An Introduction to Simplicial Homotopy Theory
Andre Joyal Myles Tierney
Universite du Quebeca Montreal Rutgers University
Preliminary Version, August 5, 1999
ii
This book is intended mainly for graduate students, and will include abs
tract homotopical algebra as well as a concrete treatment of the basic homo
topy theory of simplicial sets. This first chapter establishes, with a new proo*
*f,
the classical Quillen model structure on simplicial sets. Further chapters will
treat covering spaces and the fundamental groupoid (including the Van Kampen
Theorem), bundles and classifying spaces, simplicial groups and groupoids and
the DwyerKan Theorem, K(ss; n)0s and Postnikov towers, Quillen model struc
tures on diagrams and sheaves, homotopy limits and colimits, and bisimplicial
sets. There will be appendices on basic category theory, cartesian closed cate
gories and compactly generated spaces, introductory topos theory with torsors
and descent, CWcomplexes and geometric realization, and abstract homotopy
theory.
Chapter 1
The homotopy theory of
simplicial sets
In this chapter we introduce simplicial sets and study their basic homotopy
theory. A simplicial set is a combinatorial model of a topological space formed
by gluing simplices together along their faces. This topological space, called *
*the
geometric realization of the simplicial set, is defined in section 1. Its prope*
*rties
are established in Appendix D. In section 2 we discuss the nerve of a small
category.
The rest of the chapter is concerned with developing the basic ingredients
of homotopy theory in the context of simplicial sets. Our principal goal is to
establish the existence of the classical Quillen homotopy structure, which will
then be applied, in various ways, throughout the rest of the book. Thus, we
give the general definition of a Quillen structure in section 3 and state the m*
*ain
theorem. In section 4 we study fibrations and the extremely useful concept of
anodyne extension due to Gabriel and Zisman [ ]. Section 5 is concerned with
the homotopy relation between maps. Next, section 6 contains an exposition
of the theory of minimal complexes and fibrations. These are then used in
section 7 to establish the main theorem, which is the existence of the classical
Quillen structure. The proof we give is different from those in the literature,
[ ] or [ ]; from [ ], for example, in that it is purely combinatorial, making
no use of geometric realization. However, none seems to be able to avoid the
use of minimal fibrations. In section 8, we introduce the homotopy groups
of a Kan complex, establish the long exact sequence of a fibration, and prove
Whitehead's Theorem. We treat Milnor's Theorem in section 9, which shows
that the category of Kan complexes and homotopy classes of maps is equivalent
to the category of CWcomplexes and homotopy classes of maps. Finally, in
section 10, we show that the weak equivalences we used in the proof of the
Quillen structure are the same as the classical ones.
1
2 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
1.1 Simplicial sets and their geometric realiza
tions
The simplicial category has objects [n] = {0; : :;:n} for n 0 a nonnegative
integer. A map ff : [n] ! [m] is an order preserving function.
Geometrically, an nsimplex is the convex closure of n + 1 points in general
position in a euclidean space of dimension at least n. The standard, geometric
nsimplex n is the convex closure of the standard basis e0; : :;:en of Rn+1.
Thus, the points of n consists of all combinations
Xn
p = tiei
i=0
P n
with ti 0, and i=0ti = 1. We can identify the elements of [n] with the
vertices e0; : :;:en of n. In this way a map ff : [n] ! [m] can be linearly
extended to a map ff: n ! m . That is,
Xn
ff(p) = tieff(i)
i=0
Clearly, this defines a functor r : ! T op.
A simplicial set is a functor X : op! Set. To conform with traditional
notation, when ff : [n] ! [m] we write ff* : Xm ! Xn instead of Xff: X[m] !
X[n].
Many examples arise from classical simplicial complexes. Recall that a sim
plicial complex K is a collection of nonempty, finite subsets (called simplice*
*s)
of a given set V (of vertices) such that any nonempty subset of a simplex is a
simplex. An ordering on K consists of a linear ordering O(oe) on each simplex
oe of K such that if oe0 oe then O(oe0) is the ordering on oe0 induced by O(oe).
The choice of an ordering for K determines a simplicial set by setting
Kn = {(a0; : :;:an)oe = {a0; : :;:an} is a simplex of K,
anda0 a1 : : :an in the orderingO(oe):}
For ff : [n] ! [m], ff* : Km ! Kn is ff*(a0; : :;:am ) = (aff(0); : :;:aff(n)).
Remark: An ff : [n] ! [m] in can be decomposed uniquely as ff = "j,
where " : [p] ! [m] is injective, and j : [n] ! [p] is surjective. Moreover, if*
* "i:
[n  1] ! [n] is the injection which skips the value i 2 [n], and jj : [n + 1] *
*! [n]
is the surjection covering j 2 [n] twice, then " = "is: :":i1and j = jjt: :j:j1
where m is > : :>:i1 0, and 0 jt< : :<:j1 < n and m = n  t + s. The
decomposition is unique, the i's in [m] being the values not taken0by ff,0and t*
*he
j's being the elements of [m] such that ff(j) = ff(j +1). The "i sand jj ssatis*
*fy
the following relations:
1.1. SIMPLICIAL SETS AND THEIR GEOMETRIC REALIZATIONS 3
"j"i = "i"j1 i < j
jjji = jijj+18 i j
< "ijj1 i < j
jj"i = : id i = j ori = j + 1
"i1jj i > j + 1
Thus, a simplicial set X can be considered to be a graded set (Xn)n0
together with functions0di=0"i*and sj = jj*satisfying relations dual to those
satisfied by the "i sand jj s. Namely,
didj = dj1di i < j
sisj = sj+1si8 i j
< sj1di i < j
disj = : id i = j ori = j + 1
sjdi1 i > j + 1
This point of view is frequently adopted in the literature.
The category of simplicial sets is [ op; Set], which we often denote simply
by S. Again for traditional reasons, the representable functor (; [n]) is wri*
*tten
[n] and is called the standard (combinatorial) nsimplex. Conforming to this
usage, we use : ! S for the Yoneda functor, though if ff : [n] ! [m], we
write simply ff : [n] ! [m] instead of ff.
Remark: We have
[n]m = ([m]; [n]) = {(a0; : :;:am )0 ai aj n fori j}
Thus, [n] is the simplicial set associated to the simplicial complex whose n
simplices are all nonempty 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 (n1)sphere _[n] cal*
*led
the boundary of [n]. Clearly, we have
_[n]m = {ff : [n] ! [m]ff is not surjective}
_[n] can also be described as the union of the (n1)faces of [n]. That is,
[n
_[n] = i[n]
i=0
where i[n] = im("i : [n  1] ! [n]). Recall that the union is calculated
pointwise, as is any colimit (or limit) in  Set [A 4.4].
Using the universal property (A.5.4) of  Set, the functor r : ! T op
can be extended to a functor r] : S ! T op, called the geometric realization.
Following Milnor [ ], we write X instead of r]X for the geometric realization
of a simplicial set X. Thus, we have a commutative triangle
4 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
C____________//_S_
CCC ___
rCCC!!C""____
T op
where
X = lim!n
[n]!X
As in A.5.4, r] =   has a right adjoint r]. For any topological space T ,
r]T is the singular complex sT of T . That is,
(r]T )[n] = T op(n; T ) = (sT )n
Using the fact that a left adjoint preserves colimits (A.4.5), we see that the *
*geo
metric realization functor   : S ! T op is colimit preserving. A consequence
of this is that X is a CWcomplex. Furthermore, if T op is replaced by T opc
 the category of compactly generated spaces  then   is also leftexact, i.*
*e.
preserves all finite limits. See Appendix B for the basic properties of T opc, *
*and
Appendix D for CWcomplexes and the proofs of the above facts.
1.2 Simplicial sets and categories
Denote by Cat the category of small categories and functors. There is a functor
! Cat which sends [n] into [n] regarded as a category via its natural orderin*
*g.
Again, by the universal property of  Set this functor can be extended to a
functor R : S ! Cat so as to give a commutative triangle
D____________//_S
DD 
DDD 
D!!D""R
Cat
where
RX = lim![n]
[n]!X
As before, R has a right adjoint N. If A is a small category, NA is the nerve
of A and
(NA )n = Cat([n]; A)
An nsimplex x of a simplicial set X is said to be degenerate if there is
a surjection j : [n] ! [m] with m < n and an msimplex y such that x =
j * y. Otherwise, we say x is nondegenerate. Consider the case when X is the
nerve NP of a partially ordered set P . Then an nsimplex of NP is an order
1.2. SIMPLICIAL SETS AND CATEGORIES 5
preserving mapping x : [n] ! P which is nondegenerate iff it is injective. Sin*
*ce
N preserves monomorphisms, the singular nsimplex [n] ! NP associated to
x is also injective. Thus, the image of a nondegenerate nsimplex of NP is a
standard nsimplex.
Suppose P is finite. Call a totally ordered subset c of P a chain of P . Then
there are a finite number c1: :c:rof maximal chains of P , and every chain c
is contained in some ci. If ci contains ni+ 1 elements we can associate to it
a unique nondegenerate simplex xi : [ni] ! P whose image in P is ci. Each
nondegenerate simplex of NP is a face of some xi. The xi together yield a
commutative diagram
P ____//_P
1i>"
i "" f
fflfflfflffl"
B ____//_Y
is a commutative diagram in which i is a cofibration and f is a fibration, then
if i or f is a weak equivalence, there is a dotted lifting making both triangles
commute.
Q4.(Factorization) Any map f : X ! Y can be factored as
X @@____i_____//_E"
@@ """
f @@@ ""p"""
Y
where i is a cofibration and p is a fibration in two ways: one in which i is a
weak equivalence, and one in which p is a weak equivalence.
The homotopy structure is said to be proper, if, in addition, the following
axiom is satisfied.
Q5. If
X0 ____//_X
w0 w
fflffl fflffl
Y 0__f_//_Y
is a pullback diagram, in which f is a fibration and w is a weak equivalence,
then w0is a weak equivalence. Dually, the pushout of a weak equivalence by a
cofibration is a weak equivalence.
An example of a Quillen homotopy structure is obtained by taking K to be
T opc. A map f : X ! Y is a weak equivalence if ss0(f) : ss0(X) ! ss0(Y )
is a bijection, and for n 1 and x 2 X, ssn(f) : ssn(X; x) ! ssn(Y; fx) is an
isomorphism. Fibrations are Serre fibrations, i.e. maps p : E ! X with the
covering homotopy property (CHP) for each nsimplex n, n 0. This means
that if h : n x I ! X is a homotopy (I = [0; 1]), and f : n ! E is such
that pf = h0, then there is a "covering homotopy" h : n x I ! E such that
h0= f, and ph = h. Cofibrations are mappings i : A ! B having the left lifting
property (LLP) with respect to those fibrations p : E ! X which are also weak
equivalences. That is, if
A ____//_E>>"
i "" p
fflfflfflffl"
B ____//_X
8 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
is a commutative diagram where p is a fibration and a weak equivalence, then
there is a dotted lifting making both triangles commute. Details of the proof
can be found in Quillen [ ] (for the case T op which is the same as T opc), or *
*see
exercise [ ] at the end of the chapter.
Our principal example is in S, the category of simplicial sets. Here, the
weak equivalences are geometric homotopy equivalences, by which we mean a
map f : X ! Y such that f : X ! Y  is a homotopy equivalence, i.e. there
is a map f0 : Y  ! X such that f0f is homotopic to idX, and ff0 is
homotopic to idY.The cofibrations are monomorphisms.
To define the fibrations, recall that the ithface of [n] (n 1; 0 i n) is
i[n] = im("i: [n  1] ! [n]). The kthhorn of [n] is
[
k[n] = i[n]
i6=k
The geometric realization of k[n] is the union of all those (n1)dimensional
faces of n that contain the kthvertex of n. For example,
442
44
44
44
44
___________4
0 1
is the geometric realization of 1[2].
Definition 1.3.1A Kan fibration is a map p : E ! X of simplicial sets ha
ving the right lifting property (RLP) with respect to the inclusions of the hor*
*ns
k[n] ! [n] for n 1, and 0 k n.
That is, if
k[n] _____//E==
   p
  
