Cubical structures, homotopy theory

Annali di Matematica pura ed applicata (IV), Vol. CLXXVIII (2000), pp. 317-324

Cubical Structures, Homotopy Theory(*).

ROSA ANTOLINI

Abstract. - We investigate the categories of semi-cubical complexes with or without degeneracies. We prove that the Kan condition does not imply that a semi-cubical complex admits degeneracies and that, unlike the simplicial case, there is no cubical appro~cimation theorem while we prove such a theorem for semi-cubical complexes with degeracies.

Introduction.

In this paper we investigate the homotopy of the categories e + and e of semi-cubical complexes, with or without degeneracies. We call such complexes [+-se t s and I - sets, respectively.

The category e + shares most of the properties of the category of semi-simplicial sets and it is a possible alternative to it for carrying out a programme of combinatorial homotopy theory. In fact, the homotopy groups of [+-se t s originally defined by Kan in [K] using only the cubical structure are equivalent to the ones defined on the realisation and seem to be more natural than the ones defined simplicially. To continue inves- tigating the analogy between semi-cubical sets and semi-simplicial sets, in this paper we prove a cubical approximation theorem for [=]+-sets.

Unlike the simplicial case [RS] we can not establish a homotopy equivalence between the categories e and e + . In fact the realisation of the singular []+-sets of a topological space has the same homotopy type as the space itself [F] while the realisation of the singular [:]-sets of a topological space has the homotopy type of the space times ~ S 2 lAW]. In general, unlike the simplicial case, degeneracies play a crucial role in carrying out a programme of combinatorial homotopy theory; indeed, the homotopy relation defined on [::]+-sets would not even be an equivalence relation on Kan []::]-sets. In this paper we will prove that there is no Homotopy Extension Property for Kan [~-sets and the Kan condition does not imply that a [~-set admits degeneracies.

The author wishes to thank Prof. Sandro Buoncristiano for his constant help and advice.

(*) Entrata in Redazione il 1 settembre 1998. Ricevuta versione finale il 3 maggio 1999. Indirizzo dell'A.: Dipartimento di Matematica, Seconda Universit~ degli studi di Roma Tor

Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy. E-maih [email protected]

318 ROSA ANTOLINI: Cubical structures, homotopy theory

1. - N o t a t i o n s a n d d e f i n i t i o n s .

Let I ~ be the standard n-cube in R " given by I ~= [0, 1] *.

DEFINITION 1.1. - The D-category is the category whose objects are the standard n-cubes, for n = 0, 1 , . . . , and whose morphisms are generated by the maps

5~:i~__>i.+1, for i = l , . . . , n + l and e e { 0 , 1 }

defined by:

~ ( z l , . . . , z , ) = ( x l , . . . , x i - 1 , e , x / , . . . , x ~ ) .

The following relations hold:

5j r)i=di+lrSj, for j~<i and e, eoE {0, 1}.

DEFINITION 1.2. - A O-set X is a contravariant functor defined on the category [] to the category of sets,

X: [~---) Sets .

We write X. for X(I ' ) . If 2: I "--* I ~ is a morphism of [] then we write 4 * for X(4) and in particular ~ for X(SD. Then 4" is called a face map and for any o e X m 4 * ( a ) is a n-face of a.

A D-map is a natural transformation of functors. Denote by C the category of []-sets and O-maps.

DEFINITION 1.3. - The realisation, IXI, of a []-set X is given by identifying (4" x, t) o � 9

and (x, 4(t)) in the disjoint union I_[ X~ • I *, where 4 is any morphism in the []-catego- i=0

ry and the sets X, are given the discrete topology for all n.

EXAMPLE 1.4. - Associate with the tepologieal n-cube I ~ the Q-set In defined in the following way:

(I~)k = {f: Ik--->I ~ t f is a morphism in []}.

The realisation of this D-set is the n-cube with the usual topology.

A larger category, [~+, which contains [3, can be eonstructed by introducing degeneracies.

DEFINITION 1.5. - The category [3 + is the category whose objects are the standard n-cubes, I ~, n = 0, 1, ..., and whose morphisms are generated by the maps

~ : I ~ - - > I ~+1, i = l , . . . , n + l a n d e e { 0 , 1 }

and the maps

ei: I~--~I ~- l , i = 1, ..., n .

ROSA ANTOLINI: Cubical structures, homotopy theory 319

The latter are defined by:

e~(xl, . . . , x~) = (xl, . . . , xi-1, xi+l , . . . , x~).

The following relations hold:

eiej=ejei+ 1 if j<. i

{ 5~ej_l if i <j

ej S ~ = identity if i = j

5~-lej if j < i

The following lemma is a straight forward consequence of the relations above.

