Cubical homotopy theory: a beginning
J.F. Jardine*
October 22, 2002
Introduction
This paper displays a closed model structure for the category of cubical sets a*
*nd
shows that the resulting homotopy category is equivalent to the ordinary homo
topy category for topological spaces. The main results are Theorem 19, which
gives the model structure, and Theorem 29 and Corollary 30 which together
imply the equivalence of homotopy categories.
The cofibrations and weak equivalences for the theory are what one might
expect, namely levelwise inclusions and maps which induce weak equivalences
of topological spaces respectively. The closed model structure is relatively ea*
*sy
to derive, once one gets away from the preconception that fibrations should be
defined by analogy with Kan fibrations. A fibration is defined to be a map
which has the right lifting property with respect to all trivial cofibrations. *
*The
verification of the closed model axioms is essentially formal, and is displayed
here (see also [4]) as a consequence of standard tricks from localization theory
having to do with a bounded cofibration condition for countable complexes. The
equivalence of the homotopy category of cubical complexes with the ordinary
homotopy category is much more interesting, and follows from the assertion
that the cubical singular functor satisfies excision in a nonabelian sense.
There is an underlying category of models, namely the box category ,
which is used to define cubical sets in the same way that the category of ordin*
*al
numbers defines simplicial sets. This means that a cubical set X is defined as a
contravariant functor X : op ! Set on the box category, taking values in the
category of sets. The box category and its basic properties are the subject of *
*the
first section of this paper, while the first properties of cubical sets are des*
*cribed
in the second section. The closed model structure is derived in Section 3, and
appears as Theorem 19.
The assertion that the homotopy categories of cubical sets and topological
spaces (or simplicial sets) are equivalent involves the final three sections of*
* this
paper.
One needs a good subdivision operator. There is certainly an obvious sub
division_of_an_ncube,_which_is just a product of barycentric subdivisions of
*This research was supported by NSERC.
1
intervals. The subdivision sdX of a cubical set X is constructed from this
naive subdivision of the ncube in the end, but a functorial description of the
subdivision of cubes is required to make it work. This is done by showing that
the naive subdivision is isomophic to a cubical complex associated to the poset
of nondegenerate cells of the ncube.
The catch is that the standard, easy relationship between posets and sim
plicial complexes is lost in the cubical setting. Cubical complexes (meaning
subcomplexes of standard ncubes) have posets of nondegenerate cells which
have extra structure, formalized here as cubical posets. Furthermore, the cu
bical nerves of these posets are too big to be useful, but cubical posets have
"minimal" cubical nerves which are cubical complexes. The naive subdivision
of the ncube is isomorphic to the minimal cubical nerve of the poset of non
degenerate cells of the cube, and the latter is the functorial construction on
cubical complexes which gives rise to the subdivision operator for all cubical
sets. These ideas are the subject of Sections 4 and 5 of this paper.
Despite the apparent conceptual pain of the construction of the cubical sub
division functor, the functor itself is much better behaved that the subdivision
functor for simplicial sets, in that there is a canonical map fl : sdX ! X as f*
*or
simplicial sets, but there is also a natural homeomorphism h :  sdX ~=X of
the associated topological spaces, and a natural homotopy h ' fl. The natural
ity of both the map h and the homotopy effectively does away with the necessity
for showing that every cubical set can be refined by a cubical complex; this is
quite unlike the corresponding situation for simplicial sets, where one needs to
show that every simplicial set can be refined by a simplicial complex, via doub*
*le
subdivision. The proof of the cubical excision theorem (Theorem 27) makes di
rect use of these constructions, and then the comparison of homotopy categories
(Theorem 29, Corollary 30) follows relatively quickly. These results are proved
in Section 6.
I should say that none of this went exactly according to plan. The idea at
the outset (and this view has been generally held) was that one should be able
to develop the homotopy theory of cubical sets by analogy with the homotopy
theory of simplicial sets. Unfortunately for that point of view (see Remark
8), the topological realization functor does not preserve products, even up to
weak equivalence, and this has the ultimate effect of breaking the analogue of
the theory of minimal fibrations. In fact, the standard ncells n are not even
contractible within the category of cubical sets even though their realizations
are hypercubes. This phenomenon can be partially fixed by adding the Brown
Higgins connections [3] as an auxilliary set of degeneracies; this works for a
long time (there's even a closed model structure), but then one sees finally th*
*at
connections do not respect the subdivision operator. Connections are important
and one can do a lot with them, but it appears that they will have to be
addressed within the homotopy theory of cubical complexes from a more subtle
point of view.
There is a a theory of combinatorial fibrations, which is defined by obvious
analogy with the theory of Kan fibrations of simplicial sets, but is not displa*
*yed
2
here. The analogy goes far enough to produce a decently behaved theory of
combinatorial homotopy groups. There is a Milnor theorem which asserts that
the canonical map X ! SX induces an isomorphism between the combina
torial homotopy groups of a fibrant cubical set X and the homotopy groups of
the associated space X, but one would like to have this statement hold more
generally for all combinatorially fibrant objects. Its proof should have some
thing to do with a cubical approximation theorem, meaning a suitable analogue
of simplicial approximation. Cubical approximation has not been proved, and
it may not yet even have a suitable expression _ it appears to be one of the
äh rd" things that will be possible to properly state and prove only once sub
stantial portions of the rest of the homotopy theory of cubical sets are proper*
*ly
developed.
There's a final punch line: one can go back and develop the homotopy the
ory of simplicial sets by analogy with the results given here. The closed model
structure for simplicial sets is much easier to derive from this point of view,*
* and
one can prove an exact analogue of the excision result given here for simplicial
sets once one understands (and successfully proves) that simplicial approxima
tion is really about showing that an arbitrary simplicial set can be refined by*
* a
simplicial complex up to weak equivalence (see [5], [6]) _ this is a much more
delicate statement than the approximation technique that is used here. This
collection of ideas will be the subject of a future paper.
This paper was completed while I was a member of the Isaac Newton Insti
tute for Mathematical Sciences during the Fall of 2002. I would like to thank
that institution for its hospitality and support.
Contents
1 The box category 4
2 Cubical sets 9
3 The closed model structure 18
4 Cubical posets 21
5 Cubical subdivision 26
6 Cubical excision 32
3
1 The box category
Write n_= {1, 2, . .,.n}, and let 1n be the nfold product of copies of the cat*
*egory
1 defined by the ordinal number 1 = {0, 1} of the same name. Write 10 for the
category consisting of one object and one morphism.
A face functor (d, ffli) : 1m ! 1n is defined by an ordered inclusion d : m_*
*! n_
and a set of elements ffli 2 {0, 1}, i 2 n_ m_. The corresponding functor is
specified by the diagrams
(d,ffli)
1m _____//CC1n
CCC pri
diCCC!!fflffl
1
where diis the projection prd1(i)if i is in the image of d, and diis the const*
*ant
functor at ffli for i 2 n_ m_.
A degeneracy functor s = sd : 1n ! 1k is specified by an ordered inclusion
d : k_! n_. In effect, the diagram
sd
1r _____//AA1k
AA pr
prd(i)AAA i
__Afflffl
1
is required to commute.
There is an isomorphism of posets
~=
n : 1n ! P(n_)
which is defined by associating to the ntuple ffl = (ffl1, . .,.ffln) the subs*
*et
n(ffl) = {i  ffli= 1}
of the set n_= {1, . .,.n}.
Suppose that (d, ffli) : 1m ! 1n is a face functor, and consider the composi*
*te
poset morphism
1n (d,ffli)!1m m! P(n_).
Suppose that A = m (d, ffli)(0, . .,.0) and that B = m (d, ffli)(1, . .,.1). *
*Write
[A, B] for the subposet of P(m_) consisting of all subsets C such that A C *
*B.
The poset [A, B] is often called the interval between A and B. Then one can
show that there is a commutative diagram of poset morphisms
(d,ffli)
1n ______//_1m
n ~= ~=m
fflffl fflffl
P(n_)_d*_//_P(m_)
4
where the poset morphism d* is defined by C 7! d(C)[B. Note that the odered
inclusion d : n_! m_determines a bijection n_~=B  A, and that d* induces a
poset isomorphism P(n_) ~=[A, B].
An ordered inclusion d : k_! n_can be identified with a subset A n_of order
k in the obvious way, and any degeneracy sd : 1n ! 1k sits in a commutative
diagram
sd
1n _______//1k
n~= ~=k
fflffl fflffl
P(n_)F P(k_)
FF 
FFF ~=
F##Ffflffl
P(A)
where the indicated isomorphism is determined by a canonical order preserving
bijection k_~=A and the morphism P(n_) ! P(A) is defined by C 7! C \ A.
In the definition of both face and degeneracy functors, the ordered inclusio*
*ns
can be replaced by choices of subsets. In effect, a subset of n_having kelemen*
*ts
determines a unique ordered inclusion k_ n_.
Consider the composite functor
1m (d,ffli)!1n s!1k
There is a pullback diagram of order preserving functions
m___d__//n_OOOO
s0 s
 
