Simplicial approximation
J.F. Jardine
December 11, 2002
Introduction
The purpose of this paper is to display a different approach to the construction
of the homotopy theory of simplicial sets and the corresponding equivalence with
the homotopy theory of topological spaces. This approach is an alternative to
existing published proofs [4],[11], but is of a more classical flavour in that *
*it
depends heavily on simplicial approximation techniques.
The verification of the closed model axioms for simplicial sets has a reputa
tion for being one of the most difficult proofs in abstract homotopy theory. In
essence, that difficulty is a consequence of the traditional approach of derivi*
*ng
the model structure and the equivalence of the homotopy theories of simplicial
sets and topological spaces simultaneously. The method displayed here starts
with using an idea from localization theory (specifically, a bounded cofibration
condition) to show that the cofibrations and weak equivalences of simplicial se*
*ts,
as we've always known them, together generate a model structure for simplicial
sets which is quite easy to derive (Theorem 6).
The fibrations for the theory are those maps which have the right lifting
property with respect to all maps which are simultaneously cofibrations and
weak equivalences. This is the correct model structure, but it is produced at
the cost of initially forgetting about Kan fibrations. Putting the Kan fibratio*
*ns
back into the theory in the usual way, and deriving the equivalence of homotopy
categories is the subject of the rest of the paper. The equivalence of the comb*
*i
natorial and topological approaches to constructing homotopy theory is really
the central issue of interest, and is the true source of the observed difficult*
*y.
Recovering the Kan fibrations and their basic properties as part of the theo*
*ry
is done in a way which avoids the usual theory of minimal fibrations. Histori
cally, the theory of minimal fibrations has been one of the two known general
techniques for recovering information about the homotopy types of realizations
of simplicial sets. The other is simplicial approximation. I have chosen to dis
play the simplicial approximation method here, partly for its own sake, but
also because of a collection of existing and expected analogies for the homotopy
theory of cubical sets [7].
Simplicial approximation theory is a part of the classical literature [1],[2*
*],
but it was never developed in a way that was systematic enough to lead to
1
results about model structures. That gap is addressed here: the theory of the
subdivision and dual subdivision is developed, both for simplicial complexes and
simplicial sets, in Sections 2 and 3, and the fundamental result that the double
subdivision of a simplicial set factors through a polyhedral complex in the same
homotopy type (Lemma 14 and Proposition 15) appears in Section 4. The
simplicial approximation theory for simplicial sets is most succintly expressed
here in Theorem 17 and Corollary 18.
The double subdivision result is the basis for everything that follows, in
cluding excision (Theorem 20), which leads directly to the equivalence of the
homotopy categories of simplicial sets and topological spaces in Theorem 22
and Corollary 23. The Milnor Theorem which asserts that the combinatorial
homotopy groups of a fibrant simplicial set coincide with the ordinary homotopy
groups of its topological realization (Theorem 30) is proved in Section 6, in t*
*he
presence of a combinatorial proof of the assertion that the subdivision functors
preserve anodyne extensions (Lemma 26).
One of the more interesting outcomes of the present development is that,
with appropriately sharp simplicial approximation tools in hand, the subdivi
sions of a finite simplicial set behave like coverings. In particular, from th*
*is
point of view, every simplicial set is locally a Kan complex (Lemma 31), and
the methods for manipulating homotopy types then follow almost by exact anal
ogy with the theory of locally fibrant simplicial sheaves or presheaves [5], [6*
*]. In
that same language, we can show that every fibration which is a weak equiva
lence has the öl cal right lifting property" with respect to all inclusions of *
*finite
simplicial sets (Lemma 33), and then this becomes the main idea leading to
the coincidence of fibrations as defined here and Kan fibrations (Corollary 36).
The same collection of techniques almost immediately implies the Quillen result
(Theorem 37) that the realization of a Kan fibration is a Serre fibration. The *
*de
velopment of Kan's Ex1 functor (Lemma 39, Theorem 40) is also accomplished
from this point of view in a simple and conceptual way.
This paper is not a complete exposition, even of the basic homotopy theory
of simplicial sets. I have chosen to rely on existing published references for
the development of the simplicial (or combinatorial) homotopy groups of Kan
complexes [4], [9], and of other basic constructions such as long exact sequenc*
*es
in simplicial homotopy groups for fibre sequences of Kan complexes, as well as
the standard theory of anodyne extensions. Other required combinatorial tools
which are not easily recovered from the literature are developed here.
This paper was written while I was a member of the Isaac Newton Institute
for Mathematical Sciences during the Fall of 2002. I would like to thank that
institution for its hospitality and support.
2
Contents
1 Closed model structure 3
2 Subdivision operators 7
3 Classical simplicial approximation 9
4 Approximation results for simplicial sets 13
5 Excision 21
6 The Milnor Theorem 26
7 Kan fibrations 31
1 Closed model structure
Say that a map f : X ! Y of simplicial sets is a weak equivalence if the
induced map f* : X ! Y  of topological realizations is a weak equivalence.
A cofibration of simplicial sets is a monomorphism, and a fibration is a map
which has the right lifting property with respect to all trivial cofibrations. *
*All
fibrations are Kan fibrations in the usual sense; it comes out later (Corollary
36) that all Kan fibrations are fibrations. As usual, we say that a fibration
(respectively cofibration) is trivial if it is also a weak equivalence.
Lemma 1. Suppose that X is a simplicial set with at most countably many non
degerate simplices. Then the set of path components ß0X and all homotopy
groups ßi(X, x) of the realization of X are countable.
Proof. The statement about path components is trivial. We can assume that X
is connected to prove the statement about the homotopy groups, with respect
to a fixed base point x 2 X0.
The fundamental group ß1(X, x) is countable, by the Van Kampen theo
rem. The space X plainly has countable homology groups
H*(X, Z) ~=H*(X, Z)
in all degrees.
Suppose that the continuous map p : Y ! Z is a Serre fibration with
connected base Z such that Z and the fibre F have countable integral homology
groups in all degrees, and such that ß1Z is countable. Then a Serre spectral
sequence argument (with twisted coefficients) shows that the homology groups
H*(Y, Z) are countable in all degrees.
This last statement applies in particular to the universal cover p : Y1 ! X
of the realization X. Then the Hurewicz theorem (in its classical form _ see
[15], for example) implies that
ß2X ~=ß2Y1 ~=H2(Y1, Z)
3
is countable.
Inductively, one shows that the nconnected covers Yn ! X have countable
homology, and in particular countable groups
ßn+1X ~=ßn+1Yn ~=Hn+1(Yn, Z).
___
The class of trivial cofibrations of simplicial sets satisfies a bounded cof*
*ibra
tion condition:
Lemma 2. Suppose that A is a countable simplicial set, and that there is a
diagram
X
i
fflffl
A _____//Y
of simplicial set maps in which i is a trivial cofibration. Then there is a cou*
*ntable
subcomplex D Y such that A ! Y factors through D, and such that the map
D \ Y ! D is a trivial cofibration.
Proof. We can assume that A is a connected subcomplex of Y . The homotopy
groups ßi(A) are countable by Lemma 1.
Suppose that x is a vertex of A = B0. Then there is a finite connected
subcomplex Lx Y whichScontains a homotopy x ! i(y) where y is a vertex
