Notes on simplicial homotopy theory [lecture notes]

119 Pages·2008·0.561 MB·English

by  André Joyal, Myles Tierney

Notes on simplicial homotopy theory
Andr´e 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 sim-
plicial 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 <nandm=n−t+s.Thedecompositionisunique,
1 t 1
the i(cid:48)s in [m] being the values not taken by α, and the j(cid:48)s being the elements of
[m] such that α(j)=α(j+1). The εi and ηj satisfy the following relations:
εjεi =εiεj−1 i<j
ηjηi =ηiηj+1 i≤j

εiηj−1 i<j
ηjεi = id i=j or i=j+1
εi−1ηj i>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 i<j
sjsi =sisj+1 i≤j

disj−1 i<j
sjdi = id i=j or i=j+1
di−1sj i>j+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<j
ηj−1ηi =ηiηj i<j
ηjεi =εiηj−1 i<j
ηiεi =id
ηiεj =εj−1ηi i<j−1
ηiεi+1 =id i=j−1.
6 Chapter 1. Simplicial sets
Note that we have interchanged i and j in the next to last equation. Now
in such a diagram, the square of η’s is an absolute, equational pushout, and the
square of ε’s is an absolute, equational pullback. In fact we have
Proposition 1.2.1. Let
C(cid:79)(cid:79)(cid:111)(cid:111) p4 (cid:47)(cid:47)A(cid:79)(cid:79)
s4
p3 s3 s1 p1
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:111)(cid:111) s2 (cid:47)(cid:47)
B D
p2
be a diagram in a category A in which
s s =s s
4 1 3 2
p p =p p
1 4 2 3
p s =s p
4 3 1 2
p s =id i=1,2,3,4
i i
p s =s p
3 4 2 1
or, instead of the last equation, p = p , s = s and p s = id. Then the square
1 2 1 2 3 4
of s’s is a pullback, and the square of p’s is a pushout.
Proof. We prove the pushout statement, as it is this we need, and leave the
pullback part as an exercise. Thus, let g : A → X and g : B → X satisfy
1 2
g p = g p . Then, g = g p s = g p s , so g s = g p s s = g p s s = g s
1 4 2 3 1 1 4 4 2 3 4 1 1 2 3 4 1 2 3 3 2 2 2
as a map from D to X. Furthermore, (g s )p = g p s = g p s = g and
1 1 2 1 4 3 2 3 3 2
(g s )p =g p s =g p s =g .Themapg s =g s isunique,forifa: D →X
2 2 1 2 3 4 1 4 4 1 1 1 2 2
satisfiesg =ap andg =ap ,theng s =ap s =aandg s =ap s =a.Ifwe
1 1 2 2 1 1 1 1 2 2 2 2
areinthealternatesituationp =p ,s =s andp s =id,theng =g p s =g
1 2 1 2 3 4 1 2 3 4 2
as above, and g = g s p , g = g s p and g s = g s is unique again as
2 1 1 2 1 2 2 1 1 1 2 2
above. (cid:3)
If i=j, then
ηi (cid:47)(cid:47)
[n+1] [n]
ηi id[n]
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
[n] [n]
id[n]
isapushout—sinceηi issurjective—andequationalsinceηi hasasection.There
is a similar pullback for εi. Notice, however, that the pullback
[0]
ε1
(cid:15)(cid:15)
(cid:47)(cid:47)
[0] [1]
ε0