r_____//_k_
d0
and there is a corresponding commutative diagram of face and degeneracy func
tors
(d,ffli)
1m _____//1n (1)
s0 s
fflffl fflffl
1r(d0,ffls(i))//_1k
The sets of face and degeneracy functors are each closed under composition,
and degeneracy functors can be öm ved past" face functors according to the
recipe specified above.
We shall write d = (d, ffli) for face functors in the following, except in p*
*laces
where the ambiguity could cause confusion.
5
Lemma 1. Suppose given a commutative diagram
1m __s__//1n
s0 d
fflffl0 fflffl
1n __d0_//1k
composed of face functors d, d0 and degeneracies s, s0. Then d = d0 and s = s0.
Proof. There is a face functor ~dwhich is a section of s. Write ` = s0~d: 1n ! *
*1n0.
The functor d0 has a left inverse given by a degeneracy, and is therefore a
monomorphism. Then
d0` = d0s0~d= dsd~= d,
while
d0`s = ds = d0s0
so that `s = s0. The functor ` is also the unique functor which makes the
diagram
1m __s__//1n
__
s0`____ d
fflffl""fflffl__0
1n __d0_//1k
commute. There is similarly a uniquely determined functor `0: 1n0! 1n which
makes the diagram
1m __s__//1n==
0___
s0`___ d
fflffl0_fflffl_
1n __d0_//1k
commute. It follows that the functor ` is an isomorphism of categories. In
particular, n = n0.
The functor ` has a factorization
1n ______`~=____//B1n==
BBB ____
pBBB!!B___~_
1r
where p is a degeneracy functor and ~ is a face functor. Then p is a monomor
phism as well as an epimorphism. If r < n then
i i
p(ffl1, . .,.0, . .,.ffln) = p(ffl1, . .,.1, . .,.ffln)
for i =2r_and p cannot be a monomorphism. It follows that r = n and p = 1,
since there is only one orderpreserving monomorphism n_! n_. It also follows_
that ~ = 1, and hence that ` = 1. __
6
The box category is the subcategory of the category of small categories
which is generated by the face and degeneracy functors. Its objects consist of *
*the
categories 1k, k 0, and it follows from Lemma 1 that a morphism ` : 1n ! 1m
in can be uniquely written as a composite
1nB______`_____//_1m==
BBB ____
sBBB__B___d_
1k
where s is a degeneracy functor and d is a face functor. Morphisms in the box
category are also called cubical functors.
The pair (i, ffl) consisting of i 2 n_and ffl 2 {0, 1} determines a unique f*
*ace
functor d(i,ffl): 1n1 ! 1n, defined by
d(i,ffl)(fl1, . .,.fln1) = (fl1, . .,.iffl, . .,.fln1).
Suppose that i < j. Then there is a commutative diagram of face functors
d(i,ffl1)//_n1
1n2 1 (2)
d(j1,ffl2) d(j,ffl2)
fflffl fflffl
1n1 d(i,ffl1)//_1n
if n 2. If i = j there is a diagram
;_______//1n1 (3)
  (i,1)
 d
fflffl fflffl
1n1 d(i,0)//_1n1
The degeneracy functor sj : 1n ! 1n1 is the projection which forgets the
jth factor, so that
sj(fl1, . .,.fln) = (fl1, . .,.flj1, flj+1, . .,.fln)
Write s1 : 1 ! 10 for the obvious map to the terminal object 10 in the box
category .
Then there are relations
sjsi= sisj+1, if i .j (4)
Similarly,
sjd(j,ffl)= 1, (5)
7
and there are commutative diagrams
(i,ffl)
1n _d____//1n+1 if i < j (6)
sj1 sj
fflffl fflffl
1n1 d(i,ffl)//_1n
and
(i+1,ffl)
1n d____//_1n+1 if i j. (7)
sj sj
fflffl fflffl
1n1 d(i,ffl)//_1n
The projections
(ffl1, . .,.ffln+k) prL7!(ffl1, . .,.ffln)
and pr
(ffl1, . .,.ffln+k) 7!R(ffln+1, . .,.ffln+k)
are degeneracy functors. Thus, any morphism ` : 1r ! 1n+k is uniquely deter
mined by the composites prL` and prR `. That said, 1n+k is not the categorical
product of 1n and 1k in the box category : one sees this by observing that
the diagonal functor : 1 ! 12 is not a face functor.
What can be said along these lines is the following:
Lemma 2. The diagrams (2), (6) and (7) are pullbacks in the box category.
Proof. A box morphism ff : 1r ! 1n factors through the face d(i,ffl): 1n1 ! 1n
if and only if the images ff(x) = (ff1(x), . .,.ffn(x)) have the form ffi(x)_= *
*ffl for
all x 2 1r. __
A poset morphism fl : P(n_) ! P(m_) is said to be cubical if the morphism
fl* : 1n ! 1m defined by the diagram
fl*
1n ______//_1m
n ~= ~=m
fflffl fflffl
P(n_)__fl//_P(m_)
is cubical in the sense that it is a morphism of the box~category .
Observe that there is a poset isomorphism `F : P(F ) =!P(F )op defined by
B 7! Bc.
Suppose that the face functor d : P(k_) ! P(n_) is defined by the interval
[A, B] P(n_), so that there is an ordered set isomorphism k_~=B  A which
8
defines an ordered inclusion d : k_! n_, and the functor d : P(k_) ! P(n_) is
defined by C 7! A [ d(C). In particular d factors canonically as the composite
~=
P(k_) ! [A, B] P(n_)
where the displayed isomorphism is induced by the ordered set isomorphism
k_~=B  A.
There is a commutative diagram
~=
P(k_)op____//_[A,OB]op__//P(n_)opOOO
`~= ~=`
 
P(k_)___~=//_[Bc, Ac]___//P(n_)
where the morphisms along the top are induced by the factorization of the
original poset morphism d, and the isomorphism P(k_) ~=[Bc, Ac] arises from
the identity AcBc = B A in P(n_). The point in checking the commutativity
of this diagram is that, for any C k_, we have (A[d(Cc))c = Ac\d(Cc)c. Also
d(Cc) t d(C) = B  A so that d(Cc)c = A t (d(C) t Bc). Thus, d(Cc)c\ Ac =
d(C) [ Bc.
Suppose that the subset A of n_defines an ordered inclusion A : k_! n_,
which in turn induces a degeneracy functor s : P(n_) ! P(A) ~=P(k_) given by
C 7! C \ A. Then the following diagram of functors commutes
op
P(n_)op_s___//P(A)opOOOO
`~= ~=`
 
P(n_)___s__//_P(A)
The point is that the complement of C \ A in A is the intersection Cc \ A.
We have proved the following:
Lemma 3. Suppose that the poset morphism ! : P(n_) ! P(m_) is cubical, and
let !* : P(n_) ! P(m_) be defined by the requirement that the diagram
op
P(n_)op_!__//_P(m_)opOOOO
`~= ~=`
 
