Notes on simplicial homotopy theory André Joyal and Myles Tierney The notes contained in this booklet were printed directly from files supplied by the authors before the course. Contents Introduction 1 1 Simplicial sets 3 1.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The skeleton of a simplicial set . . . . . . . . . . . . . . . . . . . . 5 1.3 Geometric realization and the fundamental category . . . . . . . . 9 1.4 The nerve of a partially ordered set . . . . . . . . . . . . . . . . . . 14 2 Quillen homotopy structures 17 2.1 Homotopy structures . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The Quillen structure on groupoids . . . . . . . . . . . . . . . . . . 20 2.3 Cofibration and fibration structures. . . . . . . . . . . . . . . . . . 24 2.4 The fundamental groupoid and the Van Kampen theorem . . . . . 30 2.5 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 The homotopy theory of simplicial sets 37 3.1 Anodyne extensions and fibrations . . . . . . . . . . . . . . . . . . 37 3.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Minimal complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 The Quillen homotopy structure . . . . . . . . . . . . . . . . . . . 59 4 Homotopy groups and Milnor’s Theorem 65 4.1 Pointed simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 A particular weak equivalence . . . . . . . . . . . . . . . . . . . . . 72 4.4 The long exact sequence and Whitehead’s Theorem. . . . . . . . . 75 4.5 Milnor’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Some remarks on weak equivalences . . . . . . . . . . . . . . . . . 79 4.7 The Quillen model structure on Top . . . . . . . . . . . . . . . . . 81 c iii iv Contents A Quillen Model Structures 85 A.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 A.2 Weak factorisation systems . . . . . . . . . . . . . . . . . . . . . . 87 A.3 Quillen model structures . . . . . . . . . . . . . . . . . . . . . . . . 90 A.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.5 The homotopy category . . . . . . . . . . . . . . . . . . . . . . . . 95 A.6 Quillen functors and equivalences . . . . . . . . . . . . . . . . . . . 102 A.7 Monodial model categories . . . . . . . . . . . . . . . . . . . . . . . 108 References 115 Introduction These notes were used by the second author in a course on simplicial homotopy theorygivenattheCRMinFebruary2008inpreparationfortheadvancedcourses on simplicial methods in higher categories that followed. They form the first four chapters of a book on simplicial homotopy theory, which we are currently preparing. What appears here as Appendix A on Quillen model structures will, in fact, form a new chapter 2. The material in the present chapter 2 will be moved else- where. The second author apologizes for the resulting organizational and nota- tional confusion, citing lack of time as his only excuse. BothauthorswarmlythanktheCRMfortheirhospitalityduringthisperiod. 1 2 Introduction Chapter 1 Simplicial sets This chapter introduces simplicial sets. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. This topologicalspace,calledthegeometricrealizationofthesimplicialset,isdefinedin section 3. Also in section 3, we introduce the fundamental category of a simplicial set, and the nerve of a small category. Section 2 is concerned with the skeleton decompositionofasimplicialset.Finally,insection4wegivepresentationsofthe nerves of some useful partially ordered sets. 1.1 Definitions and examples The simplicial category ∆ has objects [n] = {0,...,n} for n ≥ 0 a nonnegative integer. A map α: [n]→[m] is an