fflffl fflffl
[n] _____//X
is a commutative diagram with n 1, and 0 k n, then there is a dotted
lifting making both triangles commute. We express this by saying "any horn in
E which can be filled in X, can be filled in E". For example, if p : E ! X is a
Serre fibration in T opc, then sp : sE ! sX is a Kan fibration in S. This is so,
because a diagram
k[n] ____//_sE
 sp
 
fflffl fflffl
[n] _____//sX
1.4. ANODYNE EXTENSIONS AND FIBRATIONS 9
is equivalent to a diagram
k[n]____//_E<>"
i k"" p
fflfflfflffl"
B _idB//_B
But then i is a retract of j, so i is anodyne. 
1.4. ANODYNE EXTENSIONS AND FIBRATIONS 13
Theorem 1.4.2 Any map f : X ! Y of S can be factored as
X @@____i_____//_E"
@@ """
f @@@ ""p"""
Y
where i is a monomorphism, and p is a trivial fibration.
Proof: Repeat the proof of Theorem 1.4.1 using the family ( _[n] ! [n]n 0)
instead of the family (k[n] ! [n]0 k n; n 1). 
Definition 1.4.2A simplicial set X is called a Kan complex if X ! 1 is a
fibration.
As an example, we have the singular complex sT of any topological space
T . Another important example is provided by the following theorem.
Theorem 1.4.3 (Moore) Any group G in S is a Kan complex.
Theorem 1.4.3, together with the following lemma, provides many examples
of fibrations.
Lemma 1.4.1 The property of being a fibration, or trivial fibration, is local.
That is, if p : E ! X and there exists a surjective map q : Y ! X such that in
the pullback
E0 ____//_E
p0 p
fflffl fflffl
Y __q_//_X
p0is a fibration, or trivial fibration, then p is a fibration, or trivial fibra*
*tion.
The straightforward proof is left as an excersise.
Definition 1.4.3A bundle with fiber F in S is a mapping p : E ! X such
that for each nsimplex [n] ! X of X, there is an isomorphism OE in
[n] x F ______OE_____//_[n] xX E
JJ ss
JJJ sss
ss1JJ$$JJ yysss1ssss
[n]
14 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
P
Let Y = [n]!X [n]. Then the cannonical map Y ! X is surjective.
Since ss1 : Y x F ! Y is clearly a fibration when F is a Kan complex, it follows
from Lemma 1.4.1 that a bundle with Kan fiber F is a fibration. In particular,
a principal Gbundle, which is a bundle with fiber a group G, is a fibration.
Proof of Theorem 1.4.3: We give a new proof of the theorem, which perhaps
involves less extensive use of the simplicial identities than the classical one.
Thus, let G be a group and let f : k[n] ! G. We want to extend f to [n]
and we proceed by induction on n. The case n = 1 is obvious, since each k[1]
is [0] and a retract of [1]. For the inductive step, let k;k1[n] be k[n] with
the (k  1)st face removed (if k = 0 use 0;1[n]). Then there is a commutative
diagram of inclusions
k;k1[n]_____//k1[n  1] x [1]
 
 
fflffl fflffl
[n] ________//[n  1] x [1]
whose geometric realizations in dimension 3 look like
jjjjj?? ??________________jjj_????
jjjjj ??? ??? jjjjj ??????
jjjjj ??? ??? jjjjjjj ??????
_____________________jjjjTTTTT?"//_ ?_____________________jjjjTTTTT"??"
TTTT """ """ TTTTTT """
TTTTT """ """ TTTTT """
TTTTT"" ""________________TTT_""
 