P(n_)__!*__//_P(m_)
commutes. Then the functor !* is cubical.
2 Cubical sets
A cubical set X is a contravariant setvalued functor X : op ! Set. Write
Xn = X(1n), and call this set the set of ncells of X. A morphism f : X ! Y
9
of cubical sets is a natural transformation of functors, and we have a category
cSet of cubical sets.
The standard ncell n is the contravariant functor on the box category
which is represented by 1n. Thus, n has mcells given by
nm= hom (1m , 1n).
There is a cell category # X for a cubical set X which is defined by analo*
*gy
with the simplex category of a simplicial set. Then objects of # X are the
morphisms oe : n ! X (equivalently ncells of X, as n varies), and a morphism
is a commutative triangle of cubical set morphisms
n RRR
 RR((R
 m6X6m
fflfflmmmm
m
There is a covariant simplicial setvalued functor ! S
1n 7! B(1n) = ( 1)xn
which is defined by the categorical nerve construction. This functor can be used
to define a cubical singular functor S : S ! cSet, where
S(Y )n = hom S(( 1)xn , Y ).
This functor has a left adjoint (called realization or triangulation) X 7! X,
where
X = lim!( 1)xn .
n!X
Here, the colimit is indexed by members of the cell category # X for X.
There are similarly defined realization and singular functors
  : cSet ø Top : S
relating cubical sets and topological spaces, and of course realization is left
adjoint to the singular functor in that context as well.
Remark 4. There is no notational distinction between the singular functors
defined on topological spaces and simplicial sets, and no distinction between
the corresponding realization functors. We shall rely on the context to tell
them apart.
Example 5. Suppose that C is a small category. The cubical nerve B (C) is
the cubical set whose ncells are all functors of the form 1n ! C, and whose
structure maps B (C)n ! B (C)m are induced by precomposition with box
category morphisms 1m ! 1n. Observe that there is a natural isomorphism
B (C) ~=S(BC),
where BC is the standard nerve for the category C in the category of simplicial
sets.
10
In a cubical set X, write d(i,ffl)for the function Xn ! Xn1 which is induced
by the functor d(i,ffl), and call this function a face map. Similarly, the deg*
*en
eracies sj : Xn ! Xn+1 are the functions which are induced by the functors
sj : 1n+1 ! 1n. Say that a cell oe 2 Xn is degenerate if it is the image of some
sj, and is nondegenerate otherwise.
Define the nskeleton sknX for a cubical set X to be the subcomplex which
is generated by the kcells Xk for 0 k n.
Lemma 6. A map f : sknX ! Y of cubical sets is completely determined by
the restrictions f : Xk ! Yk for 0 k n,
Proof. We want to show that the maps f : Xk ! Yk extend uniquely to a
morphism f* : sknX ! Y . Suppose that z 2 sknXn+1. Then z is degenerate,
so that z = six for some x 2 Xn, and it must be that f*(z) = sif(x) if the
extension exists. Suppose that z is degenerate in two ways, so that also z = sjy
for some i < j and y 2 Xn. Then
x = d(i,0)six = d(i,0)sjy = sj1d(i,0)y,
while
sjsi(d(i,0)y) = sisj1(d(i,0)y) = six = sjy.
All degeneracies are injective, so that y = sid(i,0)y, and
sif(x) = sisj1d(i,0)f(y) = sjsid(i,0)f(y) = sjf(y).
Inductively, the map f* : skn(X)r ! Yr for r = k is completely determined_by
the maps for r < k in the same way. __
It follows that there are pushout diagrams
F
@ n _____//sk X
x2NXn n1
 
 
F fflffl fflffl
n ______//_skX
x2NXn n
where NXn denotes the nondegenerate part of Xn, and @ n = skn1 n. In
other words, there is a good notion of skeletal decomposition for cubical sets.
The object @ n is the subcomplex of the standard ncell which is generated
by the faces d(i,ffl): n1 ! n. It follows from the fact that the diagram (2)
is a pullback in the box category that there is a coequalizer
G G
n2 ' n1 ! @ n
(ffl1,ffl2) (i,ffl)
0 i 2 by a simple com
binatorial argument, while there is a single nondegenerate 2cell given by the
isomorphism of categories 12 ! 1 x 1 (NB: this is a product of box category
morphisms, namely the product of left and right projections, but the isomor
phism does not define 12 as a categorical product in the box category _ see
the discussion of 1skeleta below). It follows that there is a pushout of cubic*
*al
complexes
@ 2 _____//sk1( 1 x 1)
 
 
fflffl fflffl
2 ________// 1 x 1
and hence a pushout of simplicial sets
@ 2_____// sk1( 1 x 1)
 