LEMMA 1.6. - Every morphism 4: I k__, I ~ in [] § factors uniquely as ~b i ~ ~b 2 with ~b 1 injective and ~b2 surjective. �9

DEFINITION 1 . 7 . - A [:]+-set X is a contravariant functor defined on the category []+ to the category of sets,

X: []+ ---->Sets.

We write X~ for X(I ~). If 4: I '~ ~ I m is a morphism in [] § then we write 4 * for X(4 ) and in particular tt i for X(ei) and, as above, ~ for X(/t~). I f 4 e [ ] + is injective then 4" is a face map and for any aeXm 4*(a) is an n-face of a. I f 4 is surjective 4* is called a degeneracy operator and for any oeXm 4*(0) is a degeneracy of a or, more simply, a de- generate cube.

Denote by C § the category of []+-sets and maps.

DEFINITION 1.8. - The realisation, IX~, of a D+-set X is given by identifying

(4*x, t) and (x, 4(t)) in the disjoint union I~ X, • I ~, where 4 is any morphism in the i=O

D+-category and the sets X~ are given the discrete topology for all n . The best known example of a D+-set is the singular [~+-set S(Y), derived from any

topological space Y. This is defined so that S(Y)p = y1p, and for any aeS(Y)p and any morphism 4: I ~--* I m of [] +, 4 * a = a4.

THEOREM 1.9 [F]. - Given a topological space X, then IS(X) I and X are homotopical- ly equivalent. �9

DEFINITION 1.10. - A D+-set is Kan if for each collection o f 2 n - 1 (n - 1)-cubes x~, j = 1, . . . , n, (o = 0 , 1, and ( j , w) ~ (i, s) such that ~ x j ~ = 3~j x~+l, i f j<-k and ~ x j ~ = =~y-lx~, i f j > k , then there exists xeX~ such that ~fx=x~' , ( j , oJ) ~ (i, s).

Denote by C~ the full subcategory of e + generated by the [:]+-sets which are K a n .

DEFINITION 1.11. - Between the categories C and e + two functors are defined.


The forgetful functor

F : e+ ---~ e

is defined in the obvious manner. The functor

G: C--~ C +

is defined by: G ( X ) k = {(~t, a)laeXr,/~: Ik---~I ~ a surjective morphism in •+} and by setting A * (/~, a) = (~b 2, ~b ~ (a)) where ~b = 2it and ~ = ~bl~ 2 is the unique factoring of Lemma 1.6.

For any []-set X, IX[ and [GX[ have the same cellular s t ructure and they are homeomorphic.

2 . - C u b i c a l E x t e n s i o n T h e o r e m .

To prove the Cubical extension Theorem we will use the combinatorial definition of homotopy groups of Kan [] § given in [4]. The proof that these homotopy groups are equivalent to the homotopy groups of the realisation is only an adaptation of the well known simplicial case and therefore is left to the reader [K2], [M].

DEFINITION 2 .1 . - Let X be a Kan � 9 Two n-cubes a, a' e X , , n >1 0 are homotopic (written a - a ' ) if ~ a = ~ a ' , i = 1, . . .n , e = 0, 1 and if there exists an n + 1-cube veXn+x such that ~ v = a , ~ v = a ' and ~ [ v = t t l ~ _ l a = t t l ~ _ l a ' , i = 1, ..., n, e= = 0 , 1 and (i , ~) ~ (1, 0) , (1 , 1).

Homotopy is an equivalence relation on the n-cubes of X, n/> 0.

DEFINITION 2 .2 . - Let (X, * ) be a based Kan [:]+-set, then denote by t ~ ( X , * ) the subset of X~ given by

- * i = l , n , e = 0 , 1 } Y2n(X, * ) = { a e X ~ [ ~ i a - n _ l , ...,

Then define the n-th homototrg group

tg~(X, * ) z~(X , * ) - , n >~ l .

DEFINITION 2.3. -- Le t (X, * ) be a based Kan [:]+-set and [a], [v] e z . ( X , * ), n I> 1. By the Kan condition there exists an (n + 1)-cube ~ e X . + 1 such that 8~ ~ = ~, a ~ + 1 ~ = v and ~ =*, ( i , t) ~ (n, 1) , (n + 1, 0 ) , (n + 1, 1). Then define [a][z] = [ ~ + 1 ~].

PROPOSITION 2.4 [K]. - The correspondence [a], [v] ~ [v][v] is a well defined product and it gives z . ( X , * ) a group structure, i f n I> 1. Fu r the rmore z n ( X , * ) is abelian i f n > l .