 
 
fflffl fflffl
jjjjj?? ??________________jjj_??
jjjjj ??? ??? jjjjj ???
jjjjj  ???  ??? jjjjjjj  ???
_____________________jjjjTTTTT?"//_ ?_____________________jjjjTTTTT"?"
TTTT  """  """ TTTTTT  """
TTTTT """ """ TTTTT """
TTTTT"" ""________________TTT_""
Now f restricted to k;k1[n] can be extended to k1[n1]x[1], since the
inclusion k;k1[n] ! k1[n  1] x [1] is an anodyne extension of dimension
n  1. Moreover, this extension can be further extended to [n  1] x [1] by
exponential adjointness and induction, since G[1] is a group. Restricting this
last extension to [n] we see that f restricted to k;k1[n] can be extended to
[n].
We are thus in the following situation. We have two subcomplexes k;k1[n]
and [n1] of [n] where the inclusion [n1] ! [n] is "k1. Furthermore,
1.5. HOMOTOPY 15
we have a map f : k[n] = k;k1[n] [ [n  1] ! G whose restriction to
k;k1[n] can be extended to [n]. Now put r = jk1 : [n] ! [n  1]. Then
r[n  1] = id and it is easy to see that r : [n] ! [n  1] ! k[n] maps
k;k1[n] into itself. The following Lemma then completes the proof.
Lemma 1.4.2 Let A and B be subcomplexes of C, and r : C ! B a mapping
such that r = id on B and r : C ! B ! A [ B maps A into itself. Let
f : A [ B ! G where G is a group. Then if fA can be extended to C, f can be
extended to C
Proof: Extend fA to g : C ! G, and define h : C ! G by h(x) =
g(x)g(r(x))1f(r(x)). 
1.5 Homotopy
Definition 1.5.1Let X be a simplicial set. In the coequaliser
_d0_//_
X1 ____//_X0___//_ss0(X)
d1
ss0(X) is called the set of connected components of X.
We remark that this ss0(X) is the same as the set of connected components
of X considered as a setvalued functor and defined in A.5, i.e. ss0(X) = lim!*
*X.
For the proof, see excercise [ ] at the end of the chapter.
Let us write the relation on X0 determined by d0 and d1 as x ~ y, saying
"x is connected to y by a path". That is, writing I for [1] and (0) I,
respectively (1) I, for the images of "1 : [0] ! [1] and "0 : [0] ! [1],
then x ~ y iff there is a map ff : I ! X such that ff(0) = x, and ff(1) = y.
In general, x ~ y is not an equivalence relation, so that if we denote the map
X0 ! ss0(X) by x 7! xthen x= yiff there is a "path of length n' connecting x
and y", i.e, we have a diagram of the form
x ! x1 x2 ! x3. .x.n1! y
x ~ y is an equivalence relation, however, when X is a Kan complex. For
suppose ff : I ! X and fi : I ! X are such that ff(1) = fi(0). Let s : 1[2] ! X
be the unique map such that s"0 = fi and s"2 = ff. A picture is given by
442
44
fl 44fi4
44
44
___________
0 ff 1
16 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
If X is Kan, there is a t : [2] ! X extending s, and fl = t"1 connects ff(0) to
fi(1), showing transitivity. Using 0[2] and the constant (degenerate) path ff(0)
as in
42
4
ff(0) 44fi
4
___________4
0 ff 1
yields the symmetry.
A useful fact is the following.
Proposition 1.5.1If X and Y are simplicial sets, the canonical mapping
ss0(X x Y ) ! ss0(X) x ss0(Y )
is a bijection.
Proof:_ The canonical mapping, in the notation defined above, is given by
(x; y)7! (x; y). It is clearly surjective. If (x1; y1) = (x2; y2) then x1= x2, *
*and
y1= y2. Thus, there is a path of length n connecting x1 to x2, and a path
of length m connecting_y1_to_y2._By using constant paths, we we may assume
n = m, so that (x1; y1)= (x2; y2)and the map is injective. 
Definition 1.5.2If f; g : X ! Y , we say f is homotopic to g if there is a map
h : X x I ! Y such that h0 = f and h1 = g.
Clearly, we can interpret a homotopy h as a path in Y X such that h(0) = f
and h(1) = g. As above, the relation of homotopy among maps is not an equi
valence relation in general (see excercise [ ] for an example), but it is when *
*Y X
is Kan. We will show below that Y X is Kan when Y is, which, by adjointness,
amounts to showing that each k[n] x X ! [n] x X is anodyne.
We denote by ho(S) the category of Kan complexes and homotopy classes of
maps. That is, its objects are Kan complexes, and the set of morphisms between
two Kan complexes X and Y is [X; Y ] = ss0(Y X). Composition is defined as
follows. Given X, Y and Z, there is a map Y X x ZY ! ZX , which is the
exponential transpose of the mapping
Y Xx ZY x X ' ZY x Y Xx X idxev!ZY x Y ev!Z
The composition in ho(S), [X; Y ] x [Y; Z] ! [X; Z], is obtained by applying ss0
to this map, using Proposition 1.5.1.
Notice that an isomorphism of ho(S) is a homotopy equivalence, i.e. X ' Y
in ho(S) iff there are mappings f : X ! Y and f0: Y ! X such that ff0~ idY
and f0f ~ idX .
1.5. HOMOTOPY 17
ho(S) has a number of different descriptions, as we will see. For example, it
is equivalent to the category of CWcomplexes and homotopy classes of maps.
Returning to the problem of showing that Y X is Kan when Y is, we will, in
fact, prove the following more general result. Let k : Y ! Z be a monomorphism
and suppose p : E ! X. Denote the pullback of Xk and pY by
(k; p)____//EY
 Y
 p
fflffl fflffl
XZ _Xk__//_XY
(k; p) is the "object of diagrams" of the form
Y ____//_E
k p
fflfflfflffl
Z ____//_X
The commutative diagram
__Ek_//Y
EZ E
PZ  pY
fflffl fflffl
XZ _Xk_//_XY
gives rise to a map kp : EZ ! (k; p), and we have
Theorem 1.5.1 If p : E ! X is a fibration, then kp : EZ ! (k; p) is a
fibration, which is trivial if either k is anodyne, or p is trivial.
Proof: Let i : A ! B be a monomorphism. The problem of finding a dotted
lifting in
A ______//EZ==

i   kp
fflffl fflffl
B _____//(k; p)
coincides with the problem of finding a dotted lifting in the adjoint transposed
diagram
(A x Z) [ (B x Y_)__//_E77o
 oo o p
 o o 
fflfflo fflffl
B x Z _________//_X
18 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
which exists for any i if p is trivial. By Theorem 1.5.2, below, the lefthand
vertical map is anodyne if either i or k is, which will complete the proof. *
* 
Taking k to be the identity yields, in particular,
Corollary 1.5.1If p : E ! X is a fibration, so is pY : EY ! XY for any Y.
This, of course, generalizes the original statement that XY is a Kan complex
when X is.
Let i : A ! B, and k : Y ! Z be monomorphisms. Then we have i x Z :
A x Z ! B x Z and B x k : B x Y ! B x Z, and we write i ? k for the inclusion
(A x Z) [ (B x Y ) ! B x Z.
Theorem 1.5.2 (GabrielZisman) If i is anodyne, so is i ? k.
For the proof of Theorem 1.5.2 we need the following auxiliary result. Recall
that we denoted by A the class of anodyne extensions, which is the saturated
class generated by all the inclusions k[n] ! [n] for n 1 and 0 k n.
Denote by a the anodyne extension (e) ! [1] and by in the inclusion _[n] !
[n]. Now let B be the saturated class generated by all the inclusions
a ? in : ((e) x [n]) [ ([1] x _[n]) ! [1] x [n]
for e = 0; 1 n 1. [1]x[n] is called a prism and ((e)x[n])[([1]x _[n])
an open prism. For example, the geometric realization of the inclusion
a ? i1 : ((1) x [1]) [ ([1] x _[1]) ! [1] x [1]
is
(0; 1) (1; 1) (0; 1) (1; 1)
___________ ___________
  
  
 _____//_  
  
  
  