 
fflffl fflffl
 2________// 1 x 1
The cell category # n has a terminal object given by the identity functor on
1n, so that there is an isomorphism
 n ~=( 1)xn .
At the same time, the definitions are rigged so that @ n coincides with the
geometric boundary of ( 1)xn . The skeleton sk1( 1x 1) has a 1cell : 1 !
12 in addition to those coming from @ 2. It follows that
 sk1( 1 x 1) ~=sk1( 1 x 1).
It follows that  1 x 1 has the homotopy type of the simplicial circle S1.
The problem with realizations of products as displayed in Remark 8 can
be fixed (following Kan [10]) as follows. The object 1n+m is not the product
1n x 1m in the box category, but there is nevertheless a functor ~x: x !
which is defined on objects by
1nx~1m = 1n+m ,
12
and is defined on morphisms by `x~fl = ` x fl.
If X and Y are cubical sets, define
X Y = lim! n+m
oe: n!X, ø: m !Y
Here, if the morphisms ` : 1n ! 1r and fl : 1m ! 1s define morphisms
` : oe ! oe0 and fl : ø ! ø0 in the box categories for X and Y respectively, th*
*en
the corresponding map 1n+m ! 1r+s is induced by `x~fl.
Note that there are isomorphisms
n m ~= n+m .
It follows that the functor Y 7! Y n has a right adjoint Z 7! Z(n), where
Z(n)r= Zr+n and has cubical structure map fl* : Z(n)r! Z(n)sdefined by
(flx~1)* : Zr+n ! Zs+n. In particular, there is an isomorphism
Y n ~= lim! m+n .
m !Y
The cubical function complex hom (Y, Z) for cubical sets Y and Z is the
cubical set defined by
hom (Y, Z)n = hom (Y n, Z).
There is a natural bijection
hom(X, hom (Y, Z)) ~=hom (X Y, Z),
which is a consequence of the identifications
hom ( n, hom (Y, Z)) = hom (Y n, Z)
and the isomorphism
Y X = lim! m+n ~= lim!Y n.
m !Y, n!X n!X
There are identifications
d(i,ffl) 1n m
n1 m _____//
~= ~=
fflffl fflffl
n+m1 _d(i,ffl)//_ n+m
and
1 d(j,ffl)n m
n m1 _____//
~= ~=
fflffl fflffl
n+m1 d(n+j,ffl)//_ n+m
13
The functor K 7! K n has a right adjoint and therefore preserves coequal
izers. Thus, if K n is the subcomplex which is generated by some list of
faces d(i,ffl): n1 ! n, the K m is isomorphic to the subcomplex of n+m
which is generated by the list of faces d(i,ffl): n+m1 ! n+m . Similarly, *
*if
L m is the subcomplex generated by faces d(j,ffl): m1 ! m , then n L
is isomorphic to the subcomplex of n+m which is generated by the list of faces
d(n+j,ffl): n+m1 ! n+m .
It follows that the induced maps @ n m ! n m and n @ m !
n m are monomorphisms of cubical sets. This implies that there are iso
morphisms
(@ n m ) [ ( n @ m ) ~=@ n+m
(un(i,ffl) m ) [ ( n @ m ) ~=un+m(i,ffl)
(@ n m ) [ ( n umi,ffl)) ~=un+mn+i,ffl.
More generally, the functors X 7! X n and Y 7! n Y preserve monomor
phisms of cubical sets.
There are isomorphisms
X Y ~= lim!  n+m 
n!X, m !Y
~= lim  n x  m 
n!X,! m !Y
~=X x Y .
In particular, there is an isomorphism of simplicial sets.
 n ~= 1xn
For any i 2 n_there is a permutation ` 2 n such that `(i) = 0. Using ` to
permute factors therefore induces a diagram
 un(i,ffl)//_ n (8)
`*~= ~=`*
fflffl fflffl
 un(0,ffl)//_ n
The relations
un(0,ffl)~=( 0 n1) [ ( 1 @ n1) 1 n1 ~= n.
imply that the simplicial set inclusion  un(0,ffl)  n can be identified u*
*p to
isomorphism with the inclusion
( 0 x  n1) [ ( 1 x @ n1)  1 x  n1,
and is therefore an anodyne extension. It follows from (8) that all induced
inclusions  un(i,ffl)  n are anodyne extensions of simplicial sets.
14
Lemma 9. Suppose that K and L are cubical sets. Then the function KkxLl!
(K L)k+l defined by sending the pair (oe, ø) to the cell oe ø : k l! K L
is an injection. If k = l = 0 this function is a bijection.
Proof. The map n0x m0! ( n m )0 is plainly a bijection, on account of the
canonical isomorphism n m ~= m+n . The map K0 x m0! (K m )0
is a bijection, since this map is a colimit of maps n0x m0 ! ( n m )0,
indexed over the cells n ! K of K. The map K0x L0 ! (K L)0 is a colimit
of maps K0 x m0! (K m )0, indexed over the cells m ! L of L, and is
therefore a bijection.
We know that the functor K 7! K L preserves monics, and that there is
a canonical isomorphism ~
c : K =!K 0.
Suppose that oe1, oe2 : k ! K and ø1, ø2 : l ! L are cells of K and L,
respectively, such that oe1 ø1 = oe2 ø2. There are commutative diagrams
oei //
k __________K
c ~= ~=c
fflffloei 1 fflffl
k 0 _____//K 0
1 0  1øi(0)
fflffl fflffl
k l oe1_ø1//_K L
Here 0 denotes the vertex (0, . .,.0) of l.
Note that oe1(0) ø1(0) = oe2(0) ø2(0), so that oe1(0) = oe2(0) and ø1(0)*
* =
ø2(0). Observe also that the maps 1 ø1(0) = 1 ø2(0) are monomorphisms. It
follows that there is a monomorphism ff = (1 øi(0))c such that
ffoe1 = (oe1 ø1)(1 0)c = (oe2 ø2)(1 0)c = ffoe2,
so that oe1 = oe2. Similarly ø1 = ø2. ___
Write NKn for the set of nondegenerate cells of a cubical set K in degree
n.
Corollary 10. The map KkxLl! (K L)l+k restricts to an injection NKkx
NLl! N(K L)k+l.
Proof. Take (oe, ø) 2 Kk x Ll. Any degeneracy functor s : k l ! n can
be written as
s1 s2 n n
k lN_____//N 1 2
NNN ~
sNNNNN =
N&&fflffl
n
15
where si is either a degeneracy functor or an identity for i = 1, 2 and at least
one of the si is not the identity. There are face functors di : n1 ! k and
d2 : n2 ! l such that sidi = 1. It follows that oe ø = s1d1oe s2d2ø, and
hence that oe = s1d1oe and ø = s2d2ø. Thus if oe ø is degenerate then one of
the cells oe and ø must be degenerate. In particular, there is an induced funct*
*ion
NKk x NLl ! N(K L)k+l. This function is the restriction of an injection,_
and is therefore injective. __
Observe as well that the induced function
G
(NKk x NLnk) ! N(K L)n
0 k n
is surjective. In effect, the corresponding function
G
(Kk x Lnk) ! (K L)n
0 k n
is surjective,
The ideas in the proof of Lemma 6 can also be used to show the following:
Lemma 11. Suppose that x and y are degenerate ncells of a cubical set X
which have the same boundary in the sense that d(i,ffl)x = d(i,ffl)y for all i *
*and ffl.
Then x = y.
Proof. Suppose that x = siu and y = sjv for some i < j. Then
u = d(i,0)siu = d(i,0)sjv = sj1d(i,0)v,
while
siu = sisj1d(i,0)v = sjsid(i,0)v.
Then
d(j,0)siu = d(j,0)sjv,
so that
sid(i,0)v = v.
It follows that
siu = sjsid(i,0)v = sjv.
so that x = y. ___
Lemma 12. Suppose that x, y : n ! X are ncells of a cubical set X such
that the induced simplicial set maps x*; y* :  n ! X coincide. Then x = y.
Proof. The inclusion sknX X induces a monomorphism  sknX ! X,
so that we can assume that X = sknX. We may further suppose that X is
generated by the subcomplex skn1X together with the ncells x and y.
The proof is by induction on n. The assumption that x* = y* therefore
guarantees that x and y have the same boundary in the sense that d(i,ffl)x =
16
d(i,ffl)y for all i and ffl. Thus if x and y are both degenerate, then x = y by
Lemma 11.
Suppose that y is nondegenerate, and write X0 for the smallest subcomplex
of X containing skn1X and x. Write i : X0 ! X for the inclusion of the
subcomplex X0 in X.
If x 6= y, then y is not in X0. Also, the intersection \ X0 = skn1,
where denotes the subcomplex of X which is generated by y. This means
that there is a pushout diagram
@ n ____//_X0
 
 
fflffl fflffl
n __y___//X
The assumption that x* = y* implies that the dotted arrow lifting exists in the
solid arrow pushout diagram
@ n ____//_X0;;_
____
 x*______
 ______ i*
fflffl____fflffl
 n__y*__//X
making it commute. The map i* is an inclusion which is not surjective, since
the solid arrow diagram is a pushout. But the existence of the dotted arrow *
*__
forces i* to be surjective. This is a contradiction, so x = y. *
*__
Corollary 13. Suppose that f : X ! Y is a map of cubical sets such that the
induced simplicial set map f* : X ! Y  is a monomorphism. Then f is a
monomorphism of cubical sets.
Proposition 14. Suppose that f : X ! Y is a map of cubical sets such that
the induced simplicial set map f* : X ! Y  is an isomorphism. Then f is an
isomorphism of cubical sets.
Proof. The map f is a monomorphism of cubical sets by Corollary 13. If f is
not surjective, there is a nondegenerate cell x : n ! Y of smallest dimension
which is not in X. It follows that f is a composite of monomorphisms
X f0!X0 f1!Y
where X0 is obtained from X by attaching the ncell x in the sense that there
is a pushout diagram
@ n _____//_X
 
 f0
fflffl fflffl
n __x__//_X0
17
The triangulation functor X 7! X preserves monomorphisms and pushouts so
that the induced map f* : X ! Y  is a composite of monomorphisms f1*f0*,
and there is a pushout diagram
@ n_____//_X
 