DEFINITION 2.5. - Le t X be a Kan []+-set and L a Kan [:]+-subset. Two n-cubes a, a '~X~,n>~O such that a~a, a ~ a ' e L ~ _ l are homotopic relatively to L (written


o ~L ~ i f 8~ ~ ---- 8 e O ' , i = 1, . . . , n , e = 0, 1, (i, e) # (1, 1) and if there exists an n-cube e Ls such that ~: 1 1 0, 81 O ~ 81 in L and if there exists an n + 1-cube v e Xs + 1 such that

8~ 8 ~ v = ~ 8~v= ~, and 8 [ ' g = [ . t l S ~ - l O ' = / - / 1 8 ~ _ l O ' , for (i, e) ;~ (1, 0),(1, 1),(2, 1). Homotopy relatively to L is an equivalence relation.

DEFINITION 2 . 6 . - Let (X, L, * ) be a based Kan [::]+-pair, then denote by t9 n(X, L, * ) the subset of Xs given by

~9~(X, L , * ) = { o e X ~ l ~ o = *~_~, i = 1 , . . . , n , e = 0 , 1,( i , e ) # ( 1 , 1); 81oeLn-~} .

Then define the n-th homotopy group relative to L

tgs(X, L, * ) a s ( X , L , * ) = , n >~ 2 .

~ L

DEFINITION 2.7. - Let (X,L, *) be a based Kan D+-pair and [~ *), n ~> 2. By the Kan condition there exists an n-cube 0 e L . such that 81 _ 10 = 8~ o and 8~ = 8~v, 8~0 = * , ( i , e) ~ (n - 1, 1),(n, 0) , (n , 1). By the Kan condition there also exists an (n + 1 )-cube ~ �9 X~ + 1 such that 81 ~ = ~ 8~ + 1 ~ = v, 8~ ~ = 0, and 8[ ~ = * , (i, e) ;~ (1, 1),(n, 1) ,(n + 1, 0) , (n + 1, 1). Then define [~ = [81+1~].

PROPOSITION 2.8 [ g ] . - The correspondence [~ Iv] ~ [~ is a well defined product and it gives an(X , L, * ) a group structure, if n I> 2. Fur thermore a s ( X , L, * ) is abelian if n > 2. �9

The simplicial techniques adapt to prove the following result.

THEOREM 2.9. - Le t (X, * ) be a based Kan D+-set. Then there is an isomorphism

as(V(X)):aAX,*) aASlXi,Sl*l)=aAIXl, l*l), n=O, 1,...

where the map y~(X):(X, * ) - - + ( S ] X ] , S I * J ) is such that ya(X)(an)(U~)= = l o s , u l. �9

For each based Kan [:]+-pair (X, L, * ) there is a long exact sequence

�9 . .an+l(X , L , * ) - - + a , ( L , * ) - + a n ( X , * ) - + a , ( X , L , * )--+...

where i . and j , are induced by the inclusions and 5 is defined by 5[a] = [Sa].


COROLLARY 2.10. - Let (X, L, * ) be a based Kan [=]+-pair. Then there is an isomorphism

zn(~p(X)): z~,(X, L, * ) - o z , ( S l X l , SIL l , Si* I) =

= z ~ ( I X I , ILl, I*1), n = 0 , 1 , . . . �9

Now we want to prove a theorem analogous to the Simplicial Extension Theorem [S] for the cubical case.

To prove this result degeneracies are essential, in fact the counterexample below proves that Kan []-sets do not have this property.

Consider the singular semi-cubical complex of a point, FS(xo). This is a Kan [:3-set. Let I 1 be the I3-set defined above, whose realisation is the unit interval with its usual topology.

Consider the constant map [I~ [--* IFS(xo)[. This map cannot be homotopic rel. [ @0, 0 [ to the only realised map which gives a generator of z i ([FS( * ) I) -- Z [J].

If we add degeneracies then we can prove the following theorem

THEOREM 2.11. - Let (Y, L) be a pair of R+-sets and X a Kan D+-set. For any map f: [Y[--* IX[ such that f] ILl = [gl, where g: L-->X is a D+-map, then there exists a [~+-map f ' : Y--* X and a homotopy

H: f - - If' lrellLI �9

PROOF. - By induction over the skeletons it is.enough to prove the theorem in the case Y is the K]+-set I~+ and L is its boundary ~§

The map

I, Ig+ I, * Ig(g§ I, *) ,

represents an element in z~( [X[ , Ig(~§ I, �9 ). By corollary 2.10 there is an isomorphism ~ ( ] Z [ , Ig(I~+) [, * ) - * a n ( X , g(~§ * ), then there exists f ':(Ic~§247 * )--) --* Of, g(Pa§ * ), such that I f ] = [[f ' I]- �9

3. - T h e f u n c t o r Q: e + - - , e ~ .

We defined the homotopy groups of a Kan D+-set X using only the cubical structure. In order to extend the definitions of all homotopy notions defined on the category e~ to the category e + we define a functor Q: e + - * e~ such that

i) Q sends homotopic maps into homotopic maps

ii) If X~ C + then there is a homotopy equivalence [q]: ]Q(X) I ~ IX[

iii) If X~ e~ then there is a O+-homotopy equivalence q: Q(X)--*X.