of X. Write C1 = A [ ( xLx). Suppose that w, z are vertices of C1 \ X which
are homotopic in C1. Then there is a finiteSconnected subcomplex Kw,z X
such that w ' z in Kw,z. Let B1 = C1[ ( w,zKw,z). Then every vertex of A is
homotopic to a vertex of C1 \ X inside C1, and any two vertices z, w 2 C1 \ X
which are homotopic in C1 are also homotopic in B1\ X. Observe also that the
maps B0 C1 B1 are ß0 isomorphisms.
Repeat this process countably many times to find a sequence
A = B0 C1 B1 C2 B2 . . .
S
of countable subcomplexes of Y . Set B = Bi. Then B is a countable sub
complex of Y such that ß0(B \ X) ~=ß0(B) ~=ß0(A) = *.
Pick x 2 B \ X. The same argument (which does not disturb the connectiv
ity) can now be repeated for the countable list of elements in all higher homot*
*opy_
groups ßq(B, x), to produce the desired countable subcomplex D Y . __
Lemma 3. Suppose that p : X ! Y is a map of simplicial sets which has the
right lifting property with respect to all inclusions @ n n. Then p is a weak
equivalence.
4
Proof. The map p is a homotopy equivalence, by a standard argument. In effect,
there is a commutative diagram
;_____//_X??___
 ____p___
 ___i___
fflffl_fflffl__
Y _1Y_//_Y
and then a commutative diagram
(1X ,ip)
X t X _____//_X;;____
 ________
 _______p
fflfflH__fflffl__
X x 1 poe__//Y
so that pi = 1Y and then H is a homotopy 1X ' ip. Here, oe : X x 1 ! X_is
the projection onto X. __
Lemma 4. Every map f : X ! Y of simplicial sets has factorizations
>Z>A
i """ AAAp
""" AA
"" f A__A
X _____________//BBY>>"
BBB """"
j BB__B"""q
W
where i is a trivial cofibration and p is a fibration, and j is a cofibration a*
*nd q
is a trivial fibration.
Proof. A standard transfinite small object argument based on Lemma 2 pro
duces the factorization f = p . i. Also, f has a factorization f = q . j, where*
* j is
a cofibration and q has the right lifting property with respect to all inclusio*
*ns_
@ n n. But then q is a trivial fibration on account of Lemma 3. __
Lemma 5. Every trivial fibration p : X ! Y has the right lifting property with
respect to all inclusions @ n n.
Proof. Find a factorization
j
X ____//_BW
BBB q
pBBB__Bfflffl
Y
5
where j is a cofibration and the fibration q has the right lifting property with
respect to all @ n n. Then q is a trivial fibration by Lemma 3, so that j is
a trivial cofibration. The lifting r exists in the diagram
1X
X _____//X>>"
j r""""p
fflffl"fflffl"
Z __q__//Y
It follows that p is a retract of q, and so p has the desired lifting property.*
* ___
Theorem 6. With these definitions, the category S of simplicial sets satisfies
the axioms for a closed simplicial model category.
Proof. The axioms CM1 , CM2 and CM3 have trivial verifications. The fac
torization axiom CM5 is a consequence of Lemma 4, while the axiom CM4 is
a consequence of Lemma 5.
The function spaces hom (X, Y ) are exactly as we know them: an nsimplex
of this simplicial set is a map X x n ! Y .
If i : A ! B and j : C ! D are cofibrations, then the induced map
(B x C) [AxC (A x D) ! B x D
is a cofibration, which is trivial if either i or j is trivial. The first part *
*of the
statement is obvious set theory, while the second part follows from the fact_th*
*at
the realization functor preserves products. __
Lemma 7. Suppose given a pushout diagram
g
A _____//C
i 
fflfflfflffl
B __g*_//D
where i is a cofibration and g is a weak equivalence. Then g* is a weak equiva
lence.
Proof. All simplicial sets are cofibrant, and this result follows from the stan*
*dard_
formalism for categories of cofibrant objects [4, II.8.5]. *
*__
The other axiom for properness, which says that weak equivalences are stable
under pullback along fibrations, is proved in Corollary 38.
6
2 Subdivision operators
Write NX for the poset of nondegenerate simplices of a simplicial set X, or
dered by the face relationship. Here "x is a face of y" means that the subcompl*
*ex
of X which is generated by x is a subcomplex of . Let BX = BNX denote
its classifying space. Any simplex x 2 X can be written uniquely as x = s(y)
where s is an iterated degeneracy and y is nondegenerate. It follows that any
simplicial set map f : X ! Y determines a functor f* : NX ! NY where f*(x)
is uniquely determined by f(x) = t . f*(x) with t an iterated degeneracy and
f*(x) nondegenerate.
Say that a simplicial set K is a polyhedral complex if K is a subcomplex of
BP for some poset P . The simplices of a polyhedral complex K are completely
determined by their vertices; in this case the nondegenerate simplices of K are
precisely those simplices x for which the list (vix) of vertices of x consists *
*of
distinct elements.
If P is a poset there is a map fl : BBP ! BP which is best descibed
categorically as the functor fl : NBP ! P which sends a nondegenerate simplex
x : n ! P to the element x(n) 2 P . This is the socalled äl st vertex map", and
is natural in poset morphisms P ! Q. In particular all ordinal number maps
` : m ! n induce commutative diagrams of simplicial set maps
`*
B m _____//B n
fl fl
fflffl fflffl
m ___`___// n
Similarly, if K BP is a polyhedral complex then flK takes values in K by
the commutativity of all diagrams
x*
B n ____//_BBP
fl fl
fflffl fflffl
n ___x__//_BP
arising from simplices x of K.
For a general simplicial set X, we write
sdX = lim!B n,
n!X
where the colimit is indexed over the simplex category of X. The object sdX
is called the subdivision of X. The maps fl : B n ! n together determine a
natural map fl : sdX ! X. Note that there is an isomorphism sd n ~=B n.
Suppose that x is a nondegenerate simplex of X. Then the inclusion
X induces an isomorphism N = \ NX. Every simplicial set X is a
colimit of the subcomplexes generated by nondegenerate simplices x. Also
7
the canonical maps sd n ~=B n ! BX which are induced by all simplices of
X together induce a natural map
ß : sdX ! BX.
The map ß is surjective, since every nondegenerate simplex x (and any string
of its faces) is in the image of some simplex oe : n ! X.
It follows that there is a commutative diagram
lim!sd_~=_// (1)
x2NX sdX
ß* ß
fflffl fflffl
lim!B_____//
x2NX BX
The bottom horizontal map lim!xB ! BX is surjective, because any string
x0 . . .xn of nondegenerate simplices of X is in the image of the correspond
ing string of nondegenerate simplices of the subcomplex . If ff 2 B
and fi 2 B map to the same element of BXn they are both images of
a string fl 2 B( \ )n. This element fl is in the image of some map
Bn ! B( \ )n. Thus there is a i 2 Bn which maps to both ff
and fi. It follows that ff and fi represent the same element in lim!xB, and*
* so
the map lim!xB ! BX is an isomorphism.
Lemma 8. The map ß : sdX ! BX is surjective in all degrees, and is a
bijection on vertices. Consequently, two simplices u, v 2 sdXn have the same
image in BX if and only if they have the same vertices.
Proof. We have already seen that ß is surjective.
For every vertex v 2 sdX there is a unique nondegenerate nsimplex x 2 X
of minimal dimension (the carrier of v) such that v lifts to a vertex of sd n
under the map x* : sd n ! sdX. Observe that
v = x*([0, 1, . .,.n])
by the minimality of dimension of x. We see from the diagram
sd nH
HH
x*  Hx*HHHH
fflfflH##
sdX __ß__//_BX
that ß(v) = . It follows that the function v 7! ß(v) = is injective. *
* ___
Let K be a polyhedral complex with imbedding K BP for some poset P .
Every nondegenerate simplex x of K can be represented by a monomorphism
of posets x : n ! P and hence determines a simplicial set monomorphism
8
x : n ! K. In particular, the map x induces an isomorphism n ~= K.
It follows from the comparison in the diagram (1) that the map ß : sdK ! BK
is an isomorphism for all polyhedral complexes K.
Suppose that L is obtained from K by attaching a nondegenerate nsimplex.
The induced diagram
sd @ n _____//sdK
i i*
fflffl fflffl
sd n ______//sdL
is a pushout, in which the maps i and i* are monomorphisms of simplicial sets.
It follows in particular that the subdivision functor sdpreserves monomorphisms
as well as pushouts (sd has a right adjoint).
Let C and D be subcomplexes of a simplicial set X such that X = C [ D.
Then the diagram of monomorphisms
N(C \ D) ____//_ND
 