 f0*
fflffl fflffl
 n _x*__//_X0
of simplicial set maps. Then the monomorphism @ n !  n is not surjective,
so that f0* is not surjective, and so f* is not surjective. This is a contradic*
*tion,_
so that f must be a surjective map of cubical sets. __
3 The closed model structure
The purpose of this section is to display a closed model structure for the cate*
*gory
of cubical sets. The homotopy category associated to this model structure will
later be shown to be equivalent to the standard homotopy category of topological
spaces.
Basically, if you want to show that a particular category has a closed model
structure, you must define three classes of morphisms in that category, namely
weak equivalences, cofibrations and fibrations, and then show that they satisfy
the five Quillen closed model axioms CM1 through CM5. The axiom CM1
is a completeness axiom which says that certain limits and colimits exist. The
weak equivalence axiom CM2 says that if any two of the composable maps
f and g and their composite fg are weak equivalences, then so is the third.
The retract axiom CM4 says that all of the three defined classes of maps are
closed under retraction. Finally the factorization axiom CM5 says that any
morphism in the category can be factored as a composite of a fibration with a
trivial cofibration, and as a composite of a trivial fibration and a cofibratio*
*n.
Here "trivial" has the standard meaning: a trivial fibration is a morphism which
is both a fibration and a weak equivalence, and a trivial cofibration is a map
which is both a cofibration and a weak equivalence.
A map f : X ! Y of cubical sets is said to be a weak equivalence if the
induced map f* : X ! Y  is a weak equivalence of topological spaces (or of
simplicial sets). A cofibration i : A ! B of cubical sets is a levelwise inclus*
*ion.
A map p : Z ! W of cubical complexes is said to be a fibration if it has the
right lifting property with respect to all maps which are both cofibrations and
weak equivalences.
The category of cubical sets certainly has all limits and colimits, so the t*
*he
axiom CM1 is satisfied. The weak equivalence axiom CM2 is a consequence
of the corresponding statement for topological spaces, and the retraction axiom
CM3 is a trivial consequence of the definitions.
For the factorization axiom, we need to show two things:
18
Lemma 15. A map p : X ! Y is a map which has the right lifting property
with respect to all inclusions @ n ! n. Then p is a fibration and a weak
equivalence.
Proof. If p has the right lifting property with respect to all inclusions @ n *
* n
then p has the right lifting property with respect to all inclusions, and is th*
*erefore
a fibration.
In fact, the map p is a homotopy equivalence of cubical sets, by the standard
argument: the map p has a section s : Y ! X since there is a commutative
diagram
;_____//_X??___
 r____p___
 ______
fflffl_fflffl__
Y __1_//_Y
and then rp ' 1 because there is a commutative diagram
(rp,1)
X @ 1 ____//_X;;____
 _________
 ___H____p
fflffl____fflffl
X 1 pcX___//Y
where cX : X 1 ! X is the constant homotopy at the identity on X. It
follows that the induced map p* : X ! Y  is a homotopy equivalence of_
simplicial sets. __
Lemma 16. There is a set A of trivial cofibrations A B such that a map
p : X ! Y is a fibration if and only if it has the right lifting property with
respect to all maps in A.
Lemma 16 is a formal consequence of Lemma 17, in that Lemma 17 implies
that the set A of trivial cofibrations of countable cubical sets does the job.
Lemma 17. Suppose that A is a countable cubical set, and that there is a
diagram
X
i
fflffl
A _____//Y
of cubical set maps in which i is a trivial cofibration. Then there is a counta*
*ble
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, since countable simplicial sets
have countable homotopy groups (any countable simplicial set has a countable
fibrant model, by the way that the small object argument works).
19
Suppose that x is a vertex of A = B0. Then there is a finite connected
subcomplex Lx Y such that Lx containsSa 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 . __
In the presence of Lemma 16, a standard transfinite small object argument
produces a factorization
f
X ____________//_@@Y??~
@@ ~~~
i@@__@~~~p
Z
with p a fibration and i a trivial cofibration for any map f : X ! Y of cubical
sets. A completely standard small object argument, together with Lemma 15,
shows that any map f : X ! Y has a factorization
f
X _____________//BBY>>"
BBB """"
j BB__B"""q
W
with j a cofibration and q a trivial fibration. Lemmas 15 and 16 therefore imply
the factorization axiom CM5.
Lemma 15 has a converse, with a formal proof:
Lemma 18. 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
20
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 15, 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.*
* ___
The axiom CM4 follows. We have proved the following:
Theorem 19. With the definitions of weak equivalence, cofibration and fibration
given above, the category cSet of cubical sets satisfies the axioms for a closed
model category.
The cubical set category is a closed cubical model category, in the sense th*
*at
if i : A ! B is a cofibration and p : X ! Y is a fibration, then the induced map
of cubical function complexes
(i*, p*) : hom (B, X) ! hom (A, X) xhom (A,Yh)om (B, Y )
is a fibration which is also a weak equivalence of cubical sets if either i or *
*p is
a weak equivalence. This is a consequence of the observation that if j : C ! D
is a second cofibration of cubical sets, then the induced map
B C [A C A D ! B D
is a cofibration which is a weak equivalence if either i or j is a weak equival*
*ence.
In effect, the triangulation functor reflects cofibrations by Corollary 13, and
reflects weak equivalences by definition.
It is also clear that the class of weak equivalences is stable under pushout
along cofibrations. This is half of the assertion that the model structure for
cubical sets is proper. The other half of the properness assertion, namely that
weak equivalences are stable under pullback along fibrations, remains to be
verified.
4 Cubical posets
Recall that an interval [A, B] Q in a poset Q is a subposet consisting of all
objects C such that A C B.
Say that a poset P is cubical if the following hold:
~=
1)there is a fixed poset isomorphism f : 1k ! [A, B] for all nonempty
intervals [A, B] of P
21
2)any inclusion i : [A, B] [C, D] of nonempty intervals induces a box
category morphism i* : 1k ! 1lsuch that the following diagram commutes
i* //
1k ________1l
f~= ~=f
fflffl fflffl
[A, B]__i_//[C, D]
3)intervals are closed under intersection in P .
In the presence of the poset isomorphism f : 1k ~= [A, B], say that k is the
dimension of [A, B]. We shall say that the isomorphisms f are parameterizations
of the intervals; they are a necessary part of the structure.
The power set poset 1n ~=P(n_) is a standard example. In that case, any
nonempty interval [A, B] P(n_) determines a unique ordered set isomorphism
~=
d : k_!B  A n_which then determines a parameterization
1k k!P(k_) d*![;, B  A] ~=[A, B]
This is the standard parameterization of an interval [A, B] P(n_), and will
always be used. Note the equality
[A1, B1] \ [A2, B2] = [A1 [ A2, B1 \ B2]
so that the set of intervals of P(n_) is closed under intersection.
In some sense, the conditions 1)3) together mean that a cubical poset P
has a covering by power sets.
Cubical posets P have "minimal" cubical nerves Bm P . The easiest way to
define Bm P as a cubical set is to decree that
Bm P = lim! k,
[A,B]
where the colimit is indexed over the poset of nonempty intervals in P and
the indicated colimit is for the functor [A, B] 7! k, where f : 1k ! [A, B] is
the poset isomorphism required by the structure. In particular, the poset of
intervals in 1n ~=P(n_) has a unique maximal element [;, n_], so that there is a
canonical isomorphis
Bm P(n_) ~= n.
Alternatively, it is easily seen that there is a coequalizer
G G
r ' k ! Bm P.
[A,B]\[C,D]6=; [A,B]6=;
Here, r is the dimension of the intersection [A, B]\[C, D] and k is the dimensi*
*on
of [A, B].
22
The intervals ~
1k =![A, B] P
in a cubical poset P determine cells oe[A,B]: k ! B P of the cubical nerve
B P . The construction of these cells respects inclusion of intervals, and the*
*re
fore determines a canonical natural map
jP : Bm P ! B P.
An important special case of this construction is the standard map j : n !
B (1n) which associates to an mcell (ie. a box category morphism) ` : 1m !
1n the corresponding functor ` : 1m ! 1n _ in other words j forgets the fact
that the functor ` is a box category morphism. The map j : n ! B (1n)
is plainly a monomorphism (this is the first step to a general story: all cells
oe[A,B]: k ! B P are monomorphisms). It is also easy to see that any face
map d : k ! n determines a pullback diagram
j k
k _____//B (1 ) (9)
d d*
fflffl fflffl
n ___j_//B (1n)
is a pullback diagram in the category of cubical sets. In effect, if fl : 1s ! *
*1k
is a functor such that the composite dfl is a cubical functor, then there is a
degeneracy functor s : 1n ! 1k such that sd = 1 and so fl = sdfl is a cubical
functor.
Now suppose that
[E, F ]____//[A, B]
 
 
fflffl fflffl
[C, D]______//_P
is a pullback diagram of nonempty intervals in a cubical poset P , so that
[E, F ] = [A, B] \ [C, D], and let
r ______// k (10)
 
 
fflffl fflffl
l _____//B P
be the corresponding diagram of cells. Then the diagram (10) factors as a
23
diagram
r _________________//GGGk
 GGjG GGGG
 GG GGG
 G##G G##
 B (1r)_____________//_B (1k)
  
