion . . . . . . . . . . . . 143 4.4 Derived functors of pronilpotent completion . . . . . . . . . . . . . . 145 4.5 Comparison to the Gersten–Swan K-theory . . . . . . . . . . . . . . 146 A Topological and geometric background 149 1 Connections on principal bundles . . . . . . . . . . . . . . . . . . . . . . . . 149 2 Coverings of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3 Homotopy fibration sequences and homotopy fibers . . . . . . . . . . . . . . 154 v vi 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 the finite totally ordered sets [n] := {0,1,...,n}, n ∈ N, as objects, and order-preserving functions as morphisms (i.e. the functions f: [n] → [m] such that i (cid:54) j ⇒ f(i) (cid:54) 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 cSets. There are two distinguished classes of morphisms in ∆: di: [n−1] (cid:44)→ [n] (n (cid:62) 1, 0 (cid:54) i (cid:54) n) sj: [n+1] (cid:16) [n] (n (cid:62) 0, 0 (cid:54) j (cid:54) n) called the face maps and degeneracy maps respectively. Informally, di skips the value i in its image and sj doubles the value j. More precisely, they are defined by (cid:40) (cid:40) k if k < i k if k (cid:54) j di(k) = sj(k) = k+1 if k (cid:62) i k−1 if k > j Theorem 1.1.5. Every morphism f ∈ Hom ([n],[m]) can be decomposed in a unique way ∆ as f = di1di2···dirsj1···sjs where m = n−s+r and i < ··· < i and j < ··· < j . 1 r 1 s 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 {0,1 (cid:55)→ 0;2,3 (cid:55)→ 1}. One can easily check that f = s0◦s2. Corollary 1.1.7. For any f ∈ Hom ([n],[m]), there is a unique factorization ∆ s (cid:47)(cid:47)(cid:47)(cid:47) (cid:31)(cid:127) d (cid:47)(cid:47) f: [n] [k] [m] where s is surjective and d injective. Corollary 1.1.8. The category ∆ can be presented by {di} and {sj} as generators with the following relations: djdi = didj−1 i < j sjsi = sisj+1 i (cid:54) j (1.1)  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∗ = {Xn}n(cid:62)0 is equivalent to giving a family of sets {X } equipped with morphisms d : X → X and s : X → X satisfying n i n n−1 i n n+1 d d = d d i < j i j j−1 i s s = s s i (cid:54) j (1.2) i j j+1 i  s d if i < j  j−1 i  d s = id if i = j or i = j +1 i j  s d if i > j +1 j i−1 The relation between (1.1) and (1.2) is given by d = X(di) and s = X(si). i i 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 ∈ Ob(sSets) is (cid:111)(cid:111) (cid:111)(cid:111) (cid:47)(cid:47) (cid:47)(cid:47) (cid:111)(cid:111) X (cid:111)(cid:111) X (cid:47)(cid:47)X ... 0 1 (cid:111)(cid:111) 2 with solid arrows denoting the maps d and dashed arrows denoting the maps s . i j Definition 1.1.11. For a simplicial set X the elements of X := X[n] ∈ Sets are called ∗ n n-simplices. 2 Definition 1.1.12. For a simplicial set X , an n-simplex x ∈ X is called degenerate if ∗ n x ∈ Im(s : X → X ) for some j. The set of degenerate n-simplices is given by j n−1 n n−1 (cid:91) {degenerate n-simplices} = s (X ) ⊆ X (1.3) j n−1 n j=0 An alternative way to describe degenerate simplices is provided by the following Lemma 1.1.13. The set of degenerate n-simplices in X is given by ∗ (cid:91) X(f)(X ) = colim X(f)(X ). (1.4) k k f:[n](cid:16)[k] f:[n](cid:16)[k] f(cid:54)=id Proof. Exercise. Remark 1.1.14. Although formula (1.3) looks simpler, formula (1.4) has advantage over (1.3) as it can be used to define degeneracies in simplicial objects of an arbitrary category C (at least, when the latter has small colimits). Definition 1.1.15. If X = {X } is a simplicial set, and Y ⊆ X is a family of subsets, n n n ∀n (cid:62) 0, then we call Y = {Y } a simplicial subset of X if n • Y forms a simplicial set, • the inclusion Y (cid:44)→ 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 1.1.16. 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 1.1.17. 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 cC. 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 n-dimensional geometric simplex is the topological space (cid:40) n (cid:41) (cid:88) ∆n = (x ,...,x ) ∈ Rn+1: x = 1,x (cid:62) 0 0 n i i i=0 3 Thus ∆0 is a point, ∆1 is an interval, ∆2 is an equilateral triangle, ∆3 is a filled tetrahedron, etc. We will label the vertices of ∆n as e ,...,e . Then any point x ∈ ∆n 0 n (cid:80) can be written as a linear combination x = x e , where x ∈ R are called the barycentric i i i coordinates of x. To any subset α ⊂ [n] = {0,1,...,n} we can associate a simplex ∆ ⊂ ∆n, which is the α convex hull of vertices e with i ∈ α. The sub-simplex ∆ is also called α-face of ∆n. Thus i α we have a bijection {subsets of [n]} ↔ {faces of ∆n}. Definition 1.2.2. By a finite polyhedron we mean a topological spaces X homeomorphic to a union of faces of a simplex ∆n: X (cid:39) S = ∆ ∪···∪∆ ⊂ ∆n. 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. ∀σ ∈ X,∅ =(cid:54) τ ⊂ σ ⇒ τ ∈ X. (cid:83) Note that the set V is not necessarily finite, nor V = σ. To realize a simplicial σ∈X complex X, we define an R-vector space V spanned by elements of V, and for every σ ∈ X we define a simplex ∆ ⊂ V to be the convex hull of σ ⊂ V ⊂ V. σ Definition 1.2.4. The realization |X| of a simplicial complex X is the union (cid:91) |X| = ∆ ⊂ V σ σ∈X equipped with induced topology from V, i.e. U ⊂ |X| is open if and only if U ∩∆ is open in σ ∆ for all σ ∈ X. σ Definition 1.2.5. A polyhedron is a topological space homeomorphic to the realization |X| of some simplicial complex X. The quotient of a simplicial complex by a simplicial subcomplex may not be a simplicial complex again. Simplicial sets are then a generalization of simplicial complexes which “capture in full the homotopy theory of spaces”. We will make this statement precise later when we will discuss Quillen equivalences, and in particular the Quillen equivalence between topological spaces and simplicial sets. To any simplicial complex one can associate a simplicial set in the following way. Suppose we have a simplicial complex (X,V) with V being totally ordered. Define SS (X) to be a ∗ simplicial set having SS (X) the set of ordered (n+1)-tuples (v ,...,v ), where v ∈ V are n 0 n i 4