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 non-abelian 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_n-cube,_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 n-cube 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 non-degenerate cells of the n-cube. 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 n-cubes) have posets of non-degenerate 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 n-cube 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 n-cells 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 ! S|X| 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 n-fold 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 prd-1(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 n-tuple 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 k-elemen* *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|| fflffl|0 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|| fflffl|0_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 order-preserving 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): 1n-1 ! 1n, defined by d(i,ffl)(fl1, . .,.fln-1) = (fl1, . .,.iffl, . .,.fln-1). Suppose that i < j. Then there is a commutative diagram of face functors d(i,ffl1)//_n-1 1n-2 1 (2) d(j-1,ffl2)|| |d(j,ffl2)| fflffl| fflffl| 1n-1 d(i,ffl1)//_1n if n 2. If i = j there is a diagram ;_______//1n-1 (3) | | (i,1) | |d fflffl| fflffl| 1n-1 d(i,0)//_1n-1 The degeneracy functor sj : 1n ! 1n-1 is the projection which forgets the jth factor, so that sj(fl1, . .,.fln) = (fl1, . .,.flj-1, 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) sj-1|| |sj| fflffl| fflffl| 1n-1 d(i,ffl)//_1n and (i+1,ffl) 1n d____//_1n+1 if i j. (7) sj|| |sj| fflffl| fflffl| 1n-1 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): 1n-1 ! 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 Ac-Bc = 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 set-valued functor X : op ! Set. Write Xn = X(1n), and call this set the set of n-cells 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 n-cell n is the contravariant functor on the box category which is represented by 1n. Thus, n has m-cells 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 n-cells 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 set-valued 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 n-cells 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 ! Xn-1 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 non-degenerate otherwise. Define the n-skeleton sknX for a cubical set X to be the subcomplex which is generated by the k-cells 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 = sj-1d(i,0)y, while sjsi(d(i,0)y) = sisj-1(d(i,0)y) = six = sjy. All degeneracies are injective, so that y = sid(i,0)y, and sif(x) = sisj-1d(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 n-1 | | | | F fflffl| fflffl|| n ______//_skX x2NXn n where NXn denotes the non-degenerate part of Xn, and @ n = skn-1 n. In other words, there is a good notion of skeletal decomposition for cubical sets. The object @ n is the subcomplex of the standard n-cell which is generated by the faces d(i,ffl): n-1 ! n. It follows from the fact that the diagram (2) is a pullback in the box category that there is a coequalizer G G n-2 ' n-1 ! @ n (ffl1,ffl2) (i,ffl) 0 i 2 by a simple com- binatorial argument, while there is a single non-degenerate 2-cell 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 1-skeleta 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 1-cell : 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 n-1 m _____// ~=|| |~=| fflffl| fflffl| n+m-1 _d(i,ffl)//_ n+m and 1 d(j,ffl)n m n m-1 _____// ~=|| |~=| fflffl| fflffl| n+m-1 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): n-1 ! n, the K m is isomorphic to the subcomplex of n+m which is generated by the list of faces d(i,ffl): n+m-1 ! n+m . Similarly, * *if L m is the subcomplex generated by faces d(j,ffl): m-1 ! 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+m-1 ! 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| ~=| 1|xn 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 n-1) [ ( 1 @ n-1) 1 n-1 ~= n. imply that the simplicial set inclusion | un(0,ffl)| | n| can be identified u* *p to isomorphism with the inclusion (| 0| x | n-1|) [ (| 1| x |@ n-1|) | 1| x | n-1|, 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|| fflffl|oei 1 fflffl| k 0 _____//K 0 1 0 || 1|øi(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 non-degenerate 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 NLn-k) ! N(K L)n 0 k n is surjective. In effect, the corresponding function G (Kk x Ln-k) ! (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 n-cells 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 = sj-1d(i,0)v, while siu = sisj-1d(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 n-cells 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 skn-1X together with the n-cells 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 non-degenerate, and write X0 for the smallest subcomplex of X containing skn-1X 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 = skn-1, 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 non-degenerate 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 n-cell 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 non-empty intervals [A, B] of P 21 2)any inclusion i : [A, B] [C, D] of non-empty 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 non-empty 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 non-empty 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 m-cell (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 non-empty 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 non-empty 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 non-degenerate 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 non-degenerate 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 non-empty. 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 non-degenerate 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 non-degenerate cells oe : k K and ø : m K is again a non-degenerate 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 non-degenerate 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 prd-1(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|| fl|n| 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 1|xn fl*|| |flxn*| fflffl| fflffl| | n| ___~=__//_| 1|xn 30 Any interval [A, B] of dimension n in P(m_) induces a diagram [A,B]* m | sd n|_______//| sd | ~=|| |~=| fflffl| fflffl| | sd 1|xn[A,B]*//_| sd 1|xm in which the map [A, B]* : | sd 1|xn ! | sd 1|xm is defined by the composites | sd 1|xn -[A,B]*---!| sd 1|xm 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 prd-1(i): | sd 1|xn ! | 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 1|xn__d*_//| sd 1|xr where the composite * pri | sd 1|xn -d! | sd 1|xr --! | sd 1| is the projection prd(i). It follows that the product homeomorphisms hxn : | sd 1|xn ! | 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 1|xn x | 1| ! | 1|xn 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 K0-j!S| sdNK0| Sh---!S|K0| ----! 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* sdnK0-j!S| sdnK0| Sh---!S|K0| --! S(Y ) lifts to S(U1) [ S(U2). Then I claim that the composite n+1 Sfi0* sdn+1K0-j!S| sdn+1K0| Sh----!S|K0| --! 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)* sdnK0-j!S| sdnK0| Sh---!S|K0| -----! 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| ! S|L0| -----! S(Y ) lifts to S(U1) [ S(U2) by a standard Lebesgue number argument. Now consider the diagram S|@ r|_____//S|K0| (15) | || | |SfiK0*| fflffl| fflffl|fflffl| 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 L0-j!S| sdNL0| Sh---!S|L0| ----! S(Y ). Thus, we can suppose that we've found the requisite number N. The com- posite N SfiK0* sdN K0-j!S| sdNK0| Sh---!S|K0| ----! S(Y ). is naturally homotopic to the composite N fi sdN K0-fl-!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| sdNK|ShN_//_S|K|_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 ßq|X| ~= lim-!ßq|K|. 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 ßq|K|. 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 ßq|S(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 ßq|L|. 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 ßq|S(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| ____//_S|X| | | | | F fflffl| fflffl| iS| n| ______//S|Y | 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|@ | ________S|X| ' || III '|| ' fflffl|F fflffl| S(V \ U) ____//_S(U \ ( i| n|))__//S(U) | | | | I | II | fflffl| F fflffl| fflffl| S(V )____'_____//iS| n| ________//_S|Y | 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(| skn-1X|)| ! | skn-1X| is a weak equivalence for all simplici* *al sets X. But then the induced diagram F n x2NXn |S(|@ |)|____//|S(| skn-1X|)| | | | | 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 ! 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