 
fflffl fflffl
NC _______//_NX
is a pullback and a pushout of partially ordered sets, and the diagram
B(C \ D) ____//_BD (2)
 
 
fflffl fflffl
BC _______//_BX
is a pullback and a pushout of simplicial sets.
There is a homeomorphism h :  sd n !  n, which is the affinePmap that
takes a vertex oe = {v0, . .,.vk} to the barycentre boe= _1_k+1vi. There is a
convex homotopy H : h ' fl which is defined by H(ff, t) = th(ff)+(1t)fl(ff*
*).
The homeomorphism h and the homotopy H respect inclusions of simplices.
Instances of the map h and homotopy H can therefore be patched together to
give a homeomorphism ~
h :  sdK =!K
and a homotopy
H : h ' fl
for each polyhedral complex K. The homeomorphism h and the homotopy H
both commute with inclusions of polyhedral complexes.
3 Classical simplicial approximation
In this section, "simplicial complex" has the classical meaning: a simplicial
complex K is a set of nonempty subsets of some vertex set V which is closed
9
under taking subsets. In the presence of a total order (V, ) on V , a simplici*
*al
complex K determines a unique polyhedral subcomplex K BV in which an
nsimplex oe 2 BV is in K if and only if its set of vertices forms a simplex of
the simplicial complex K.
Any map of simplicial complexes f : K ! L in the traditional sense deter
mines a simplicial set map f : K ! L by first imposing an orientation on the
vertices of L, and then by choosing a compatible orientation on the vertices of
K. It is usually, however, better to observe that a simplicial complex map f
induces a map f* : NK ! NL on the corresponding posets of simplices, and
hence induces a map f* : BNK ! BNL of the associated subdivisions.
Suppose given maps of simplicial complexes
K __ff_//X
i 
fflffl
L
where i is a cofibration (or monomorphism) and L is finite. Suppose further
that there is a continuous map f : L ! X such that the diagram
K__ff*//_X==
__
i* ____
fflfflf__
L
commutes. There is a subdivision sdnL of L such that in the composite
n f
 sdnL h!L ! X,
every simplex oe  sdnL maps into the star st(v) of some vertex v 2 X.
Recall that st(v) for a vertex v can be characterized as an open subset of
X by
st(v) = X  Xv,
where Xv is the subcomplex of X consisting of those simplices which do not
have v as a vertex.P One can also characterize st(v) as the set of those linear
combinations ffvv 2 X such that ffv 6= 0. Note that the star st(v) of a
vertex v is convex.
The homeomorphism h :  sdK ! K is defined on vertices by sending oe to
the barycentre boe2 oe. Observe that if oe0 . . .oen is a simplex of sdKPand
v is a vertex of somePoei then the image of any affine linear combination ffi*
*oei
is the affine sum ffiboeiof the barycentres. Then since v appears nontrivial*
*ly
in boeiit must appear nontrivially in the sum of the barycentres. This means
that h(st(oe)) st(fl(oe)), where fl : sdK ! K is the last vertex map. In other
words fl is a simplicial approximation of the homeomorphism h, as defined by
Spanier [14].
10
It follows that fln is a simplicial approximation of hn; in effect,
hn(st(v)) hn1(st(fl(v)) hn2(st(fl2(v)) . . .
There is a corresponding convex homotopy H : fln ! hn defined by
H(x, t) = (1  t)fln(x) + thn(x)
which exists precisely because fln is a simplicial approximation of hn.
The point is now that the composite
n f
 sdnL h!L ! X,
admits a simplicial approximation for n sufficiently large since fhn(st(v))
st(OE(w)) for some vertex OE(w) of X, and the assignment w 7! OE(w) defines a
simplicial complex map OE : sdnL ! sdX ! X whose realization OE* is ho
motopic to fhn by a convex homotopy no matter how the individual vertices
OE(w) are chosen subject to the condition on stars above. In particular, the
function w 7! OE(w) can be chosen to extend the vertex map underlying the
simplicial complex map fffln. It follows that there is a simplicial complex map
OE : sdnL ! X such that the diagram of simplicial complex maps
fln ff
sdnK ____//_K____//X66mmm
mmm
i* mmmmmm
fflfflmOEmm
sdn L
commutes, and such that OE ' fhn via a homotopy H0 that extends the homo
topy ffH : fffln ! ffhn.
The homotopy fH : ffln ! fhn also extends the homotopy ffH. It follows
that there is a commutative diagram
(s0ffH,(fH,H0 ))
( sdnK x 2) [ ( sdnL x _22)_______//X44iiii
iii
 iiiii
 iiiiiKi
fflffliii
 sdnL x 2
Then the composite
2 n K
 sdnL x 1 1xd! sd L x 2 ! X
is a homotopy from OE to the composite ffln rel  sdnK, and we have proved
Theorem 9. Suppose given simplicial complex maps
K __ff_//X
i
fflffl
L
11
where i is an inclusion and L is finite. Suppose that f : L ! X is a contin*
*uous
map such that fi = ff. Then there is a commutative diagram of simplicial
complex maps
fln ff
sdnK ____//_K____//X66mmm
mmm
i mmmmmm
fflfflmOEmm
sdn L
such that OE ' ffln rel  sdnK.
One final wrinkle: the maps in the statement of Theorem 9 are simplicial
complex maps which may not reflect the orientations of the underlying simplicial
set maps. One gets around this by subdividing one more time: the corresponding
diagram
Nfln Nff
N sdnK _____//NK ____//_NX55jjj
jjjj
Ni  jjjjjj
fflffljjjNOE
N sdnL
of poset morphisms of nondegenerate simplices certainly commutes, and hence
induces a commutative diagram of simplicial set maps
BNfln BNff
BN sdnK _____//BNK _____//BNX44iii
iiii
BNi  iiiiiii
fflffliiiiBNOE
BN sdnL
It follows that there is a commutative diagram of simplicial set maps
fln+1// ff //
sdn+1K _____K _____6X6mmmm
mmm
i mmmmm
fflfflmOEflmm
sdn+1 L
provided that the original maps ff and i are themselves morphisms of simplicial
sets. Finally,there is a homotopy OE ' ffln rel  sdnK, so that OEfl ' f*
*fln+1
rel  sdn+1K. We have proved the following:
Corollary 10. Suppose given simplicial set maps
K __ff_//X
i
fflffl
L
12
between polyhedral complexes, where i is a cofibration and L is finite. Suppose
that f : L ! X is a continuous map such that fi = ff. Then there is a
commutative diagram of simplicial set maps
fln ff
sdnK ____//_K____//X66mmm
mmm
i mmmmmm
fflfflmOEmm
sdn L
such that OE ' ffln rel  sdnK.
4 Approximation results for simplicial sets
Note that sd( n) = C sd(@ n), where in general CK denotes the cone on a
simplicial set K. This is a consequence of the following
Lemma 11. Suppose that P is a poset, and that CP is the poset cone, which
is constructed from P by formally adjoining a terminal object. Then there is an
isomorphism BCP ~=CBP .
Proof. Any functor fl : n ! CP determines a pullback diagram
k ______//P
 
 
fflffl fflffl
n _____//CP
where k is the maximum vertex in n which maps into P . It follows that
BCPn = BPn t BPn1 t . .t.BP0 t {*},
where the indicated vertex * corresponds to functors n ! CP which take all
vertices into the cone point. The simplicial structure maps do the obvious thing
under this set of identification, and so BCP is isomorphic to CBP (see_[4],
p.193). __
Following [2], say that a simplicial set X is regular if for every nondegen*
*erate
simplex ff of X the diagram
d0ff//
n1 _____ (3)
d0 
fflffl fflffl
n ___ff__//
is a pushout.
It is an immediate consequence of the definition (and the fact that trivial
fibrations are closed under pushout) that all subcomplexes of a regular
simplicial set X are weakly equivalent to a point. We also have the following:
13
Lemma 12. Suppose that X is a simplicial set such that all subcomplexes
which are generated by nondegenerate simplices ff are contractible. Then the
canonical map ß : sdX ! BX is a weak equivalence.
Proof. We argue along the sequence of pushout diagrams
F
ff2NnX@____//skn1X
 
 
F fflffl fflffl
ff2NnX______//_sknX
The property that all nondegenerate simplices of X generate contractible sub
complexes is shared by all subcomplexes of X, so inductively we can assume
that the natural maps ß : sd@ ! B@ and ß : sdskn1X ! B skn1are
weak equivalences.
But the comparison map fl : sd ! is a weak equivalence, and
is contractible by assumption. At the same time B is a cone on B@ by
Lemma 11, so the comparison ß : sd ! B is a weak equivalence for all
nondegenerate simplices ff. The gluing lemma (see also (2)) therefore implies_
that the map ß : sdsknX ! B sknX is a weak equivalence. __
Corollary 13. The canonical map ß : sdX ! BX is a weak equivalence for
all regular simplicial sets X.
Write N*K for the poset of nondegenerate simplices of K, with the opposite
order, and write B*K = BN*K for the corresponding polyhedral complex. The
cosimplicial space n 7! B* n determines a functorial simplicial set
sd*X = lim!B* n,
n!X
and the "first vertex maps" fl* : B* n ! n together determine a functorial
map fl* : sd*X ! X. Similarly, the maps B* n ! B*X induced by the
simplices n ! K of K together determine a natural simplicial set map ß* :
sd* X ! B*X. Observe that the map ß* : sd* n ! B* n is an isomorphism.
We shall say that sd*X is the dual subdivision of the simplicial set X.
Lemma 14. The simplicial set sd*X is regular, for all simplicial sets X.
Proof. Suppose that ff is a nondegenerate nsimplex of sd*X. Then there is
a unique nondegenerate rsimplex y of X of minimal dimension (the carrier of
ff) and a unique nondegenerate nsimplex oe 2 sd* r such that the classifying
map ff : n ! sd*X factors as the composite
n oe!sd* r y*!sd*X.
This follows from the fact that the functor sd*preserves pushouts and monomor
phisms. Observe that oe(0) = [0, 1, . .r.], for otherwise oe 2 sd@ r and r is n*
*ot
minimal.
14
The composite diagram
n1 ____//_sd*@ r____//sd*@ (4)
d0   
fflffl fflffl fflffl
n __oe__//_sd* r_____//_sd*
is a pullback (note that all vertical maps are monomorphisms), and the diagram
(3) factors through (4) via the diagram of monomorphisms
____//sd*@
 