  
  
fflffl  
l  
GGG  
GG  
GGG  
G##fflffl fflffl
B (1l)______________//_B P
where all of the indicated square are pullbacks, and the map j is a monomor
phism. It follows that the diagram (10) is a pullback in cubical sets.
The subobject X of B P which is generated by the intervals oe[A,B]: k !
B P is in fact covered by those intervals since (10) is a pullback, and it fol*
*lows
that there is a coequalizer
G G
r ' k ! X.
[A,B]\[C,D]6=; [A,B]6=;
The map jP : Bm P ! B P factors through an isomorphism Bm P ~= X by
comparison of coequalizers, so that jP is a monomorphism. We have proved
Lemma 20. Suppose that P is a cubical poset. Then the canonical map jP :
Bm P ! B P is a monomorphism, so that Bm P can be identified with the
subobject of the cubical nerve B P which is generated by nonempty intervals.
Lemma 21. Suppose that P is a cubical poset. Then there is an isomorphism
of simplicial sets
Bm P  ~=BP.
Proof. The intervals cover P , so there is a coequalizer diagram of simplicial *
*sets
G G
B(1r) ' B(1k) ! BP.
[A,B]\[C,D]6=; [A,B]6=;
There are natural canonical isomorphisms
~= n
 n ! B(1 )
which together induce a comparison of coequalizer diagrams
F r_____//_F k
[A,B]\[C,D] _____//_[A,B] ____//_Bm_P 
__
  _____
  ____
F fflffl ____//_F fflffl fflffl____
[A,B]\[C,D]B(1r)___//_[A,B]B(1k)______//BP
so that the induced dotted arrow is an isomorphism. ___
24
~=
One can show that the isomorphism Bm P  ! BP coincides with the com
posite
Bm P  i*!B P  ~=S(BP ) ffl!BP,
where S denotes the cubical singular functor S : S ! cSets.
A cubical poset morphism g : P ! Q is a poset morphism which respects
the cubical structure of intervals in the sense that in all diagrams
f i
1k ___~=__//_[A,_B]____//_P
g* g g
fflffl~= fflffl fflffl
1l____f//_[g(A), g(B)]i//_Q
the uniquely determined functor g* : 1k ! 1l is a cubical functor. All cubical
functors ` : 1n ! 1m are cubical poset morphisms.
Suppose that P and Q are cubical posets, and consider the product poset
P x Q. Any interval [(A1, A2), (B1, B2)] has the form
[(A1, A2), (B1, B2)] ~=[A1, B1] x [A2, B2],
~= ~=
and so the parameterizations 1r ! [A1, B1] and 1s ! [A2, B2] together induce
a parameterization ~
1r+s =![(A1, A2), (B1, B2)].
It's plain from these identifications that any inclusion of intervals in P x Q *
*is a
cubical morphism, and of course intervals in P xQ are closed under intersection.
In particular, the product poset P x Q is a cubical poset. It is also clear that
the projections P x Q ! P and P x Q ! Q are cubical poset morphisms.
Recall that the minimal nerve Bm P of a cubical poset P is defined by the
identification
Bm P = lim! k
[A,B]
where the limit is indexed over the intervals [A, B] P and k is the dimen
sion of [A, B]. Write [A, B] : k ! Bm P for the canonical cubical set map
corresponding to the interval [A, B]. It follows that there is an isomorphism
Bm P Bm Q ~= lim! r s,
[A1,B1],[A2,B2]
where [A1, B1] and [A2, B2] vary over the intervals with corresponding dimen
sions r, s of P and Q respectively. The composites
r s ~= r+s [(A1,A2),(B1,B2)]!Bm (P x Q)
determine a cubical set map
: Bm P Bm Q ! Bm (P x Q) (11)
25
The construction can plainly be reversed, and it follows that is an isomor
phism. The isomorphism is natural with respect to cubical poset morphisms
in both variables.
5 Cubical subdivision
Write N n for the poset of nondegenerate cells in the cubical complex n.
Observe that an object oe of N n can be identified with a coface (d, ffl) : 1k *
*! 1n,
and hence with an interval [A, B] P(n_). Here, A is identified with the image
of (0, . .,.0) under (d, ffl), while B is the image of (1, . .,.1).
Write NP(n_) for the poset of intervals in P(n_). We have just displayed a
poset isomorphism
N n ~=NP(n_).
Under this identification, a face relation ø oe between nondegenerate cells
corresponds to an inclusion of intervals [C, D] [A, B], where A C D B.
The corresponding interval [[C, D], [A, B]] in the poset N n can be identified
up to isomorphism with the product poset [A, C]opx [D, B] (with C D), via
the map (E, F ) 7! [E, F ]. There is a parameterization
1s x 1t! [Cc, Ac] x [D, B] `x1![A, C]opx [D, B] ~=[[C, D], [A, B]]
which arises from the standard parameterizations for [Cc, Ac] and [D, B] and
the canonical isomorphism ` : [Cc, Ac] ! [A, C]op
An intersection
[[C1, D1], [A1, B1]] \ [[C2, D2], [A2, B2]]
of intervals in N n consists of intervals [E, F ] such that
E 2 [A1, C1] \ [A2, C2] = [A1 [ A2, C1 \ C2]
and
F 2 [D1, B1] \ [D2, B2] = [D1 [ D2, B1 \ B2].
It follows that the displayed intersection is equal to the interval
[[C1 \ C2, D1 [ D2], [A1 [ A2, B1 \ B2]].
This interval can be empty, of course.
Any cubical morphism ` : P(n_) ! P(m_) restricts to cubical morphisms
` : [E, F ] ! [`(E), `(F )]. There is a commutative diagram
[[C, D],O[A,OB]]`*__//[[`(C), `(D)],O[`(A),O`(B)]] (12)
~= ~=
 