___________ ___________
(0; 0) (1; 0) (0; 0) (1; 0)
Finally, we denote by C the saturated class generated by all inclusions
a ? m : ((e) x Y ) [ ([1] x X) ! [1] x Y
where m : X ! Y is a monomorphism of S and e = 0; 1.
Theorem 1.5.3 B A and A = C
With Theorem 1.5.3, whose proof we give shortly, we can complete the proof
of Theorem 1.5.2.
1.5. HOMOTOPY 19
Proof of Theorem 1.5.2: Let k : Y ! Z be an arbitrary monomorphism
of S, and denote by D the class of all monomorphisms i : A ! B such that
i ? k : (A x Z) [ (B x Y ) ! B x Z is anodyne. D is clearly saturated, so it
suffices to show that C D since A = C.
Thus, let m0: Y 0! Z0be a monomorphism of S, and consider the inclusion
a ? m0: ((e) x Z0) [ ([1] x Y 0) ! [1] x Z0 of C. Then
(a ? m0) ? k : (((e) x Z0) [ ([1] x Y 0)) x Z [ ([1] x Z0) x Y ! ([1] x Z0) x Z
is isomorphic to
a ? (m0? k) : ((e) x Z0x Z) [ [1] x (Y 0x Z [ Z0x Y ) ! [1] x (Z0x Z)
which is in C, and hence anodyne. It follows that a ? m0is in D, which proves
the theorem. 
Proof of Theorem 1.5.3:
B A: Taking e = 1, we want to show the inclusion
a ? in : ((1) x [n]) [ ([1] x _[n]) ! [1] x [n]
is anodyne. From section 2, we know that the top dimensional nondegenerate
simplices of [1] x [n] correspond, under the nerve N, to the injective order
preserving maps oej : [n + 1] ! [1] x [n] whose images are the maximal chains
((0; 0); : :;:(0; j); (1; j); : :;:(1; n))
for 0 j n. As to the faces of the oej, we see that dj+1oej = dj+1oej+1 with
image
((0; 0); : :;:(0; j); (1; j + 1); : :;:(1; n))
for 0 j n. Also, dioej 2 [1] x _[n] for i 6= j; j + 1, d0oe0 2 (1) x [n] and
dn+1oen 2 (0) x [n].
So, we first attach oe0 to the open prism ((1) x [n]) [ ([1] x _[n]) along
1[n + 1] since all the faces dioe0 except d1oe0 are already there. Next we atta*
*ch
oe1 along 2[n + 1] since now d1oe1 = d1oe0 is there, and only d2oe1 is lacking.*
* In
general, we attach oej along j+1[n + 1] since djoej = djoej1 was attached the
step before, and only dj+1oej is lacking. Thus, we see that the inclusion of the
open prism in the prism is a composite of n+1 pushouts of horns. For example,
the filling of the inclusion
a ? i1 : ((1) x [1]) [ ([1] x _[1]) ! [1] x [1]
above proceeds as follows:
20 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
(0; 1) (1; 1) (0; 1) (1; 1) (0; 1) (1; 1)
___________ ___________"" ___________""
 ""   "" 
 ""   oe1 "" 
 _____//_ """  _____//_  """ 
 ""   "" 
 "" oe   "" oe 
 "" 0   "" 0 
___________ ___________"" ___________""
(0; 0) (1; 0) (0; 0) (1; 0) (0; 0) (1; 0)
When e = 0, we first attach oen, and work backwards to oe0.
C A: We proved above that a ? in is anodyne. Thus, a ? in has the left lifting
property (LLP) with respect to any fibration p : E ! Z. By adjointness, in
has the LLP with respect to ap. But then any member of the saturated class
generated by the in, i.e. any monomorphism m : X ! Y , has the LLP with
respect to ap. Thus, again by adjointness, a ? m has the LLP with respect to
p, and hence is anodyne.
A C: For 0 k < n, let sk : [n] ! [1] x [n] be the injection sk(i) = (1; i)
and rk : [1] x [n] ! [n] the surjection given by rk(1; i) = i, and rk(0; i) = i
for i k, rk(0; i) = k i k. Clearly, rksk = id[n]. It is easy to check that
Nsk : [n] ! [1] x [n] induces a map k[n] ! ((0) x [n]) [ ([1] x k[n])
and Nrk : [1] x [n] ! [n] a map ((0) x [n]) [ ([1] x k[n]) ! k[n]. It
follows that we have a retract
k[n] ____//_((0) x [n]) [ ([1] x k[n])//_k[n]
  
  
fflffl fflffl fflffl
[n] ____________//[1] x [n]___________//[n]
The middle vertical map is in C, so the horns k[n] ! [n] for k < n are in C.
For k = n, or in general k > 0, we use the inclusion uk : [n] ! [1] x [n] given
by uk(i) = (0; i) and the retraction vk : [1] x [n] ! [n] given by vk(0; i) = i,
vk(1; i) = k if i k and vk(1; i) = i for i k. 
An important consequence of Theorem 1.5.2 is the covering homotopy ex
tension property (CHEP) for fibrations, which is the statement of the following
proposition.
Proposition 1.5.2Let p : E ! X be a fibration, and h : Z x I ! X a
homotopy. Suppose that Y ! Z is a monomorphism, and h0: Y x I ! E is a
lifting of h to E on Y x I, i.e. the diagram
1.5. HOMOTOPY 21
0
Y x I_h___//E
 
 p
fflffl fflffl
Z x I_h___//X
commutes. Suppose further that f : Z x (e) ! E is a lifting of he(e = 0; 1) to
E, i.e.
Z x (e)_f__//_E
 p
 
fflffl fflffl
Z x I__h__//_X
commutes. Then there is a homotopy h: Z x I ! E, which lifts h, i.e. ph = h,
agrees with h0on Y x I, and is such that he= f.
Proof: The given data provides a commutative diagram
(Y x I) [ (Z x (e))_//_E77o
 hoo o p
 oo 
fflfflo fflffl
Z x I____h_____//X
which has a dotted lifting hby Theorem 1.5.2, since (e) ! I is anodyne. 
We establish now several applications of the CHEP, which will be useful
later. To begin, let i : A ! B be a monomorphism of S.
Definition 1.5.3A is said to be a strong deformation retract of B if there is
a retraction r : B ! A and a homotopy h : B x I ! B such that ri = idA,
h0 = idB , h1 = ir, and h is "stationary on A", meaning
A x III__//_B x_Ih_//_B<>"
i "" p
fflfflfflffl"
B __b_//_X
is given with p a fibration, consider
ss1 a
A x I_____//A____//E
ixI i p
fflffl fflfflfflffl
B x I_h___//B_b__//X
A lifting of bh at 1 is provided by ar : B ! E, so lift all of bh by the CHEP,
and take the value of the lifted homotopy at 0 as a dotted filler in the origin*
*al
diagram. i is then anodyne by Corollary 1.4.1 
Proposition 1.5.5A fibration p : E ! X is trivial iff p is the dual of a strong
deformation retraction. That is, iff there is an s : X ! E and h : E x I ! E
such that ps = idX , h0 = idE, h1 = sp, and
E x I _h__//_E
ss1 p
fflffl fflffl
E ___p__//_X
1.5. HOMOTOPY 23
commutes.
Proof: If p is trivial, construct s and h as dotted liftings in
0 _____//E>>
 s "" 
 " p
fflfflfflffl"
X __=_//_X
and
(idX;sp)
E x ((0) + (1))__//E88q
 h qq p
 q q 
fflfflq fflffl
E x I __pss1__//_X
On the other hand, if p is the dual of a strong deformation retraction as above,
and a diagram
A __a__//E
i p
fflfflfflffl
B __b__//X
is given, with i an arbitrary monomorphism, lift provisionally by sb, then adju*
*st
by the CHEP as in Proposition 1.5.4. 
We can use Proposition 1.5.5 to obtain
Proposition 1.5.6A fibration p : E ! X is trivial iff p is a homotopy equi
valence.
Proof: Suppose p is a homotopy equivalence, i.e. there is a map s : X ! E
together with homotopies k : X x I ! X and h : E x I ! E such that k0 = ps,
k1 = idX , h0 = sp, h1 = idE. First, let k0be a lifting in the diagram
X x (0)s___//E;;w
 k0ww p
 w 
fflfflw fflffl
X x I _k___//_X
Then s0= k01is such that ps0= idX , so we may assume from the beginning that
ps = idX . Now we have two maps I ! EE . Namely the adjoint transposes of
h and sph, which agree at 0, giving a diagram
24 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
0[2]__a__//EE<<
 ffyy E
 y p
fflffly fflffl
[2] b=phj1//_XE
Since p is a fibration, we can find a dotted filler ff. Then, ff"0 = h0is a hom*
*otopy
between idE and sp, which is fiberwise, i.e.
0
E x I h___//_E
ss1 p
fflffl fflffl
E ___p__//_X
commutes. Thus, p is trivial by Proposition 1.5.5. The converse follows imme
diately from Proposition 1.5.5. 
Proposition 1.5.7Let p : E ! X be a fibration, and i : A ! X a monomor
phism. If A is a strong deformation retract of X, then in
p1(A)_____//E
 p
 