 
fflffl fflffl
_____//_sd*
It follows that the diagram (3) is a pullback.
If two simplices v, w of n map to the same simplex in , then oe(v)
and oe(w) map to the same simplex of sd*. But then oe(v) = oe(w) or both
simplices lift to sd*@ r, since sd*preserves pushouts and monomorphisms. If
oe(v) = oe(w) then v = w since oe is a nondegenerate simplex of the polyhedral
complex sd* r. Otherwise, oe(v) and oe(w) both lift to sd*@ r, and so v and
w are in the image of d0. Thus all identifications arising from the epimorphism
n ! take place inside the image of d0 : n1 ! n, and the square_(4)
is a pushout. __
Proposition 15. Suppose that X is a regular simplicial set. Then the dotted
arrow exists in the diagram
sdX __ß__//BX__
____
OE __________
fflffl______
X
making it commute.
Proof. All subcomplexes of a regular simplicial set are regular, so it's enough
to show (see the comparison (1)) that the dotted arrow exists in the diagram
sd_ß__//B_
_____
OE ___________
fflffl_____
for a nondegenerate simplex ff, subject to the obvious inductive assumption on
15
the dimension of ff: we assume that there is a commutative diagram
sd_ß__//B
ss
OE sssss
fflfflOE*yyss
Consider the pushout diagram
d0ff//
n1 _____
d0 
fflffl fflffl
n ___ff__//
Then given nondegenerate simplices u, v of n1r, = in if*
* and
only if either u = v or u, v 2 d0 n1 and = in .
Suppose given two strings u1 . . .uk and v1 . . .vk of nondegenerate
simplices of n such that = in for 1 i k. We want to
show that these elements of (sd n)k map to the same element of under the
composite map
sd n OE! n ff!.
If this is true for all such pairs of strings, then there is an induced commuta*
*tive
diagram of simplicial set maps
ff*
sd n _____//B
OE OE*
fflffl fflffl
n ___ff__//
and the Proposition is proved.
We assume inductively that the corresponding diagram
d0ff*//
sd n1 _____ B
OE OE*
fflffl fflffl
n ___d0ff__//
exists for d0ff.
Set i = k +1 if all uiand viare in d0 n1. Otherwise, let i be the minimum
index such that ui and vi are not in d0 n1. Observe that a nondegenerate
simplex w of n is outside d0 n1 if and only if 0 is a vertex of w.
If i = k + 1 the strings u1 . . .uk and v1 . . .vk are both in the
image of the map d0*: sd n1 ! sd n, and can therefore be interpreted as
16
elements of sd n1 which map to the same element of B. These strings
therefore map to the same element in , and hence to the same element of
.
If i = 0 the strings are equal, and hence map to the same element of .
Suppose that 0 < i < k + 1. Then the simplices uj = vj have more than one
vertex (including 0), and so the last vertices of uj and d0uj coincide for j *
*i.
It follows that the strings
u1 . . .ui1 d0ui . . .d0uk
and
v1 . . .vi1 d0vi . . .d0vk
determine elements of sd n1 having the same images under the map OE :
sd n ! n as the respective original strings. These strings also map to the
same element of B since d0uj = d0vj for j i. The strings u1 . . .uk_
and v1 . . .vk therefore map to the same element of . __
Lemma 16. Suppose given a diagram
A __ff//_X
i f
fflffl fflffl
B __fi_//Y
in which i is a cofibration and f is a weak equivalence between objects which a*
*re
fibrant and cofibrant. Then there is a map ` : B ! X such that ` . i = ff and
f . ` is homotopic to fi rel A.
Proof. The weak equivalence f has a factorization
j
X _____//@@Z
@@ q
f@@@__fflffl
Y
where q is a trivial fibration and j is a trivial cofibration. The object Z is *
*both
cofibrant and fibrant, so there is a map ß : Z ! X such that ß . j = 1X and
j . ß ' 1Z rel X. Form the diagram
jff
A _____//Z>>~
~~
i !~~ q
fflffl~fflffl~
B __fi_//Y
Then the required lift B ! X is ß . !. ___
17
Theorem 17. Suppose given maps of simplicial sets
A __ff_//X
i
fflffl
B
where i is a cofibration of polyhedral complexes and B is finite, and suppose t*
*hat
there is a commutative diagram of continuous maps
ff
A_____//X>>
__
i____
fflfflf__
B
Then there is a diagram of simplicial set maps
fl*flm ff
sdmsd*A ____//_A____//X66lll
lll
i* llllll
fflfflllOEl
sdmsd*B
such that
OE ' ffl*flm  :  sdmsd*B ! X
rel  sdmsd*A
Proof. The simplicial set sd*X is regular (Lemma 14), and there is a (natural)
commutative diagram
sd sd*X __c__//B sd*X
rr
fl rrrrr
fflffl~flxxrrr
sd*X
by Proposition 15. On account of Lemma 16, there is a continuous map f~:
 sdsd*B !  sdsd*X such that the diagram
ff
 sdsd*A_____//8sdsd*X8
qqq
i qqqqq
fflfflf~qqq
 sdsd*B
commutes and such that fl*flf~' ffl*fl rel  sdsd*A. Now consider the dia
18
gram
cff*
 sdsd*A_____//B8sd*X8
qqq
i* qqqq
fflfflcf~qqq
 sdsd*B
Then by applying Corollary 10 to the continuous map cf~the polyhedral com
plex map cff* and the cofibration of polyhedral complexes i*, we see that there
is a diagram of simplicial set maps
fln cff*
sdnsdsd*A _____//sdsd*A_____//B4sd*X4hhh
hhhh
i* hhhhhhhh
fflfflhhhh _
sdnsdsd*B
such that _ ' cf~fln rel  sdnsdsd*A. It follows that
fl*~fl_ ' fl*~flcf~fln = fl*flf~fln ' ffl*flfln.
Thus OE = fl*~fl_ is the required map of simplicial sets, where m = n + 1. *
*___
Corollary 18. Suppose given maps of simplicial sets
A __ff_//X
i
fflffl
B
where i is a cofibration and B is finite, and suppose that there is a commutati*
*ve
diagram of continuous maps
ff
A_____//X>>
__
i____
fflfflf__
B
Then there is a diagram of simplicial set maps
fl*flfl*flmff
sdm sd*sdsd*A _____//A____//_X55kkkk
kkk
i* kkkkkk
fflfflkkkOEk
sdm sd*sdsd*B
such that
OE ' ffl*flfl*flm  :  sdmsd*sd sd*B ! X
rel  sdmsd*sd sd*A
19
Proof. The cofibration i induces a cofibration of polyhedral complexes
i* : B sd*A ! B sd*B.
The simplicial set maps
~fl fl* ff
B sd*A _____//sd*A_____//A____//X
i*
fflffl
B sd*B
and the composite continuous map
B sd*B ~fl! sd*B fl*!B f!X
satisfy the conditions of Theorem 17. ___
Suppose that K is a polyhedral complex, and recall that NK denotes the
poset of nondegenerate simplices of K with face relations, with nerve BK =
BNK ~=sd K. Recall also that N*K = (NK)op is the dual poset; it has the
same objects as NK, namely the nondegenerate simplices of K, but with the
reverse ordering. The nerve BN*K coincides with the dual subdivision sd*K
of K.
The poset NBK of nondegenerate simplices of BK has as objects all strings
oe : oe0 < oe1 < . .<.oeq (5)
of strings of nondegenerate simplices of K with no repeats. The face relation
in NBK corresponds to inclusion of strings. The poset NB*K has as objects
all strings
ø0 > ø1 > . .>.øp
of nondegenerate simplices of K with no repeats, with the face relation again
given by inclusion of substrings. Reversing the order of strings defines is a p*
*oset
isomorphism ~
OEK : NBK =!NB*K
which is natural in polyhedral complexes K. The poset isomorphism OEK induces
a natural isomorphism
~=
K : sdsdK ! sdsd*K
of associated nerves.
The composite
sdsd n fl!sd n fl! n
is induced by the poset morphisms
NBN n fl!N n fl!n
20
which are defined by successive application of the last vertex map. Thus, this
composite sends the object oe (as in (5) to oeq(m) 2 n, where the poset inclusi*
*on
oeq : m ! n defines the msimplex oeq 2 n. The composite of poset morphisms
NBN n OE!NBN* n fl!N* n fl*!n
(where fl* is the first vertex map) sends the object oe to the element oe0(0) 2*
* n.
There is a relation oe0(0) oeq(m) in the poset n which is associated to all s*
*uch
objects oe. These relations define a homotopy NBN n x 1 ! n from fl*flOE
to flfl. The maps and the homotopy respect all ordinal number morphisms
` : m ! n.
It follows, by applying the nerve construction that there is an explicit sim
plicial homotopy H : sdsd n x 1 ! n from fl*fl * to flfl, and that this
homotopy is natural in ordinal number maps. Glueing together instances of the
isomorphisms * : sdsd( n) ! sdsd*( n) along the simplex for a simplicial
set X therefore determines an isomorphism
~=
X : sdsdX ! sdsd*X (6)
and a natural homotopy
H : sdsdX x 1 ! X (7)
from the composite
sd sdX X!~=sdsd*X fl!sd*X fl*!X
to the composite
sdsdX fl!sdX fl!X.
5 Excision
Lemma 19. Suppose that U1 and U2 are open subsets of a topological space
Y such that Y = U1 [ U2. Suppose given a commutative diagram of pointed
simplicial set maps
K __ff_//S(U1) [ S(U2)
i 
fflffl fflffl
L ____fi___//S(Y )
where i is an inclusion of finite polyhedral complexes. Then for some n the
composite diagram
fln ff
sdnK ____//_K____//S(U1) [ S(U2)
i* 
fflffl fflffl
sdnL _fln_//_L____fi__//S(Y )
21
is pointed homotopic to a diagram
sdnK _____//S(U1)8[8S(U2)___
______
i* ____________
fflffl_______ fflffl
sdnL _________//S(Y )
admitting the indicated lifting.
Proof. There is an n such that the composite
n Sfi*
sdn L j!S sdnL Sh!SL ! SY
factors uniquely through a map ~fi: sdnL ! S(U1) [ S(U2), where fi* : L ! Y
is the adjoint of fi.
Suppose that r K is a nondegenerate simplex of K. The diagram
n
 sdn r_h__//_ r
i* i*
fflffl fflffl
 sdnL_hn__//L
is homotopic to the diagram
fln r
 sdn r____//_ 
i* i*
fflffl fflffl
 sdnL_fln//_L
and the homotopies of such diagrams respect inclusions between nondegenerate
simplices of K. Thus, each composite diagram
fln ff
sdn r ____//_ r____//S(U1) [ S(U2)
i* 
fflffl fflffl
sdnL _fln__//L____fi___//S(Y )
is homotopic to a diagram
j n rShn rSff*
sdn r _____//S sd ___//_S ____//S(U1)2[2S(U2)eeee
eeeeee
i* eeeeeeeeeee 
fflffleeeeeeeeeeenf~i fflffl
sdn L __j__//_S sd LShn_//SL__Sfi*__//_S(Y )
22
and the homotopies respect inclusions between nondegenerate simplices of K.
Note that the map ff : r ! S(U1) [ S(U2) factors through some S(Ui) so that
the ä djoint" ff* is induced by a map  r ! Ui. Observe also that the maps h
and fln coincide, and the homotopy between them is constant on the vertices
of K.
It follows that the composite diagram
fln ff
sdnK ____//_K____//S(U1) [ S(U2)
i* 
fflffl fflffl
sdnL _fln_//_L____fi__//S(Y )
is pointed homotopic to a diagram
(~fii)
sdn K ____________________________//S(U1)2[2S(U2)fff
ffffff
i* ffffffffff 
fflfflffffffffffn~fi fflffl
sdn L __j__//S sd LShn//_SL_Sfi*_//_S(Y )
___
Theorem 20. Suppose that U1 and U2 are open subsets of topological space Y ,
and suppose that Y = U1 [ U2. Then the induced inclusion of simplicial sets
S(U1) [ S(U2) S(Y ) is a weak equivalence.
Proof. First of all observe that the induced function
ß0S(U1 [ U2) ! ß0S(Y )
is a bijection, by subdivision of paths.
Pick a base point x 2 Y , and let FxY denote the category of all finite
pointed subcomplexes of S(Y ) containing x, ordered by inclusion. This category
is plainly filtered, and there is an isomorphism
ßnS(Y ) ~= lim!ßnK.
K2FxY
The natural weak equivalences fl0= fl*~fl: B(sd*K) ! K resulting from Lemma
14 and Proposition 15 may be used to replace a finite simplicial set K by a fin*
*ite
polyhedral complex B(sd*K).
Suppose that [ff] 2 ßq(S(Y ), x) is carried on a finite subcomplex ! : K
S(Y ) in the sense that [ff] = !*[ff0] for some [ff0] 2 ßqK. Then it follows *
*from
Lemma 19 that there is an r 0 such that the diagram
fl0flr x
sdrB(sd* 0) _~=_//_ 0____//S(U1) [ S(U2)
  