[A, C]opx [D, B]`opx`//_[`(A), `(C)]opx [`(D), `(B)]
26
The same observation applies to inclusions
[[C1, D1], [A1, B1]] [[C2, D2], [A2, B2]]
of intervals in N n: such a map coincides up to isomorphism with a product
[A1, C1]opx [D1, B1] ! [A2, C2]opx [D2, B2]
of inclusions of intervals.
It follows in particular that the poset N n has a cubical structure, and we
define the cubical set sd n by
sd n = Bm N n.
We now know as well that any cubical set map ` : n ! m induces a morphism
of cubical posets ` : N n ! N m , and hence functorially determines a cubical
set map `* : sd n ! sd m . Note finally that the assignment [A, B] 7! B
defines a cubical poset map fl : N n ! P(n_) which respects all cubical structu*
*re
maps ` : n ! m in the sense that all diagrams of poset maps
`*
N n ____//_N m
fl fl
fflffl fflffl
P(n_)_`*__//P(m_)
commute. It follows that there are cubical set maps fl : sd n ! n which
respect all cubical set maps n ! m .
The subdivision sdX of a cubical set X is defined by
sdX = lim!sd n.
n!X
This construction is functorial in X, and there is a natural transformation
fl : sdX ! X
which is induced by the maps fl : sd n ! n.
Suppose in general that P is a cubical poset, and that Q P is a subposet
which is closed under taking subobjects in the sense that if A B and B 2 Q
then A 2 Q. Then the induced poset morphism
[A, B]Q [A, B]P
is an isomorphism if [A, B]Q is nonempty. It follows that Q is a cubical poset,
and the inclusion Q P is a morphism of cubical posets.
Example 22. Suppose that K is a cubical complex in the sense that K n
for some n, and let NK denote the poset of nondegenerate cells in K. Then
as a subposet of N n, NK is closed under taking subobjects, and is therefore
a cubical poset.
27
Suppose that K n is a cubical complex. Then the intersection of any
two nondegenerate cells oe : k K and ø : m K is again a nondegenerate
cell oe \ ø : r K, simply because this is true in n. It follows that there *
*is a
coequalizer G G
r ' k ! K
oe\ø oe
which is determined by the covering {oe : n K} arising from the collection
of nondegenerate cells. The functor K 7! sdK plainly has a right adjoint, and
therefore preserves colimits, so that the picture
G G
sd r ' sd k ! sdK
oe\ø oe
is a coequalizer.
There is a comparison of fork diagrams
F r________//F k //
oe\øsd ________//oesd ________sdK
~= ~= i
F fflffl____//_F fflffl fflffl
oe\øBm N r ____//_oeBm N k _____//Bm NK
which becomes a comparison of coequalizers in simplicial sets after triangulati*
*ng.
In effect, the poset NK is covered by the posets N k corresponding to non
degenerate cells oe : k ! K, so that the fork
G G
BN r ' BN k ! BNK
oe\ø oe
is a coequalizer of simplicial sets. Now use Lemma 21.
It follows from Proposition 14 that the induced map i : sdK ! Bm NK is
an isomorphism. There is a cubical set monomomorphism Bm NK B NK. It
follows that the cubical subdivision functor preserves monomorphisms between
cubical complexes, and this in turn implies the following:
Lemma 23. The functor X 7! sdX preserves monomorphisms of cubical sets.
Proof. Use a relative skeletal decomposition for a monomorphisms i : X !
Y , in conjunction with the fact that all induced maps sd@ n sd n_are_
monomorphisms. __
We have also proved
Lemma 24. Suppose that K n is a cubical complex. Then there is an
isomorphism ~
i : sdK =!Bm NK.
28
There is a poset isomorphism
~= xn
P(n_) ! P(1_) (13)
where the composite ~
P(n_) =!P(1_)xn pri!P(1_)
with the ith projection functor pri coincides with the degeneracy functor si :
P(n_) ! P(1_) which is defined by intersection with the subset {i}. In other
words, (
si(A) = ; if i =2A,
1_= {1} if i 2 A.
This is on account of the identification
n_~=1_t . .t.1_.
The poset isomorphism (13) induces a cubical poset isomorphism
~= xn
NP(n_) ! NP(1_)
of the corresponding posets of intervals. Any interval [A, B] of dimension n in
P(m_) induces a cubical morphism [A, B] : P(n_) ! P(m_) in the usual way, and
there is a commutative diagram
[A,B]*
NP(n_) _______//NP(m_)
~= ~=
fflffl fflffl
NP(1_)xn [A,B]*//_NP(1_)xm
To describe the bottom horizontal map, write d : n_~=B  A m_for the unique
ordered monomorphism associated to the interval [A, B]. Then the composite
NP(1_)xn [A,B]*!NP(1_)xn pri!NP(1_)
factors through the object [;, ;] if i =2B, factors through the object [1_, 1_]*
* if i 2 A
and is the projection
prd1(i)
NP(1_)xn ! NP(1_)
if i 2 B  A.
Suppose that the subset A n_determines an ordered set monomorphism
d : r_! n_via the composite
r_~=A n_
in the usual way. Then restriction along d (aka. intersection with A) induces
a cubical morphism d* : P(n_) ! P(r_) in the usual way, and this morphism
29
induces a cubical poset morphism d* : NP(n_) ! NP(r_) on the corresponding
posets of intervals. There is a corresponding commutative diagram
*
NP(n_) ___d___//NP(r_)
~= ~=
fflffl fflffl
NP(1_)xn __d*_//NP(1_)xr
and each composite
* pri
NP(1_)xn d! NP(1_)xr ! NP(1_)
coincides with the projection prd(i): NP(1_)xn ! NP(1_).
The cubical poset morphisms fln : NP(n_) ! NP(1_) respect the cubical
structure functors si. It follows that there is a commutative diagram
~=
NP(n_)_____//NP(1_)xn
fln (fl1)xn
fflffl fflffl
P(n_)__~=__//P(1_)xn
Finally, we know that the minimal nerve construction Bm takes products to
products, and that sd n = Bm NP(n_), while Bm P(n_) = n. It follows that
there are isomorphisms ~
sd n =!(sd 1) n
which respect the cubical structure functors, and that there are commutative
diagrams
~=
sd n _____//(sd 1) n

fl fln
fflffl fflffl
n ____~=_//( 1) n
There is a homemorphism h :  sd 1 !  1 which is defined by sending
the vertex [;, ;] to 0, the vertex [1_, 1_] to 1, and the vertex [;, 1_] to 1=2*
*, and then
extending linearly. The map fl* :  sd 1 !  1 is the affine map which sends
[;, ;] to 0 and the other two vertices to 1. There is plainly a convex homotopy
H : h ! fl*. Since it's convex, and h and fl* have the same effect on the verti*
*ces
[;, ;] and [1_, 1_], the homotopy H is constant on the images of these vertices.
Topological realization takes products to products, so there are commu
tative diagrams
~=
 sd n_____// sd 1xn
fl* flxn*
fflffl fflffl
 n ___~=__//_ 1xn
30
Any interval [A, B] of dimension n in P(m_) induces a diagram
[A,B]* m
 sd n_______// sd 
~= ~=
fflffl fflffl
 sd 1xn[A,B]*//_ sd 1xm
in which the map [A, B]* :  sd 1xn !  sd 1xm is defined by the composites
 sd 1xn [A,B]*! sd 1xm pri! sd 1
where pri[A, B]* factors through the vertex [;, ;] if i =2B, factors through [1*
*_, 1_]
if i 2 A and coincides with the projection prd1(i):  sd 1xn !  sd 1 if
i 2 B  A. Again, d is the unique ordered monomorphism n_~=B  A m_
which is determined by the interval [A, B].
Similarly, if A n_of order r determines the ordered set monomorphism
d : r_! n_in the usual way, then there is a commutative diagram
*
 sd n___d___// sd r
~= ~=
fflffl fflffl
 sd 1xn__d*_// sd 1xr
where the composite
* pri
 sd 1xn d!  sd 1xr !  sd 1
is the projection prd(i).
It follows that the product homeomorphisms hxn :  sd 1xn !  1 de
termine homeomorphisms hn :  sd n !  n which commute with all maps
induced by cubical set maps ` : n ! m in the sense that the diagrams
 sd n_`*_//_ sd m 
hn~= ~=hm
fflffl fflffl
 n __`*___//_ m 
commute. The homotopies
H* :  sd 1xn x  1 !  1xn
which are defined by
Hn*(t1, . .,.tn, s) = (H(t1, s), . .,.H(tn, s))
31
induce homotopies
H0n:  sd n x  1 !  n
from hn ! fln* which respect cubical set maps ` : n ! m in the sense that
the diagrams
 sd n x  1`*x1//_ sd m  x  1
H0n H0m
fflffl fflffl
 n _____`*______//_ m 
commute.
We have assembled a proof of the following
Theorem 25. There is a homeomorphism h :  sdX ! X which is natuaral
in cubical sets X, and a natural homotopy H :  sdX x  1 ! X from h to
fl*.
6 Cubical excision
Lemma 26. 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 cubical
set maps
K __ff_//S(U1) [ S(U2)
i 
fflffl fflffl
L ____fi___//S(Y )
where i is an inclusion of finite cubical sets. Then for some n 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
sdnK _____//S(U1)8[8S(U2)___
______
i* ____________
fflffl_______ fflffl
sdnL _________//S(Y )
admitting the indicated lifting.
32
Proof. Suppose that K K0 L. We begin by showing inductively that there
is an N such that the composite
N S(fi0)*
sdN K0j!S sdNK0 Sh!SK0 ! SY
factors uniquely through a map ~fi0: sdN K0 ! S(U1) [ S(U2), where fi0 is the
composite K0 L fi!S(Y ), and fi0*: K0 ! Y is the adjoint of fi0.
Note that a map f : K00! S(Y ) lifts to S(U1) [ S(U2) if and only if for
every cell oe : k ! K00the adjoint (foe)* :  k ! Y of the composite foe lifts
to U1 or U2.
Suppose that the composite
n Sfi0*
sdnK0j!S sdnK0 Sh!SK0 ! S(Y )
lifts to S(U1) [ S(U2). Then I claim that the composite
n+1 Sfi0*
sdn+1K0j!S sdn+1K0 Sh!SK0 ! S(Y ) (14)
lifts to S(U1) [ S(U2).
In effect, suppose that oe : s ! sdn+1K0 is a cell of sdn+1K0. Then the
realization oe* :  s !  sdn+1K0 is carried on a cell ø : r ! sdnK0 in the
sense that there is a commutative diagram
 s____f____//_ r
oe* ø*
fflffl~= fflffl
 sdn+1K0_h___// sdnK0
The cell ø lifts to S(U1)[S(U2) by assumption, so that its adjoint ø* :  r ! Y
factors through either U1 or U2. The adjoint oe* of oe is the composite
 s f! r ø*!Y,
so that oe* factors through either U1 or U2. It follows that the composite (14)
factors through S(U1) [ S(U2).
Suppose that L0 L is obtained from K0 by attaching a cell, so that there
is a diagram
@ r ____//_K0
 
 
fflffl fflffl
r ______//L0
Suppose further that there is some n such that the composite
n S(fiK0)*
sdnK0j!S sdnK0 Sh!SK0 ! SY
33
lifts to S(U1) [ S(U2), where fiK0 is the composite K0 L fi!S(Y ). There is a
number m such that the composite
m S(fiL0*)
sdm r j!S sdm r h!S r ! SL0 ! S(Y )
lifts to S(U1) [ S(U2) by a standard Lebesgue number argument.
Now consider the diagram
S@ r_____//SK0 (15)
 