order preserving function. Geometrically, an n-simplex is the convex closure of n+1 points in general position in a euclidean space of dimension at least n. The standard, geometric n-simplex∆ istheconvexclosureofthestandardbasise ,...,e ofRn+1.Thus, n 0 n the points of ∆ consists of all combinations n n (cid:88) p= t e i i i=0 witht ≥0,and(cid:80)n t =1.Wecanidentifytheelementsof[n]withthevertices i i=0 i e ,...,e of ∆ . In this way a map α: [n] → [m] can be linearly extended to a 0 n n map ∆ : ∆ →∆ . That is, α n m n (cid:88) ∆ (p)= t e . α i α(i) i=0 Clearly, this defines a functor r: ∆→Top. 3 4 Chapter 1. Simplicial sets A simplicial set is a functor X: ∆op → Set. To conform with traditional notation,whenα: [n]→[m]wewriteα∗: X →X insteadofX : X[m]→X[n]. m n α The elements of X are called the n-simplices of X. n Many examples arise from classical simplicial complexes. Recall that a simplicial complexK isacollectionofnon-empty,finitesubsets(calledsimplices)ofa givensetV (ofvertices)suchthatanynon-emptysubsetofasimplexisasimplex. An ordering on K consists of a linear ordering O(σ) on each simplex σ of K such that if σ(cid:48) ⊆σ then O(σ(cid:48)) is the ordering on σ(cid:48) induced by O(σ). The choice of an ordering for K determines a simplicial set by setting K ={(a ,...,a )|σ ={a ,...,a } is a simplex of K n 0 n 0 n and a ≤a ≤...≤a in the ordering O(σ).} 0 1 n For α: [n]→[m], α∗: K →K is α∗(a ,...,a )=(a ,...,a ). m n 0 m α(0) α(n) Remark. An α: [n] → [m] in ∆ can be decomposed uniquely as α = εη, where ε: [p]→[m]isinjective,andη: [n]→[p]issurjective.Moreover,ifεi: [n−1]→[n] istheinjectionwhichskipsthevaluei∈[n],andηj: [n+1]→[n]isthesurjection coveringj ∈[n]twice,thenε=εis...εi1 andη =ηjt...ηj1 wherem≥is >...> i ≥0,and0≤j <...j+1 Thus, a simplicial set X can be considered to be a graded set (X ) to- n n≥0 gether with functions di = εi∗ and sj = ηj∗ satisfying relations dual to those satisfied by the εi(cid:48)s and ηj(cid:48)s. Namely, djdi =didj−1 ij+1 This point of view is frequently adopted in the literature. 1.2. The skeleton of a simplicial set 5 Thecategoryofsimplicialsetsis[∆op,Set],whichweoftendenotesimplyby S.Againfortraditionalreasons,therepresentablefunctor∆( ,[n])iswritten∆[n] and is called the standard (combinatorial) n-simplex. Conforming to this usage, weuse∆: ∆→SfortheYonedafunctor,thoughifα: [n]→[m],wewritesimply α: ∆[n]→∆[m] instead of ∆α. Remark. We have ∆[n] =∆([m],[n])={(a ,...,a )|0≤a ≤a ≤n for i≤j} m 0 m i j Thus,∆[n]isthesimplicialsetassociatedtothesimplicialcomplexwhosesimplices are all non-empty subsets of {0,...,n} with their natural orders. The boundary of this simplicial complex has all proper subsets of {0,...,n} as simplices. Its associated simplicial set is a simplicial (n−1)-sphere ∂∆[n] called the boundary of ∆[n]. Clearly, we have ∂∆[n] ={α: [m]→[n]|α is not surjective} m ∂∆[n] can also be described as the union of the (n−1)-faces of ∆[n]. That is, n (cid:91) ∂∆[n]= ∂i∆[n] i=0 where ∂i∆[n] = im(εi: ∆[n − 1] → ∆[n]). Recall that the union is calculated pointwise, as is any colimit (or limit) in [∆op,Set] [Appendix A 4.4]. 1.2 The skeleton of a simplicial set The relations between the (cid:15)i and ηj of section 1 can be expressed by a diagram ηj (cid:47)(cid:47) [n](cid:111)(cid:111) [n−1] (cid:79)(cid:79) (cid:79)(cid:79) εj ηi εi εi ηi (cid:15)(cid:15) (cid:15)(cid:15) [n−1](cid:111)(cid:111) εj−1 (cid:47)(cid:47)[n−2] ηj−1 in ∆, for n≥2, in which εjεi =εiεj−1 i

Notes on simplicial homotopy theory [lecture notes]

Similar Notes on simplicial homotopy theory [lecture notes]