 x i
fflffl fflffl fflffl
sdr B(sd*K) _fl0flr//_K__!____//_S(Y )
23
is pointed homotopic to a diagram
sdrB(sd* 0) __x_//_S(U1)6[6S(U2)__
_____
 __________
 ___oe______ i
fflffl______ fflffl
sdrB(sd*K) _________//_S(Y )
in which the indicated lift oe exists. But fl0flr is a weak equivalence, so th*
*at
[ff0] = (fl0flr)*[ff00] for some ff00. But then [ff] = !*(fl0flr)*[ff00] = i*oe*
**[ff00] so that
i* is surjective on homotopy groups.
Suppose that [fi] 2 ßqS(U1) [ S(U2) is carried on the subcomplex K
S(U1)[S(U2) and suppose that i*[fi] = 0. Then there is a commutative diagram
of simplicial set inclusions
i1
K _____//S(U1) [ S(U2)
j 
fflffl fflffl
L ____i2___//S(Y )
such that [fi] 7! 0 in ßqL. There is an s 0 such that the composite diagram
0fls i1
sdsB(sd*K) _fl__//K_____//S(U1) [ S(U2)
j* i
fflffl fflffl
sdsB(sd*L) _fl0fls//_L__i2__//S(Y )
is pointed homotopic to a diagram
i01
sdsB(sd*K) _____//S(U1)6[6S(U2)__
________
j* ___ø________i___
fflffl_______ fflffl
sdsB(sd*L) __i0_____//S(Y )
2
in which the indicated lifting exists. Again, the maps fl0fls are weak equivale*
*nces,
so that [fi] = (fl0fls)*[fi0] for some [fi0] 2 ßq sdsB(sd*K) and
i1*[fi] = i1*(fl0fls)*[fi0] = i01*[fi0] = ø*j*[fi0].
Finally, (fl0fls)*j*[fi0] = j*[fi] = 0 so that j*[fi0] = 0 in ßq sdsB(sd*L)_a*
*nd so
i1*[fi] = 0 in ßqS(U1) [ S(U2). __
The category S of simplicial sets is a category of cofibrant objects for a
homotopy theory, for which the cofibrations are inclusions of simplicial sets
24
and the weak equivalences are those maps f : X ! Y which induce weak
equivalences f* : X ! Y  of CW complexes. As such, it has most of the usual
formal calculus of homotopy cocartesian diagrams (specifically II.8.5 and II.8.8
of [4]).
Lemma 21. Suppose that the diagram
F n1 _____//
i S X
 
 
F fflffl fflffl
i en ______//_Y
is a pushout in the category of CW complexes. Then the diagram
F n1 _____//
i S(S ) S(X)
 
 
F fflffl fflffl
iS(en) ______//_S(Y )
is a homotopy cocartesian diagram of simplicial sets.
Proof. The usual classical arguments say that one can find an open subset U
Y such that X U and this inclusion is a homotopy equivalence. The set U is
constructed by fattening up each sphere Sn1 to an open subset Uiof the ncell
en (by radial projection) such that Sn1 Ui is a homotopy equivalence. We
can therefore assume that the inclusion
G G
Sn1 ( en) \ U
i i
is a homotopy equivalence. We can also assume that there is an open subset
Vi en such that the inclusion is a homotopy equivalence, such that Vi\Ui Ui
is a homotopy equivalence, and such that en = Vi[ Ui. The net result is a
commutative diagram
F n1 ______//_
iS(S ) S(X)
'  III '
' fflfflF fflffl
S(V \ U)_____//S(U \ ( ien))___//S(U)
  