fflffl fflffl
A ___i__//_X
p1(A) is a strong deformation retract of E.
Proof: Let h : X x I ! X denote the deformation of X into A. Then we have
two commutative diagrams
E x (0)______id_______//E
 p
 
fflffl fflffl
E x I_pxid//_X x_Ih__//X
and
p1(A) x Iss1//_p1(A)___//E
 p
 
fflffl fflffl
E x I__pxid_//X x Ih___//X
These provide a diagram
1.5. HOMOTOPY 25
(p1(A) x I) [ (E x (0))____________//E33hhh
 kh h h h p
 h hh 
fflfflhhh fflffl
E x I_____pxid___//_X x_Ih__//X
A dotted lifting k, which exists by the CHEP, provides a deformation of E into
p1(A). 
As applications of Proposition 1.5.7 we have the following.
Corollary 1.5.2Let p : E ! X x I be a fibration. Denote p1(X x (0)) and
p1(X x (1)) by p0 : E0 ! X and p1 : E1 ! X. Then p0 and p1 are fiberwise
homotopy equivalent. That is, there are mappings
f
E0A___________//E1
AAA """"
p0AAAA ""p1""""
X
and
g
E1A___________//E0
AAA """"
p1AAAA ""p0""""
X
together with homotopies h : E0 x I ! E0 and k : E1 x I ! E1 such that
h0 = idE0, h1 = gf, k0 = idE1, k1 = fg and the diagrams
E0x I _h___//E0
ss1 p0
fflffl fflffl
E0 __p0__//_X
and
E1x I _k___//E1
ss1 p1
fflffl fflffl
E1 __p1__//_X
commute.
Proof: X x (0) and X x (1) are both strong deformation retracts of X x I.
Thus, by Proposition 1.5.7, E0 and E1 are strong deformation retracts of E.
26 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
The inclusions and retractions of E0 and E1 yield homotopy equivalences f and
g. It follows easily from the proof of Proposition 1.5.7 that the homotopies h
and k are fiberwise. 
Corollary 1.5.3Let p : E ! Y be a fibration and f; g : X ! Y two homotopic
maps Then the pullbacks f*(E) and g*(E) are fiberwise homotopy equivalent.
Proof: It suffices to consider the case of a homotopy h : X x I ! Y such that
h0 = f and h1 = g. For this, take the pullback
h*(E) ____//_E
q  p
fflffl fflffl
X x I _h__//_Y
and apply Corollary 1.5.2 to q. 
Corollary 1.5.4Let X be connected and p : E ! X a fibration. Then any two
fibers of p are homotopy equivalent.
Proof: It is enough to show that the fibers over the endpoints of any path
ff : I ! X are homotopy equivalent. For this, apply Corollary 1.5.3 to the
inclusion of the endpoints of ff. 
1.6 Minimal complexes
Let X be a simplicial set and x; y : [n] ! X two nsimplices of X such
that x _[n] = y _[n] = a. We say x is homotopic to y mod _[n], written
x ~ y mod _[n], if there is a homotopy h : [n] x I ! X such that h0 = x,
h1 = y and h is "stationary on _[n]", meaning
_[n] x Iss1//__[n]
 
 a
fflffl fflffl
[n] x I _h___//_X
commutes. It is easy to see that x ~ y mod _[n] is an equivalence relation when
X is a Kan complex.
Definition 1.6.1Let X be a Kan complex. X is said to be minimal if x ~
y mod _[n] entails x = y.
Our main goal in this section is to prove the following theorem, and its
corresponding version for fibrations.
1.6. MINIMAL COMPLEXES 27
Theorem 1.6.1 Let X be a Kan complex. Then there exists a strong deforma
tion retract X0 of X which is minimal.
In the proof of Theorem 1.6.1 we will need a lemma.
Lemma 1.6.1 Let x and y be two degenerate nsimplices of a simplicial set X.
Then x _[n] = y _[n] implies x = y.
Proof: Let x = sidix and y = sjdjy. If i = j, we are done. If, say, i < j, we
have x = sidix = sidiy = sidisjdjy = sisj1didjy = sjsididjy. Thus, x = sjz,
where z = sididjy. Hence, djx = djsjz = z and x = sjdjx = sjdjy = y. 
Proof of Theorem 1..1: We construct X0 skeleton by skeleton. For Sk0X0
take one representative in each equivalence class of ss0(X). Suppose we have de
fined Skn1X0. To define SknX0we take one representative in each equivalence
class among those nsimplices of X whose restrictions to _[n] are contained
in Skn1X0, choosing a degenerate one wherever possible. Lemma 1.6.1 shows
that X0ncontains all degenerate simplices from X0n1.
For the deformation retraction, suppose we have h : Skn1X x I ! X.
Consider the pushout
P _ P
e(X)n [n] x I___//_e(X)n[n] x I
 
 
fflffl fflffl
Skn1X x I_________//SknX x I
To extend h to SknX x I we must define it on each [n] x I consistent with its
given value on _[n] x I. Thus, let x 2 e(X)n. Since h is already defined on the
boundary of x, we have an open prism
"""________h________""_
"""  """
"""  """ 
????_ _______________"?x???
??  ?? 
?? ??
?___________________?_
in X whose (n  1)simplices in the open end belong to X0. Since X is Kan, we
can fill the prism obtaining a new nsimplex y at the other end whose boundary
is in X0. Now take a homotopy mod _[n] to get into X0. This defines the
retraction r and the homotopy h. 
Theorem 1.6.2 Let X and Y be minimal complexes and f : X ! Y a homo
topy equivalence. Then f is an isomorphism.
The proof of Theorem 1.6.2 follows immediately from the following lemma.
Lemma 1.6.2 Let X be a minimal complex and f : X ! X a map homotopic
to idX . Then f is an isomorphism.
28 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
Proof: We show first that fn : Xn ! Xn is injective by induction on n, letting
X1 be empty. Thus, let x; x0 : [n] ! X be such that f(x) = f(x0). By
induction, x _[n] = x0 _[n] = a. Let h : X x I ! X satisfy h0 = f and
h1 = idX . Then the homotopy h(x x I) is f(x) at 0 and x at 1. Similarly,
h(x0x I) is f(x0) at 0 and x0 at 1. Since f(x) = f(x0), we obtain a map
[n] x 0[2] ! X. Let _[n] x [2] ! X be the map h(a x j1). These agree on
the intersection of their domains, so we obtain a diagram
([n] x 0[2]) [ ( _[n] x [2])_//_X66ll
 kl l l
 l l
fflffll
[n] x [2]
which has a dotted filler k by Theorem 1.5.2. k(id x "0) is a homotopy between
x and x0mod _[n]. Since X is minimal, x = x0.
Now assume that fm : Xm ! Xm is surjective for m < n, and let x : [n] !
X be an nsimplex of X. By induction, and the first part of the proof, each x"i
is uniquely of the form f(yi) for yi : [n  1] ! X. Hence, we obtain a map
y : _[n] ! X such that f(y) = x _[n] . The maps h(a x I) and x x (0) agree
on their intersections giving a diagram
([n] x (0)) [ ( _[n]_x_I)//_X66nn
 knn n
 nn
fflffln
[n] x I
which has a dotted filler k. k at 0 is x, and k at 1 is some nsimplex z. The
homotopy h(z x I) is f(z) at 0 and z at 1. Thus, as above, we obtain a diagram
([n] x 1[2]) [ ( _[n] x [2])_//_X66ll
 lll l
 l l
fflffll
[n] x [2]
which has a dotted filler l by Theorem 1.4.1. k(id x "1) is a homotopy between
x and f(z) mod _[n]. Since X is minimal, x = f(z). 
We discuss now the corresponding matters for fibrations. Thus, let p : E !
X be a map and e; e0 : [n] ! E two nsimplices of E such that e _[n] =
e0 _[n] = a, and pe = pe0= b. We say e is fiberwise homotopic to e' mod _[n],
written e ~f e0mod _[n], if there is a homotopy h : [n] x I ! E such that
h0 = e, h1 = e0, h = a on _[n] as before, meaning
1.6. MINIMAL COMPLEXES 29
_[n] x Iss1//__[n]
 