Condition i) implies that every homotopy notion on the category e + yields by the composition with the functor Q a homotopy notation on the category C~.


Condition ii) implies that any homotopy notion yielded by the composition with the functor Q is equivalent to the usual homotopy notion on the realisation.

Condition iii) implies that on the category e~ the homotopy notions induced by composition with the functor Q coincide with the original ones.

DEFINITION 3.1. - Let u: 3~)I ~(~)-->X be a morphism, where e(u) = 0, 1, n(u) >I 1, is an integer and i (u) is an integer such that 0 ~< i (u) <~ n(u). Let V be the set of all morphisms u then we have a diagram

]*]" "1 i(u), s(u) l - I n(u) V

v

z(x) ) X

O<~i<<.n. The components of Z(X) are all the morphisms u e V and v(X) is induced by the in-

to i(u), e(u) clusions of =n(u) into I~(+ u) for all u ~ V. Let Ex(X) be the amalgamated sum of the above diagram, i.e.

E x ( X ) = v Z (u ) o = v(u) a

Let w(X): X---)Ex(X) be the canonical inclusion. Define inductively E x "~ (X) = E x ( E x n - 1 (X)). We then have an infinite sequence of

inclusions

w(X)) w 1 (X)) E x 2 ( X ) w2 (x) X Ex (X) ) . . . .

LEMMA 3.2. - The direct limit Q(X) of the E x n ( X ) is a Kan [:]+-set. �9

The same construction defmes a functor Q: e-- )eK.

4. - A Kan H-set tha t does n o t admi t degenerac ies .

It is well known that any Kan A-set admits degeneracies [RS]. We want to prove that this property does not hold for Kan H-sets.

The simpllcial proof in [RS] adapts to prove the following lemma.

LEMMA 4.1. - A H-set X admits degeneracies if and only if there exists a H-retraction r: FG(X) --->X satisfying r()~, r(tt, o)) = r(tt)~, a), for all ~t,/t e [3 + , and a e X . �9

We will apply the functor Q to prove that there is a Kan [::]-set which does not admit degeneracies.


Let I~ be the D-set defined above. We will prove that Q(I 1) does not admit degeneracies, i.e. that there is no retraction r : FGQ(I~)---~Q(I~).

Assume that such a retraction exists. Le t a be a 0-cube in Q(I~), then (el, a) is a 1- cube in FGQ(I1). Assume that r(el , a) -- v, then ~l~ a) --r(~l~ a) ) - - r ( I d , a) = = a = r (a ] (e l , (~))= a~ r(el, o). This proves that there exists v e Q(I~) such that ~ v = = 3~ v = a. We want to see that this is not possible. I f such a 1-cube v e Q(I~) exists, then there is an integer p such that v e E x P ( I 1) and v ~ E x p - l ( I 1) and we have a map u: 3]' ~(~)---~Exp-I(I 1) and a commutative diagram

~ , E(u) _2, ExP- I(I~)

v(u) $ w(u)

I~ -~ SxP(I~)

I f s(u)= 0 then ~ v e E x p - l ( I ~ ) and ~ v ~ E x p - l ( I ~ ) . Therefore ~ v ~ 9Iv. How- ever we assumed that ~ v = ~ v = a, then (i, s) ;~ (1, 0). Analogously we prove that (i, s) ~ (1, 1). Then such a 1-cube v does not exist. Then there is no retraction with the required properties.

R E F E R E N C E S

[AWl

[F]

[J] [K] [K2]

[M]

[RS]

IS]

R. ANTOLINI - B. WIEST, The singular cubical set of a topological space, Proc. Camb. Phil. Soc., 126 (1999), pp. 149-154. H. FEDERER, Lectures in Algebraic Topology, it Brown University, Providence, R.I., 1962. I. M. JAMES, Reduced Product Spaces, Annals of mathematics, 62 (1955), pp. 170-197. D. M. KAN, Abstract Homotopy I, Proc. Nat. Acad. Sci., 41 (1955), pp. 1092-1096. D. M. KAN, A combinatorial definition of homotopy groups, Ann. of Maths., 67 (1958), pp. 282-312. J. P. MAY, Simplicial Objects in Algebraic Topology, Van Nostrand Co., Inc., Princeton N.J. 1967. C. P. ROURKE - B. J. SANDERSON, A-sets I: Homotopy theory, Quart. J. Math. Oxford (2), 22 (1971), pp.321-38. n. J. SANDERSON, The simplicial extension theorem, Math. Proc. Camb. Phil. Soc., 77 (1975) pp. 497-98.

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Cubical structures, homotopy theory