 I  II 
fflffl F fflffl fflffl
S(V )____'____//_iS(en)_______//S(Y )
of simplicial set homomorphisms in which all vertical maps are cofibrations and
the labelled maps are weak equivalences. The the composite diagram I + II
is homotopy cocartesian by excision (Lemma 20), so that the diagram II is
homotopy cocartesian by the usual argument. It follows that the composite_
diagram III+ IIis homotopy cocartesian, again by a standard argument. __
25
Theorem 22. The adjunction map ffl : S(T ) ! T is a weak equivalence for all
spaces T .
Proof. The functor T 7! S(T ) preserves fibrations and trivial fibrations, and
thus preserves weak equivalences since all spaces are fibrant. In particular, t*
*he
functor T 7! S(T ) preserves weak equivalences. We can therefore presume
that T is a CW complex.
All cells en are contractible spaces, so that the natural maps ffl : S(en *
*! en
are weak equivalences. If the diagram
F n1 _____//
i S X (8)
 
 
F fflffl fflffl
ien ______//_Y
is a pushout in the category of CW complexes, then it follows from Lemma 21
that the induced diagram
F n1 _____//
i S(S ) S(X) (9)
 
 
F fflffl fflffl
iS(en)______//_S(Y )
is homotopy cocartesian. It follows by induction on dimension that the maps
ffl : S(Sn1) ! Sn1 are weak equivalences. The general case follows by
comparison of the homotopy cartesian diagrams (8) and (9), and the usual_sort_
of transfinite induction. __
The following is now a consequence of Theorem 22 and a standard adjoint
ness trick:
Corollary 23. The canonical map j : X ! SX is a weak equivalence for all
simplicial sets X.
6 The Milnor Theorem
Write Sf for the full subcategory of the simplicial set category whose objects *
*are
the fibrant simplicial sets. All fibrant simplicial sets X are Kan complexes, a*
*nd
therefore have combinatorially defined homotopy groups ßn(X, x), n 1, x 2
X0, as well as sets of path components ß0X. Say that a map f : X ! Y of
fibrant objects is a combinatorial weak equivalence if it induces isomorphisms
ß0X ~=ß0Y and ßn(X, x) ~=ßn(Y, f(x)) for all n and x. Recall that any fibre
sequence
Fy __i__//X
 p
 
fflffl fflffl
0 __y__//Y
26
(ie. pullback, with p a fibration) induces a long exact sequence in homotopy
groups
. .!.ß2(Y, y) @!ß1(Fy, x) i*!ß1(X, x) p*!ß1(Y, y) @!ß0Fy i*!ß0X p*!ß0Y
for any choice of vertex x 2 Fy.
Lemma 24. A map p : X ! Y between fibrant simplicial sets is a fibration and
a combinatorial weak equivalence if and only if it has the right lifting proper*
*ty
with respect to all inclusions @ n n.
Proof. If p has the right lifting property with respect to all @ n n then it
has the right lifting property with respect to all cofibrations, and therefore *
*has
the right lifting property with respect to all trivial cofibrations. It follows*
* that
p is a fibration. The map p is also a homotopy equivalence since X and Y are
fibrant, by a standard argument, so it is a combinatorial weak equivalence.
The reverse implication is the standard argument: see [4, I.7.10], and_also
the proof of Lemma 33 below. __
Lemma 25. The category Sf of all fibrant simplicial sets, together with the
classes of all fibrations and combinatorial weak equivalences in the category,
satisfies the axioms for a category of fibrant objects for a homotopy theory.
Proof. With Lemma 24 and the closed simplicial model structure of Theorem 6
in place, the only axiom that requires proof is the weak equivalence axiom. In
other words we have only to prove that, given a commutative triangle
f
X ____//_@@Y
@@ g
h@@@__fflffl
Z
of morphisms between fibrant simplicial sets, if any two of the maps are combi
natorial weak equivalences then so is the third. This is a standard argument [4*
*,_
I.8.2], which uses a combinatorial construction of the fundamental groupoid. *
*__
I shall say that a finite anodyne extension is an inclusion K L of simplic*
*ial
sets, such that there are subcomplexes
K = K0 K1 . . .KN = L
such that there are pushout diagrams
ms______//Ki
 
 
fflffl fflffl
m _____//Ki+1
27
The notation means that Ki+1 is constructed from Ki by explicitly attaching a
simplex to a horn in Ki.
Recall [4] that a cofibration is said to be an anodyne extension if it is a
member of the saturation of the set of all inclusions nk n. In other words,
the class of anodyne extensions is generated by all inclusions of horns in simp*
*lices
under processes involving disjoint union, pushout and filtered colimit, and is
closed under retraction. All anodyne extensions are weak equivalences.
Lemma 26. The functors sd and sd*preserve finite anodyne extensions.
Proof. We will prove that the subdivision functor sd preserves finite anodyne
extensions. The corresponding statement for sd*has a similar proof.
It suffices to show that all induced maps sd nk! sd n are finite anodyne
extensions. This will be done by induction on n; the case n = 1 is obvious.
Here is the outline of the proof. It is a consequence of Lemma 11 that
sd n coincides up to isomorphism with the cone C sd@ n on sd n. The cone
functor C takes the inclusion @ r ! r to the anodyne extension r+1r+1 r+1,
and hence takes all inclusions K L of finite simplicial sets to finite anodyne
extensions CK ! CL. There is a commmutative diagram
sd nk


fflffl
C sd@ n1 _____//_C sd nk
 
 
fflffl fflffl
C sd n1 ______//C sd@ n
in which the square is a pushout since the cone and subdivision functors both
preserve pushouts. The map C sd nk! C sd@ n is therefore an anodyne
extension. It thus suffices to show that the canonical map sd nk! C sd nkis a
finite anodyne extension.
Note that nkhas a filtration by subcomplexes Fr, where Fr is generated
by the nondegenerate rsimplices which have k as a vertex. Then F0 = {k},
Fn1 = nk, and there are pushout diagrams
F r
x2F(r)r j____//_Fr1
 
 
F fflffl fflffl
r_____//_
x2F(r)r Fr
where Fr(r)denotes the set of rsimplices in Fr. In particular, the map 0 nk
arising from the inclusion of the vertex k is a finite anodyne extension. It al*
*so
follows, by induction, that the map
0 = sd 0 ! sd nk
28
which is induced by applying sd to the inclusion {k} nkis a finite anodyne
extension. __
The proof is completed in Lemma 27 below. __
Lemma 27. Suppose that v : 0 ! K is a finite anodyne extension for some
choice of vertex v in a finite complex K. Then the canonical inclusion K ! CK
is a finite anodyne extension.
Proof. Suppose given a pushout diagram
nk__ff_//K
 
 i
fflffl fflffl
n _____//L
where there is some vertex v 2 K such that the corresponding map v : 0 ! K
is finite anodyne. Assume inductively that the map N ! CN is anodyne for all
finite complexes constructed in fewer stages than L, and for all N constructed *
*by
adjoining simplices of dimension smaller than n. Then the inclusions K ! CK
and nk! C nkare both anodyne, and there are pushout diagrams
K _________//_L
 