 a
fflffl fflffl
[n] x I _h___//_X
commutes, and h is "fiberwise", meaning
[n] x I h___//_E
ss1 p
fflffl fflffl
[n] __b___//_X
commutes. As before, it is easy to see that e ~f e0mod _[n] is an equivalence
relation when p is a fibration.
Definition 1.6.2A fibration p : E ! X is said to be minimal if e ~f e0mod _[n]
entails e = e0.
Notice that minimal fibrations are stable under pullback.
Theorem 1.6.3 Let p : E ! X be a fibration. Then there is a subcomplex E0
of E such that p restricted to E0 is a minimal fibration p0: E0! X which is a
strong, fiberwise deformation retract of p.
Proof: The proof is essentially the same as that for Theorem 1.5.1, with x ~
y mod _[n] replaced by e ~f e0mod _[n]. 
Theorem 1.6.4 Let p : E ! X and q : G ! X be minimal fibrations and
f : E ! G a map such that qf = p. Then if f is a fiberwise homotopy
equivalence, f is an isomorphism.
Proof: Again, the proof is essentially the same as that for Theorem 1.5.2, with
x ~ y mod _[n] replaced by e ~f e0mod _[n]. 
Theorem 1.6.5 A minimal fibration is a bundle.
Proof: Let p : E ! X be a minimal fibration, and x : [n] ! X an nsimplex
of X. Pulling back p along x yields a minimal fibration over [n] so it suffices
to show that any minimal fibration p : E ! [n] is isomorphic to one of the
form ss1 : [n] x F ! [n].
For this, define c : [n] x [1] ! [n] as follows: c(i; 0) = i and c(i; 1) = n *
*for
0 i n. Nc = h is a homotopy [n] x I ! [n] between the identity of [n]
and the constant map at the nthvertex of [n]. From Corollary 1.5.3 it follows
that p : E ! [n] is fiberwise homotopy equivalent to ss1 : [n] x F ! [n]
where F is the fiber of p over the nthvertex of [n]. By Theorem 1.6.4, p is
isomorphic to ss1 over [n]. 
30 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
1.7 The Quillen homotopy structure
Here we establish Theorem 1.3.1, or rather a modified version thereof.
Definition 1.7.1Let f : X ! Y be a mapping in S. We say f is a weak
equivalence if for each Kan complex K, [f; K] : [Y; K] ! [X; K] is a bijection.
For example, a homotopy equivalence is a weak equivalence. Thus, a trivial
fibration is a weak equivalence by Proposition 1.5.6. Or, if i : A ! B is anody*
*ne
and K is Kan, then Ki : KB ! KA is a trivial fibration by Theorem 1.5.1,
hence a homotopy equivalence by Proposition 1.5.6. Thus, ss0(Ki) = [i; K] is a
bijection and i is a weak equivalence. Also, if f : X ! Y is a weak equivalence
with X and Y Kan, then f is a homotopy equivalence since f becomes an
isomorphism in ho(S). __
Notice_that Definition 1.7.1 is equivalent_to_saying_that if X ! X and
Y ! Y are anodyne extensions with X and Y Kan, and fis any dotted filler in
__
X _____//XO
f  O_fO
fflfflfflffl_
Y _____//Y
__
then f is a homotopy equivalence.
Now, in S we take as fibrations the Kan fibrations, as cofibrations the mono
morphisms, and as weak equivalences the ones given in Definition 1.7.1. Then
the main theorem of this chapter is the following.
Theorem 1.7.1 The fibrations, cofibrations and weak equivalences defined above
form a proper Quillen homotopy structure on S.
Remark: We will show in section 10 that the weak equivalences of Definition
1.7.1 coincide with the goemetric homotopy equivalences of section 3, so that
Theorem 1.7.1 is, in fact, the same as Theorem 1.3.1.
The proof of Theorem 1.7.1 is based on the following two propositions, which
we establish first.
Proposition 1.7.1A fibration p : E ! X is trivial iff p is a weak equivalence.
Proposition 1.7.2A cofibration i : A ! B is anodyne iff i is a weak equiva
lence.
Proof of Proposition 1.7.1: Let p : E ! X be a fibration and a weak
equivalence. By Theorem 1.6.3 there is a minimal fibration p0: E0! X which
is a_strong_deformation retract of p,_and hence also a weak equivalence. Let
X ! X be an anodyne extension with X Kan (Theorem 1.4.1). By Theorem
1.6.5, p0is_a_bundle._Using Lemma 1.7.1 below, we can extend p0uniquely to a
bundle _p: E ! X in such a way as to have a pullback
1.7. THE QUILLEN HOMOTOPY STRUCTURE 31
__
E0 ____//_E
p0 _p
fflfflfflffl_
X _____//X
__ _
with E0 ! E anodyne._Now p is a fibration since its fiber is a minimal Kan
complex, so E is Kan. Furthermore, _pis a weak equivalence since all the other
maps in the diagram are. Thus, _pis a homotopy equivalence. So, by Proposition
1.5.6, _pis a trivial fibration. It follows that p0is also trivial, hence a hom*
*otopy
equivalence. But this shows that p is also a homotopy equivalence, and thus a
trivial fibration. 
Proof of Proposition 1.7.2: Let i : A ! B be a cofibration and a weak
equivalence. Factor i as
j
A@@__________//E"
@@ """
i@@OO@""p"""
B
where j is anodyne and p is is a fibration. p is a weak equivalence since i and
j are. Thus, by Proposition 1.7.1, p is a trivial fibration. But then there is a
dotted lifting s in
j
A ____//_E>>"
i s"" p
fflfflfflffl"
B _idB//_B
so that i is a retract of j and hence anodyne. 
Lemma 1.7.1 Let A ! B be an anodyne extension and p : E ! A a bundle.
Then there is a pullback diagram
E ____//_E0
p  p0
fflfflfflffl
A _____//B
such that p0: E0! B is a bundle, and E ! E0 is anodyne. Furthermore, such
an extension p0of p is unique up to isomorphism.
Proof: Let E be the class of all monomorphisms having the unique extension
property above. We will show that E contains the horn inclusions and is satu
rated, hence contains all anodyne extensions.
32 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
For the horn inclusions, let C be a contractible simplicial set, i.e. one pro*
*vi
ded with an anodyne map c : 1 ! C. Let p : E ! C be a principal Gbundle
over C. Picking a point e : 1 ! E such that pe = c, we have a dotted lifting in
the diagram
1___e_//E??"
x "" p
fflfflfflffl"
C _idC//_C
so that p has a section and hence is trivial. In case p : E ! C is a bundle
with fiber F , Iso(C x F; E) ! C is a principal Aut(F )bundle, and hence has
a section. But such a section is a trivialization of p. Thus, any bundle over C
is trivial. In particular, any bundle p : E ! k[n] is trivial (the kth vertex is
an anodyne point), and hence can be extended uniquely as a trivial bundle over
k[n] ! [n].
E clearly contains all the isomorphisms. Let us see that its maps are stable
under pushout. Thus, let A ! B be in E, and let A ! A0be an arbitrary map.
Form the pushout
A ____//_A0
 
 
fflfflfflffl
B ____//_B0
and suppose p0: E0! A0is a bundle. Pull back p0to a bundle p : E ! A and
extend p as
E _____//G
p q
fflfflfflffl
A _____//B
with E ! G anodyne since A ! B is in E. Now form the pushout
E ____//_E0
 
 
fflfflfflffl
G ____//_G0
giving a cube
1.7. THE QUILLEN HOMOTOPY STRUCTURE 33
"E ______//_E0
"
""""" """"
G ______//_G0 
   
 fflffl fflffl
 A ___ __//A0
 "" ""
fflffl""fflffl"""""
B ______//B0
where q0 : G0 ! B0 is the natural induced map. In this cube, the lefthand
and back faces are pullbacks.Hence, by the lemma following this proof, we can
conclude that the front and righthand faces are also. Thus, the pullback of q0
over the surjection B + A0! B0 is the bundle q + p0: G + E0 ! B + A0. It
follows that q0 is a bundle and unique, for any other bundle which extends p0
must pull back over B ! B0 to q by the uniqueness of q. Clearly, E0 ! G0is
anodyne.
Now let
A_____//A0____//A
  
