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 Simplicial sets 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 A Topological and geometric background 41 1 Connections on principal bundles . 41 2 Coverings of spaces . 43 3 Homotopy fibration sequences and homotopy fibers . 46 iii iv Chapter 1 Simplicial sets 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. Note that the set V is not necessarily finite, nor V = S σ. To realize a simplicial σ2X complex X, we define an R-vector space V spanned by elements of V , and for every σ 2 X we define a simplex ∆σ ⊂ V to be the convex hull of σ ⊂ V ⊂ V. Definition 1.2.4. The realization jXj of a simplicial complex X is the union [ jXj = ∆σ ⊂ V σ2X equipped with induced topology from V, i.e. U ⊂ jXj is open if and only if U \ ∆σ is open in ∆σ for all σ 2 X. Definition 1.2.5. A polyhedron is a topological space homeomorphic to the realization jXj 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 SSn(X) the set of ordered (n + 1)-tuples (v0; : : : ; vn), where vi 2 V are vertices such that fv0; : : : ; vng 2 X. Notice that we do allow vi = vj for some i 6= j in the tuple (v0; : : : ; vn), and fv0; : : : ; vng means the underlying set of that tuple. If f : [n] ! [m] is a morphism in ∆, the induced map f∗ : SSm(X) ! SSn(X) is given by (v0; : : : ; vn) 7! (vf(0); : : : ; vf(n)).