 SfiK0*
fflffl fflfflfflffl
S r Sfi__//_S(Y )
r*
Then there is a number N such that after refinement along the maps S(hN )j
the diagram (15) lifts to a commutative diagram
sdN@ r ________//sdNK0
 
 
fflffl fflffl
sdN r _____//_S(U1) [ S(U2)
The subdivision functor preserves pushouts, so there is a uniquely determined
lift sdN L0! S(U1) [ S(U2) of the composite
N SfiL0*
sdN L0j!S sdNL0 Sh!SL0 ! S(Y ).
Thus, we can suppose that we've found the requisite number N. The com
posite
N SfiK0*
sdN K0j!S sdNK0 Sh!SK0 ! S(Y ).
is naturally homotopic to the composite
N fi
sdN K0fl!K0 L ! S(Y )
for all complexes K0 between K and L.
The map fi : K ! S(Y ) already lifts to S(U1) [ S(U2) so that there is a
commutative diagram
j // N k ShN // k S(ø)//
sdN k _____S sd  _~=__S  _____S(U1) [ S(U2) (16)
oe* oe* oe* 
fflffl fflffl~= fflffl fflffl
sdNK ___j_//_S sdNKShN_//_SK_S(ff*)//_S(Y )
for all cells oe : k ! K, where the composite
 k oe*!K ff*!Y
34
factors through some map ø :  k ! Ui. It follows that the restriction to
sdN k of the homotopy S(ff*)S(hN )j ' ffflN stays inside S(U1) [ S(U2). This
is true for all oe, so that the homotopy of the lifting sdN K ! S(U1) [ S(U2)
with the composite
N ff
sdNK fl!K ! S(U1) [ S(U2)
stays inside S(U1) [ S(U2).
It also follows from the commutativity of diagram (16) that the homotopy
of diagrams preserves base points: in particular, take the cell oe : _k_! K to *
*be
the base point 0 ! K. __
Theorem 27 (cubical excision). Suppose that Y is covered by open subsets
U1 and U2. Then the induced map of cubical sets i : S(U1) [ S(U2) S(Y ) is
a weak equivalence.
Proof. Suppose that X is a pointed cubical set. The category F*(X) of pointed
finite cubical subsets K X has all finite limits is plainly filtered, and the*
*re is
an isomorphism
ßqX ~= lim!ßqK.
K2F*(X)
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. There is an N 0
such that the diagram
flN // x //
sdN 0 _~=__ 0 _____S(U1) [ S(U2)
  
 x i
fflffl fflffl fflffl
sdN K _flN__//K____!____//S(Y )
is pointed homotopic to a diagram
__x__//
sdN 0 S(U1)7[7S(U2)____
 ___________
 ___oe_____i
fflffl_____ fflffl
sdN K _________//S(Y )
in which the indicated lift oe exists. But flN is a weak equivalence, so that
[ff0] = flN*[ff00] for some ff00. But then [ff] = !*flN*[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
35
of cubical 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 N 0 such that the composite diagram
fln// i1 //
sdNK _____K _____ S(U1) [ S(U2)
j* i
fflffl fflffl
sdN L__fln_//L____i2__//S(Y )
is pointed homotopic to a diagram
i01
sdN K ____//_S(U1)8[8S(U2)____
______
j* ___ø______i__
fflffl______ fflffl
sdNL ____i02__//S(Y )
in which the indicated lifting exists. Again, the maps fln are weak equivalence*
*s,
so that [fi] = fln*[fi0] for some [fi0] 2 ßq sdNK and
i1*[fi] = i1*flN*[fi0] = i01*[fi0] = ø*j*[fi0].
Finally, flN*j*[fi0] = j*[fi] = 0 so that j*[fi0] = 0 in ßq sdNL and_so_i1*[f*
*i] = 0
in ßqS(U1) [ S(U2). __
The category cSets of cubical sets is a category of cofibrant objects for a
homotopy theory, for which the cofibrations are inclusions of cubical sets 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 [8*
*]).
Corollary 28. Suppose that the diagram
F n //
i@ _____X
 
 
F fflffl fflffl
i n ______//Y
36
is a pushout in the category of simplicial sets. Then the diagram of cubical set
morphisms F
i S@ n ____//_SX
 
 
F fflffl fflffl
iS n ______//SY 
is homotopy cocartesian.
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 all @ n to an open subset Ui of  n (by radial
projection) such that @ n Ui is a homotopy equivalence. We can therefore
assume that the inclusion
G G
@ n (  n) \ U
i i
is a homotopy equivalence. We can also assume that there is an open subset
Vi  n such that the inclusion is a homotopy equivalence, such that Vi\Ui
Ui is a homotopy equivalence, and such that  n = Vi[ Ui. The net result is
a commutative diagram
F n //
iS@  ________SX
'  III '
' fflfflF fflffl
S(V \ U) ____//_S(U \ ( i n))__//S(U)
  
 I  II 
fflffl F fflffl fflffl
S(V )____'_____//iS n ________//_SY 
of cubical set homomorphisms in which all vertical maps are cofibrations and
the labelled maps are weak equivalences. The the composite diagram I + IIis
homotopy cocartesian by cubical excision (Theorem 27), 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. __
Theorem 29. Suppose that Y is a topological space, and let ffl : S(Y ) ! Y be
the adjunction map arising from the cubical set singular functor S and its left
adjoint   : cSets ! Top . Then the map ffl : S(Y ) ! Y is a weak equivalenc*
*e.
Proof. The cubical singular functor S : Top ! cSets preserves weak equiv
alences. In effect, all spaces are fibrant, so the standard construction which
replaces a map by a fibration can be used to show that any weak equivalence
37
f : X ! Y has a factorization
j
X ____//_@@Z
@@ ß
f@@__@fflffl
Y
where ß is a trivial fibration and j is a section of a trivial fibration Z ! X.
It is therefore enough to show that the cubical singular functor takes trivial
fibrations to weak equivalences. Finally, if ß : Z ! Y is such a trivial fibrat*
*ion,
then it has the right lifting property with respect to all inclusions @ n *
* n,
so that the induced map S(ß) : SZ ! SY has the right lifting property with
respect to all inclusions @ n n by adjointness. We already know that this
means that S(ß) is a weak equivalence (in fact, a homotopy equivalence).
The functor Z 7! S(Z) therefore preserves weak equivalences. We can thus
assume that Y = X for some simplicial set X.
The functor Z 7! S(Z) also preserves disjoint unions and filtered colimits
of CW complexes (because the spaces  n are compact). It also preserves
homotopies, and therefore preserves contractible spaces; in particular, the map
ffl : S( n) !  n is a weak equivalence for all standard simplices n.
Finally, we can induct along skeleta of simplicial sets X and suppose that
the map ffl : S( skn1X) !  skn1X is a weak equivalence for all simplici*
*al
sets X. But then the induced diagram
F n
x2NXn S(@ )____//S( skn1X)
 
 
F fflffl fflffl
x2NXn S( n)______//_S( sknX)
is homotopy cocartesian for all simplicial sets X by Corollary 28. The various
occurrences of ffl then give a comparison of homotopy cocartesian diagrams, and
the map ffl : S( sknX) !  sknX is a weak equivalence by the gluing_lemma
[8, II.8.8]. __
Corollary 30. The counit map j : X ! S(X) is a weak equivalence for any
cubical set X.
Proof. The map j is a weak equivalence of cubical sets if and only if the induc*
*ed
map j : X ! S(X) is a weak equivalence of topological spaces. This,_
however, follows from a triangle identity and Theorem 29. __
38
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39