  
fflfflfflffl fflffl
B ____//_B0__//_B
be a retract with A0! B0in E. Pushing out A0! B0along A0! A, and using
the stability of E under pushouts, we see that it is enough to consider retracts
of the form
A @
"" @@@
""" @@
""""_r_____@OO//_
B0oo__________B
i
ri = idB , with A ! B0 in E. Thus, let p : E ! A be a bundle. Extend p
to p0: E0 ! B0, then take the pullback p00: E00! B of p0along i, yielding a
diagram of pullbacks
E ____//_E0oo_E00
p  p0 p00
fflfflfflfflifflffl
A ____//_B0oo_ B
with E ! E0 anodyne. Pulling back p00back along A ! B gives the pullback
of p0along A ! B0, i.e. p. Thus, p00is a bundle extending p. But any bundle
over B which extends p pulls back along r to a bundle over B0 which extends
p, so it must be p0by uniqueness. Thus p00is unique and we have a retract
34 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
E B
"" BBBB
""" BB
""""________B//_
E0 oo__________E00
so that E ! E00is anodyne.
We leave the straightforward verification of coproducts and countable com
posites as an excercise for the reader. 
Lemma 1.7.2 Let
"E ______//_E0
"
""""" """"
G ______//_G0 
   
 fflffl fflffl
 A ___ _//_A0
 "" ""
fflffl""fflffl"""""
B ______//B0
be a commutative cube in S, whose lefthand and back faces are pullbacks. If
A ! B is a monomorphism, and the top and bottom faces are pushouts, then
the righthand and front faces are pullbacks.
Proof: It is enough to prove the lemma in the category of sets. In that case,
the righthand and front faces are
E0 ____//_E0+ (G  E)
 
 
fflffl fflffl
A0 ____//_A0+ (B  A)
and
E + (G  E)____//_E0+ (G  E)
 
 
fflffl fflffl
A + (B  A)____//_A0+ (B  A)
In these diagrams, G  E maps to B  A since the lefthand face of the cube is
a pullback. Thus, these two faces are the coproducts of
E0__id_//E0
 
 
fflfflfflfflid
A0_____//A0
1.7. THE QUILLEN HOMOTOPY STRUCTURE 35
and
0_____//(G  E)

 
 
fflffl fflffl
0_____//(B  A)
and
E ____//_E0
 
 
fflfflfflffl
A ____//_A0
and
(G  E)__id_//(G  E)
 
 
fflfflid fflffl
(B  A)_____//(B  A)
respectively. Now use the fact that the coproduct of two pullbacks in Sets is a
pullback. 
Remark: Notice that Lemma 1.7.1 shows that weak equivalences are stable
under pullback along a bundle. In fact, let w : A ! B be a weak equivalence
and p0 : E0 ! B a bundle. Factor w as w = pi where i is a cofibration and
p is a trivial fibration (Theorem 1.4.2). i is a weak equivalence since w and p
are, hence anodyne by Proposition 1.7.2. Trivial fibrations are stable under any
pullback, and anodyne extensions are stable under pullback along a bundle by
Lemma 1.7.1, so the result follows.
Proof of Theorem 1.7.1: Q1 and Q2 are clear. Q3 follows immediately from
Propositions 1.7.1 and 1.7.2, and Q4 follows from Theorems 1.4.1 and 1.4.2.
For Q5, let w : A ! B be a weak equivalence and p : E ! B a fibration.
By Theorem 1.6.3 there is a minimal fibration p0 : E0 ! B which is a strong
fiberwise deformation retract of p. Let p0: E0! A be the pullback of p along w
and p00: E00! A the pullback of p0. Then p00is a strong fiberwise deformation
retract of p0, and the map E00! E0is a weak equivalence by the remark following
Lemma 1.7.1. Thus E0! E is a weak equivalence. Dually, let
A__w__//B
i i0
fflffl fflffl
A0 _w0_//_B0
36 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
be a pushout diagram with w a weak equivalence and i a cofibration. If K is a
Kan complex, the diagram
Ki0_//_B
KB0 K
  w
Kw0 K
fflffl0 fflffl
KA __Ki_//KA
is a pullback with Ki a fibration by Theorem 1.5.3. Kw is a weak equivalence 0
(factor w as a cofibration followed by a trivial fibration to see this) so Kw *
*is a
weak equivalence by the first part of Q5. Thus w0is a weak equivalence, which
proves the theorem. 
1.8 Homotopy groups and Whitehead's Theo
rem
Let _[n] ! [n], n 1, be the inclusion of the boundary, and
_[n]_______//[n]
 
 
fflffl fflffl
1 ____b//__[n]=[n]
a pushout. If X is a Kan complex, and x 2 X0, we write ssn(X; x), n 1, for
the set of homotopy classes of maps _[n]=[n] ! X, which take b to x, modulo
homotopies which are constantly equal to x at b. At the moment, the ssn(X; x)
are simply pointed sets, the point being the class [x] of the constant map at x*
*. In
our final version we will, in fact, show combinatorially that the ssn(X; x), n *
* 1,
are groups, which are abelian for n 2. We will not use this here, however,
though we remark that in section 10 we will show that ssn(X; x) ' ssn(X; x)
so this will follow, albeit unsatisfactorially.
Let p : E ! X be a fibration with X Kan, and write i : F ! E for the
inclusion p1(x) ! E. Let e 2 F0. If ff : [n] ! X represents an element of
ssn(X; x), let fl denote a dotted filler in the diagram
0[n] _____//E==
 fl p
  
fflfflfflffl
[n] _ff__//X
where 0[n] ! E is constant at e. Then d0fl sends _[n  1] to x, and is
independent up to homotopy of the choice of fl and the choice of ff. It thus
defines a function
1.8. HOMOTOPY GROUPS AND WHITEHEAD'S THEOREM 37
@ : ssn(X; x) ! ssn1(F; e)
called the boundary map.
Theorem 1.8.1 The boundary map is a homomorphism for n 2, and the
sequence
: :!:ssn(F; e) ssni!ssn(E; e) ssnp!ssn(X; x) @!: :s:s1(X; x) @!ss0(F ) ss0i!ss0*
*(E) ss0p!ss0(X)
is exact as a sequence of pointed sets, in the sense that for each map, the set*
* of
elements of its domain which are sent to the point is the image of the preceding
map.
Again, we will prove this in detail combinatorially in the final version. We
remark only that, as above, it will follow later from Quillen's theorem that p
is a Serre fibration, which is proved in Appendix D.
Let f : X ! Y be a mapping of simplicial sets. We will call f, just in this
chapter, a homotopy isomorphism if ss0f : ss0(X) ! ss0(Y ) is a bijection, and
ssnf : ssn(X; x) ! ssn(Y; fx) is an isomorphism for n 1 and x 2 X0.
Lemma 1.8.1 Let X be a minimal Kan complex such that X ! 1 is a homotopy
isomorphism. Then X ! 1 is an isomorphism.
Proof: Clearly, X ! 1 induces an isomorphism Sk1X ! Sk11. Suppose
Skn1X ! Skn11 is an isomorphism, with inverse represented by a basepoint
x : 1 ! X. Let oe : [n] ! X be an nsimplex of X. oe _[n] = x, so oe represents
an element of ssn(X; x). ssn(X; x) = [x], so oe ~ x mod _[n]. But then oe = x.