 
fflffl fflffl
CK _____//CK [K L
and
C nk[ nk n _____//CK [K L
 
 
fflffl fflffl
C n ___________//CL
The cofibration
C nk[ nk n ! C n
is isomorphic to the anodyne extension n+1k n+1. ___
For a simplicial set X, the simplicial set Ex X has nsimplices Ex Xn =
hom (sd n, X). The functor X 7! Ex X is right adjoint to the subdivision
functor A 7! sdA. It follows from Lemma 26 that Ex X is a Kan complex if X
is a Kan complex; it is easier to see that Ex X is fibrant if X is fibrant. Wri*
*te
fl : X ! Ex X for the natural simplicial set map which is adjoing to the map
fl : sdX ! X.
Lemma 28. Suppose that X is a Kan complex. Then the map fl : X ! Ex X
is a combinatorial weak equivalence.
29
Proof. The functor Ex preserves Kan fibrations on account of Lemma 26, and
the map fl plainly induces a bijection
ß0X ~=ß0Ex X.
The functor Ex also preserves those fibrations which have the right lifting pro*
*p
erty with respect to all @ n ! n, since the subdivision functor sd preserves
inclusions of polyhedral complexes.
Pick a base point x 2 X, and construct the corresponding comparison of
fibre sequences
X ________//P X_______//X
fl fl fl
fflffl fflffl fflffl
Ex X ____//_ExP X____//_ExX
Then Ex P X is simplicially contractible, and so there is an induced diagram
~=
ß1X ________//ß0 X
 ~
 =
fflffl fflffl
ß1Ex X __~=_//ß0Ex X
It follows that the induced map ß1X ! ß1Ex X is an isomorphism for all choices
of base points in all Kan complexes X.
This construction may be iterated to show that the induced map ßnX !
ßn ExX is an isomorphism for all choices of base points in all Kan complexes_
X, and for all n 0. __
There is a similar description of a functorially constructed simplicial set
Ex *X has nsimplices Ex* Xn = hom (sd* n, X). The functor X 7! Ex* X is
right adjoint to the (dual) subdivision functor A 7! sd*A. The dual subdivi
sion functor also preserves weak equivalences, cofibrations and finite anodyne
extensions, and the natural map fl* : sd*A ! A is a weak equivalence. It fol
lows that Ex* X is a Kan complex if X is a Kan complex, and that Ex* X is
fibrant if X is fibrant. Write fl* : Y ! Ex*Y for the adjoint of the natural map
fl* : sd*Y ! Y . The proof of the following result is formally the same proof as
Lemma 28:
Lemma 29. Suppose that X is a Kan complex. Then the map fl* : X ! Ex*X
is a combinatorial weak equivalence.
Theorem 30 (Milnor Theorem). Suppose that X is a Kan complex. Then
the canonical map j : X ! S(X) induces an isomorphism
ßi(X, x) ~=ßi(X, x)
for all vertices x 2 X and for all i 0.
30
In other words, Theorem 30 asserts the existence of an isomorphism between
the combinatorial homotopy groups of a Kan complex X and the ordinary ho
motopy groups of its topological realization X.
Proof of Theorem 30.The vertical arrows in the comparison diagram
ßi(X, x)____________//_ßi(SX, x)
 
 
fflffl fflffl
ßi(Ex mEx *X, x)____//_ßi(Ex mEx *SX, x)
are isomorphisms for all m by Lemma 28 and 29. The simplicial approximation
result Theorem 17 says that any element ßi(SX, x) lifts to some element of
ßi(Ex rEx*X, x) for sufficiently large r, and that any element of ßi(X, x) whic*
*h_
maps to 0 2 ßi(SX, x) must also map to 0 in ßi(Ex sEx*X, x) for some s. __
7 Kan fibrations
Write SD (X) for either the subdivision sdX of a simplicial set X or for the
dual subdivision sd*X, and let : SD (X) ! X denote the corresponding
canonical map. Similarly, write EX (X) for either Ex X or Ex* X, and also let
: X ! EX (X) denote the adjoint map.
Here is one of the more striking consequences of simplicial approximation
(Theorem 17 or Corollary 18): every simplicial set X is a Kan complex up to
subdivision. More explicitly, we have the following:
Lemma 31. Suppose that ff : nk! X is a map of simplicial sets. Then there
is an r 0 such that ff extends to n up to subdivision in the sense that there
is a commutative diagram
r ff
SDr( nk)____//_ nk___//X66lll
 llllll
 llll
fflffllll
SD r( n)
of simplicial set maps.
Proof. All spaces are fibrant, so there is a diagram of continuous maps
ff
 nk____//X==
 zzzz
 zzf
fflfflzz
 n
Now apply Theorem 17. ___
31
Remark 32. In fact, although it's convenient to do so for the moment we
do not have to mix instances of sd and sd* in the proof of Lemma 31 _ see
the proof of Lemma 39 below. The point is that the inclusion nk n of
polyhedral complexes induces a strong deformation retraction of the associated
realizations.
Lemma 33. Suppose that p : X ! Y is a Kan fibration and a weak equivalence.
Suppose that there is a commutative diagram
@ n _ff_//_X
 
 p
fflffl fflffl
n __fi__//Y
Then there is an r 0 and a commutative diagram
r ff
SD r(@ n)_____//@ n_____//X55kkk
kkk
 kkkk 
 kkkk p
fflfflkkk fflffl
SDr( n) ___r__// n__fi_//_Y
In other words all maps which are both Kan fibrations and weak equivalences
have the right lifting property with respect to all inclusions @ n n, up to
subdivision. We will do better than that, in Theorem 34.
Proof of Lemma 33. Suppose that i : K L is an inclusion of finite polyhedral
complexes. If the diagram
K __ff//_X (10)
 
 p
fflffl fflffl
L __fi_//_Y
is homotopic up to subdivision to a diagram for which the lifting exists, then
the lifting exists for the original diagram up to subdivision.
In effect, a homotopy up to subdivision is a diagram
h1
SD k(K x 1) ____//_X
 
 p
fflffl fflffl
SDk(L x 1) _h2__//_Y
32
It starts (up to subdivision) at the original diagram if the diagram
d1* k h1
SD k(K)_____//SD(K x 1) _____//X
 
 p
fflffl fflffl
SD k(L)__d1_//_SDk(L x 1)____//_Y
* h2
coincides with the diagram
k ff
SD k(K)_____//K_____//X (11)
 
 p
fflffl fflffl
SD k(L)___k_//_L_fi_//_Y
If the lifting exists at the other end of the homotopy in the sense that there *
*is
a commutative diagram
d0* k h1
SD k(K)_____//SD(K x 1) _____//X44hhhhh
hhh
 oehhhhh 
 hhhhhh p
fflfflhhhhh fflffl
SD k(L)__d0_//_SDk(L x 1)____//_Y
* h2
then there is a commutative diagram
d1* k k (h1,oe)
SDk(K) _____//SD(K x 1) [ SD (L)_____//X77nnn
  oe0nnnnn 
 j  nnnn p
fflffl fflfflnn fflffl
SDk(L) ____d1___//_SDk(L x 1)________//_Y
* h2
The map labelled j is a finite anodyne extension by Lemma 26, so the lifting oe0
exists. The outer square diagram is the diagram (11) and the composite oe0d1*is
the required lift.
The contracting homotopy h1 : n0x 1 ! n0onto the vertex 0 extends to
a homotopy of diagrams up to subdivision from the diagram (10) to a diagram
ff1
SDk(@ n) _____//X (12)
 
 p
fflffl fflffl
SD k( n) __fi1//_Y
33
where the composite
i ff
SD k( n1) d*!SDk(@ n) !1X
factors through a fixed base point * = ff(0) for i 6= 0.
The composite
0 ff
SD k( n1) d*!SDk(@ n) !1X
represents an element [ff1d0*] 2 ßn1X, and this element maps to 0 2 ßn1X
since the diagram (12) commutes. The homotopy  SDk n1 x 1 ! X from
ff1d0* to the base point is homotopic rel boundary and after subdivision to t*
*he
realization of a simplicial map SDr(SD k( n1)x 1) ! X, which extends after
subdivision to a homotopy of diagrams
SD s(SD k(@ n) x 1)____//_X
 
 
fflffl fflffl
SDs(SD k( n) x 1)_____//_Y
from a subdivision of the diagram (12) to a diagram
_ff2//_
SD s+k@ n X
 
 p
fflffl fflffl
SD s+k n __fi2_//Y
such that ff2 maps all of SDs+k @ n to the base pont of X.
The element [fi2] 2 ßnY  lifts to an element [fl] 2 ßnX since p* : ßn*
*X !
ßnY  is an isomorphism. The map fl :  SDs+k n ! X is homotopic rel
boundary and after subdivision to the realization of a simplicial set map f :
SD s+k+l n ! X which maps SDs+k+l@ n into the base point. It follows that,
after subdivision, fi2 is homotopic rel boundary to the map pf. The homotopy
 SDs+k+l n x 1 ! Y  rel boundary is itself homotopic to the realization of
a simplicial homotopy SDm (SD s+k+l n x 1) ! Y rel boundary after further *
* __
subdivision. It follows that fi2 lifts to X rel boundary after subdivision. *
* __
Theorem 34. Suppose that p : X ! Y is a Kan fibration and a weak equiv
alence. Then p has the right lifting property with respect to all inclusions
@ n ! n.
Proof. Suppose given a diagram
@ n ____//_X
 
 p
fflffl fflffl
n __oe__//Y
34
and let x = oe(0) 2 Y . The fibre Foe(0)over oe(0) is defined by the pullback
diagram
Foe(0)____//X
 p
 