Corollary 1.8.1Let X be a Kan complex, and p : E ! X a minimal fibration.
If p is a homotopy isomorphism, then p is an isomorphism.
Proof: The homotopy exact sequence of p as above has the form
: :!:ssn+1(E; e) '!ssn+1(X; pe) ! ssn(F; e) ! ssn(E; e) '!ssn(X; pe) : : :
ss1(E; e) '!ss1(X; pe) ! ss0(F ) ! ss0(E) '!ss0(X)
It follows that ss0(F ) = [e] and ssn(F; e) = [e] for n 1. By Lemma 1.8.1, the
fiber F over each component of X is a single point, so p is a bijection. 
38 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
Theorem 1.8.2 (Whitehead) Let X and Y be Kan complexes and f : X ! Y
a homotopy isomorphism. Then f is a homotopy equivalence.
Proof: Factor f as
X @@___i_____//_E"
@@ """
f@@@ """p""
Y
where i is anodyne and p is a fibration. Y is Kan, so E is and i is a strong
deformation retract by Proposition 1.5.3. By Theorem 1.6.3, let
E0 AA_________//_E
""
AAA """
p0 AA """p"
Y
be a minimal fibration which is a strong, fiberwise deformation retract of p. S*
*ince
p induces isomorphisms on ssn for n 0 so does p0. Thus p0is an isomorphism
by Corollary 1.8.1. It follows that p is a homotopy equivalence, so the same is
true of f. 
1.9 Milnor's Theorem
Our goal in this section is to prove the following theorem.
Theorem 1.9.1 (Milnor) Let X be a Kan complex, and let jX : X ! sX be
the unit of the adjunction   a s. Then jX is a homotopy equivalence.
The proof of Theorem 1.9.1 uses some properties of the path space of X, so
we establish these first. To begin, since (0) + (1) ! I is a cofibration and X
is a Kan complex, (p0; p1) : XI ! X x X is a fibration. Let x : 1 ! X be a
basepoint. Define P X as the pullback
P X _______//XI
p1 (p0;p1)
fflffl fflffl
1 x X xxidX//_X x X
(0) ! I is anodyne, so p0 : XI ! X is a trivial fibration, again by Theorem
1.5.1. The diagram
P X _____//XI
 p
  0
fflffl fflffl
1___x__//X
1.9. MILNOR'S THEOREM 39
is a pullback, so P X ! 1 is a trivial fibration and hence a homotopy equivalen*
*ce.
Let X denote the fiber of p1 over x, and write x again for the constant path
at x. Then the homotopy exact sequence of the fibration p1 has the form
: :!:ssn(P X; x) ! ssn(X; x) @!ssn1(X; x) ! ssn1(P X; x) : : :
ss1(P X; x) ! ss1(X; x) @!ss0(X) ! ss0(P X) ! ss0(X)
where ss0(P X) = [x] and ssn(P X; x) = [x] for n 1. It follows that the
boundary map induces isomorphisms ss1(X; x) ! ss0(X) and ssn(X; x) !
ssn1(X; x) for n 2.
Proof of Theorem 1.9.1: First, let X be connected. Then any two vertices
of X can be joined by a path. But then any two points of X can be joined
by a path, since any point is in the image of the realization of a simplex, whi*
*ch
is connected. Thus X is connected, as is sX. Otherwise, X is the coproduct
of its connected components. Since s  preserves coproducts, it follows that
ss0jX : ss0(X) ! ss0(sX) is a bijection. Assume by induction that for any
Y , and any y 2 Y0 ssm jX : ssm (Y; y) ! ssm (sY ; y) is an isomorphism for
m n  1. By naturality we have a diagram
jX
X _____//sX
 
 
fflffljPfflfflX
P X ____//_sP X
p1 sp1
fflffl fflffl
X ___jX_//sX
By Quillen's theorem (see Appendix B), p1 is a Serre fibration, so sp1 is a
Kan fibration. Since P X ! 1 is a homotopy equivalence, so is sP X ! 1.
Hence we have a commutative diagram
ssn(X; x)_ssnjX//_ssn(sX; x)
'  '
fflffl fflffl
ssn1(X; x)ssn1jX//_ssn1(sX; x)
ssn1jX is an isomorphism by induction, so ssnjX is an isomorphism. By
Whitehead's Theorem, jX is a homotopy equivalence. 
40 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
An entirely similar argument, using the topological path space, shows that
if T is a topological space, then the counit "T : sT  ! T is a topological we*
*ak
equivalence. Thus, if T is a CWcomplex "T is a homotopy equivalence by
the topological Whitehead Theorem. Since   and s both clearly preserve the
homotopy relation between maps, we see that they induce an equivalence
ho(T opc)_________//ho(S)oo_
where ho(T opc) is the category of CWcomplexes and homotopy classes of maps.
1.10 Some remarks on weak equivalences
Here we collect all the possible definitions we might have given for weak equi
valence, and show they are all the same. We begin with a lemma.
Lemma 1.10.1 If j : C ! D is a mapping in T opc which has the left lifting
property with respect to the Serre fibrations, then C is a strong deformation
retreact of D.
Proof: n is a retract of n x I, so every space in T opc is fibrant. Also, if T *
*is
a space and X a simplicial set, we see easily that s(T X) ' (sT )X . From this
it follows that the singular complex of (p0; p1) : T I! T x T is a Kan fibratio*
*n,
so it itself is a Serre fibration. Now we obtain the retraction r as a lifting *
*in
idC
C _____//C>>"
j "r"
fflffl"
D
and the strong deformation h as the exponential transpose of a lifting in
___jss1
C ______//DI;;x
j x_x (p0;p1)
fflfflhxfflfflx
D _(idD;jr)//_D x D

As a consequence we obtain immediately
Proposition 1.10.1If i : A ! B is an anodyne extension then i : A ! B
and A is a strong deformation retract of B.
Proposition 1.10.2Let X be an arbitrary simplicial set. Then jX : X ! sX
is a weak equivalence.
1.10. SOME REMARKS ON WEAK EQUIVALENCES 41
__ __
Proof: Let i : X ! X be an anodyne extension with X Kan. Then in the
diagram
jX
X ____//_sX
i si
_fflffl_fflffl_
X __j__X//_sX 
__
i is a homotopy equivalence by the above, jX is a homotopy equivalence by
Milnor's Theorem, and i is a weak equivalence, so jX is a weak equivalence.

Proposition 1.10.3w : X ! Y is a weak equivalence in the sense of Defini
tion 1.7.1 iff w is a geometric homotopy equivalence.
Proof: Factor w as
X @@___i_____//_E"
@@ """
w @@@ """p""
Y
where i is a cofibration and p is a trivial fibration. Since w and p are weak
equivalences so is i. i is anodyne by Proposition 1.7.2, so i is a homotopy
equivalence. p is a homotopy equivalence by Proposition 1.5.6, so p is. Thus
w is a homotopy equivalence.
On the other hand, suppose w : X ! Y is a goemetric homotopy equivalence.
Then in the diagram
jX
X ____//_sX
w  sw
fflffl fflffl
Y __jY_//sY 
jX and jY are weak equivalences as above, and w is a homotopy equivalence,
so sw is. Thus w is a weak equivalence. 
Let w : U ! V be a topological weak equivalence. From the diagram
sw
sU_____//sV 
"U "V
fflffl fflffl
U ___w___//V
42 CHAPTER 1. THE HOMOTOPY THEORY OF SIMPLICIAL SETS
we see that sw is a weak equivalence since "U, "V and w are. By the topologic*
*al
Whitehead Theorem, it follows that sw is a homotopy equivalence. Thus sw
is a geometric homotopy equivalence, and a weak equivalence by the above.
Thus, both s and   preserve weak equivalences. Since "T : sT  ! T and
jX : X ! sX are weak equivalences, we see that s and  induce an equivalence
__________// 1
T opc[W 1]oo_______S[W ]
where W stands for the class of weak equivalences in each case. This is not
surprising, of course, since by Appendix E we have that ho(T opc) is equivalent
to T opc[W 1] and ho(S) is equivalent to S[W 1].
When X is a Kan complex, the homotopy equivalence jX : X ! sX pro
vides a bijection ss0(X) ! ss0(X) and an isomorphism ssn(X; x) ! ssn(X; x)
for n 1 and x 2 X0. As remarked above, this shows that the ssn(X; x) are
groups for n 1 and abelian for n 2, though this is certainly not the way to
see that. In any case, for X an arbitrary simplicial set we can define ssn(X; x)
as ssn(X; x) = ssn(X; x) as this is consistent with the case when X is Kan.
Finally, let f : X ! Y be a homotopy isomorphism, i.e. ss0f : ss0(X) !
ss0(Y ) is a bijection, and ssnf : ssn(X; x) ! ssn(Y; fx) is an isomorphism for
n 1 and x 2 X0. Now, if f is a geometric homotopy equivalence, f is a ho
motopy isomorphism. On the other hand, if f is a homotopy isomorphism then
f is a geometric homotopy equivalence by the topological Whitehead Theorem.
Thus, the classes of weak equivalences, geometric homotopy equivalences and
homotopy isomorphisms all coincide.