fflffl fflffl
0 _oe(0)//_Y
and the Kan complex Foe(0)has the property that all maps @ n ! Foe(0)can be
extended to a map SDr n ! Foe(0)after a suitable subdivision, by Lemma 33.
All maps r : Foe(0)! EX rFoe(0)are weak equivalences of Kan complexes,
while the extension up to subdivision property for Foe(0)implies that all eleme*
*nts
of the combinatorial homotopy group ßjFoe(0)vanish in ßjEX rFoe(0)for some
r. The Kan complex Foe(0)therefore has trivial combinatorial homotopy groups,
and is contractible.
A standard (combinatorial) result about Kan fibrations [4, I.10.6] asserts
that there is a fibrewise homotopy equivalence
Foe______`'_____//_BFoe(0)x n
BB sss
BBB sssprs
B!!B yyss
n
where Foedenotes the pullback of p over n. It follows that the induced lifting
problem
@ n _____//Foe<<___
 _____p___
 ______ *
fflffl__fflffl__
n __1__//_ n
can be solved up to homotopy of diagrams, and can therefore be solved. ___
Corollary 35. Suppose that i : A ! B is a cofibration and a weak equivalence.
Then i has the left lifting property with respect to all Kan fibrations.
Proof. The map i has a factorization
j
A ____//_@@X
@@ p
i@@__@fflffl
B
where j is anodyne and p is a Kan fibration. Then p is a weak equivalence as
well as a Kan fibration, and therefore has the right lifting property with resp*
*ect
35
to all cofibrations by Theorem 34. The lifting ` therefore exists in the diagram
j
A _____//X>>~
~~
i `~~ p
fflffl~fflffl~
B __1__//B
It follows that i is a retract of j, and so i has the left lifting property_with
respect to all Kan fibrations. __
Corollary 36. Every Kan fibration is a fibration of simplicial sets, and con
versely.
Theorem 37 (Quillen). Suppose that p : X ! Y is a fibration. Then the
realization p : X ! Y  of p is a Serre fibration.
Proof. We want to show that all lifting problems in continuous maps
 nk_ff_//X==_ (13)
___
 ______
 ______ p
fflffl___fflffl_
 n__fi_//Y 
can be solved. The idea is to show that all such problems can be solved up to
homotopy of diagrams.
We can assume, first of all, that ff(k) is a vertex of X. If it is not, ther*
*e will
be path in X from ff(k) to some vertex x 2 X, and that path extends to a
homotopy of diagrams in the usual way.
There is a simplicial set map ff0 : SD r nk! X such that the realization
ff0*:  SDr nk ! X is homotopic to ff r relative to the image of the cone
point k in X. This homotopy extends to a homotopy from fi r to a map
fi1 :  SDr n ! Y  which restricts to pff0 on  SDr nk.
There is a further subdivision SDs+r n such that the composite map fi1 s
is homotopic rel  SDs+r nk to the realization of a simplicial map
fi0: SDs+r n ! Y.
It follows that there is a homotopy of diagrams from the diagram
 s+r n ff
 SDs+r nk____// k____//X (14)
 
 p
fflffl fflffl
 SDs+r n_s+r//__nfi//_Y 
36
to the realization of the diagram of simplicial set morphisms
0 s
SD s+r nk_ff__//X;;____
____
 _____p_
 _______ 
fflffl____fflffl
SDs+r _fi0__//_Y
The indicated lift exists in the diagram of simplicial set morphisms, since p is
a fibration and the induced map SDs+r nk! SDs+r n is anodyne, by Lemma
26.
The lifting problem can therefore be solved for the diagram (14). The map
 s+r is homotopic to a homeomorphism, and the homotopy and the homeo
morphism are natural in simplicial complexes. It follows that there is a diagram
homotopy from the diagram (14) to a diagram which is isomorphic to the orig __
inal diagram (13), so the lifting problem can be solved for that diagram. _*
*_
The following result is an easy consequence of Theorem 37 and the formalism
of categories of fibrant objects [4, II.8.6]. Its proof completes the proof of *
*the
assertion that the model structure on the category of simplicial sets is proper.
Corollary 38. Suppose given a pullback diagram
f*
A xY X _____//X
 p
 
fflffl fflffl
A ___f___//_Y
where p is a fibration and f is a weak equivalence. Then the induced map
f* : A xY X ! X is a weak equivalence.
Write Ex1 X for the colimit of the system
X fl!ExX fl!Ex2X ! . . .
Write ~fl: X ! Ex1 X for the natural map. This is Kan's Ex1 construction,
applied to the simplicial set X. The following result is well known [4], but has
a remarkably easy proof in the present context.
Lemma 39. The simplicial set Ex1 X is a Kan complex.
Proof. The space  nk is a strong deformation retract of  n. By Corollary 10,
there is a commutative diagram of simplicial set homomorphisms
r
sdr nk fl__//_;nk;
ww
 ww
 www
fflfflww
sdr n
37
This means that any map ff : nk! Y sits inside a commutative diagram
r
nk__ff_//Y_fl_//_ExrY66m
 mmmmmm
 mmm
fflfflmmmm
n
for some r. This is true for all simplicial sets Y , and hence for all ExrX. *
* ___
Theorem 40. The natural map ~fl: X ! Ex1 X is a weak equivalence, for all
simplicial sets X.
Proof. The Ex1 functor preserves fibrations on account of Lemma 26, and the
map fl : X ! Ex X induces a bijection ß0X ~=ß0(Ex X) for all simplicial sets
X.
Suppose that j : X ! ~Xis a fibrant model for X, and let x 2 X be a choice
of base point. The space of paths P ~Xstarting at x 2 X~ and the fibration
ß : P ~X! ~Xdetermines a pullback diagram
j*
X xX~P ~X _____//P ~X
ß*  ß
fflffl fflffl
X ____j____//_~X
in which the map ß* is a fibration and j* is a weak equivalence by Corollary 38.
The fibre X~ for both ß and ß* is a Kan complex, so that the map ~fl: X~ !
Ex1 X~ is a weak equivalence by Lemma 28 and Theorem 30. It follows from
Theorem 37 and the method of proof of Lemma 28 that the map ~fl: X ! Ex1
is a weak equivalence if we can show that the simplicial set Ex1 (X xX~P ~X) is
weakly equivalent to a point.
It is therefore sufficient to show that Ex1 Y is weakly equivalent to a point
if the map Y ! * is a weak equivalence. The object Ex1 Y is a Kan complex
by Lemma 39, so it suffices to show that all lifting problems
@ n __ff_//Ex1Y::___
 _________
 _______
fflffl____
n
can be solved if Y is weakly equivalent to a point. By an adjointness argument,
this amounts to showing that the map ff* : sdr@ n ! Y can be extended over
n after subdivision in the sense that there is a commutative diagram
fls //r ff* //
sds+r@ n _____ sd @ n _____Y44iiiii
 iiiiiii
 iiiii
fflffliii
sds+r n
38
There is a commutative diagram
sdsd*ff* ß
sdsd*sdr@ n _______//sdsd*Y_______//B sd*Y
oo
fl*fl fl*fl oooooo
fflffl fflfflwwoooo
sdr@ n ____ff*_____//Y
on account of Lemma 14 and Proposition 15. The map ß is a weak equivalence
by Corollary 13 and Lemma 14. The map fl*fl is a weak equivalence since its
realization is homotopic to a homeomorphism. It follows that the polyhedral
complex B sd*Y is weakly equivalent to a point.
Corollary 10 and the contractibility of the space B sd*Y  together imply
that there is a commutative diagram
flt // 2 r * // r n ß sdsd*ff*//
sdtsd2sdr@ n ______sd sd @ n ______sdsd*sd @dddd______B1sd*Y1dddddd
 dddddddddddddd
 dddddddddd
fflffldddddddddd
sdtsd2sdr n
The natural homotopy (7) induces a homotopy
h : sd2sdr(@ n) x 1 ! Y
from the composite ff*fl*fl * to ff*fl2. There is an obvious map
sd2sdr(@ n x 1) ! sd2sdr(@ n) x 1
which, when composed with h, and by taking adjoints gives a homotopy from
ff : @ n ! Y to a map (ff*fl*fl *)* : @ n ! Ex1 Y which extends to a map
n ! Ex1 Y . The object Ex1 Y is a Kan complex, so the map ff extends over_
n as well, by a standard argument. __
Corollary 41. The map fl : X ! ExX is a weak equivalence for all simplicial
sets X.
Proof. The map ~fl: X ! Ex 1 X is a weak equivalence, as is the map ~fl:
Ex X ! Ex1 X, and there is a commutative diagram
~fl 1
X ______//_ExX99t
tt
fl ttt
fflffl~flttt
Ex X
___
39
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40