Category of cubical objects and category of simplicial objects

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1981)

Ad e l a Sw i a t e k ( T o r u n )

Category of cubical objects and category of simplicial objects

The set of singular simplexes of a topological space, by means of which one defines homology groups of the latter, gave rise to the theory of simplicial objects over an arbitrary category A. The same homology groups one obtains by use of the set of singular cubes, after performing normalization.

Paper [4] by D. M. Kan initiated investigations of category of cubical sets. The present paper deals with cubical objects over an arbitrary cat­

egory A.

In part I, apart from giving a definition of category of cubical objects ( □ op,A ) over arbitrary category A, we define for abelian category A the functors

N C, K C: (D op, A)->(ch, A)

((ch,A) is a category of chain complexes), we describe the «-dimensional component of a cubical object X in terms of the complex N C(X) and prove a theorem on equivalence of the category (П ор,А ) with the category of incomplete cubical objects (□^op,A ).

In (1.6) an example of a cubical group object not fulfilling Kan’s extension condition is given. In part II the homotopy relation between maps of cubical objects over A is defined. This relation is an equivalence in case of an additive category. In the category of cubical sets (Щор, Set) it is identical with the homotopy relation of [4].

In part III the functor t : □ ~ > ( A op,Set) ascribing to a cube /" the simplicial set t ( I n) associated with some triangulation of I" is defined. The functor t generates a pair of adjoint functors t , Eh{t) between the categories (Д ор, Set), ( □ op, Set).

Using the functors t , Eh(t), one obtains a pair of adjoint functors

( □ op, л Mod) £й«)

(Д ор, я Mod) ( k M oc ! denotes the category of left K-modules).

II — Roczniki PTM — Prace Matemaiyczne XXII

Furthermore the definition of the latter can be applied to any abelian category.

In (III.2) one constructs a natural transformation of functors A: N c -+ к ■ Eh(t) in order to investigate the homotopical commutativity of

the diagram '

Eh(t)

( □ op, *Mod) --- ► (Д ор, «Mod)

(ch, RMod)

The problem whether A determines a homotopical equivalence of functors is open.

The theorems are given without proofs. Proofs and other details can be found in [5].

I. C A T E G O R Y O F C U B IC A L OBJECTS

1.1. Let /" denote an «-dimensional cube;

I n = { ( *i , *2 , •••> Q : 0 < t, < 1,

= 1 , 2, . . . , n ]

for n > 0 and 1° is a one-point set.

We distinguish the maps e^: Г ~1 -* Г , rjl,: given by the formulae:

4

i ,

..

(*1 > ^2» ■ • j — 1 ? ^5

...,

! for

= 4 (ti,

...■> fii + l) = (^1

...■,

1 »

1 » • • • 5

for i = 1,2,..., n+ 1.

The above maps satisfy the following relations:

(a) 4 - i 4 ,<5 = id/«,

= 1,2,.. . , n, S = 0 , 1 , (b) Pi,5 pj,a _ pj+\,a i,ô

ьп + 1 fcn fcn + 1 fcn » i <

< 5 , ^о - = 0 , 1 , (c) 4 - i d +1 = ri- iVn,

<

(d) У*] pM — 1

wn - 1 Ц — 1 4n

9

^<

^т-4

о II

(e) yjj — pi yjj

Чп—1^п Ьп— 1 *fn — 2 >

^

+ 1 , 5 = 0 , 1 .

A category, whose objects are cubes Г and whose morphisms are maps

a: composed from maps £qô,rjlq, will be denoted by □ .

Besides, we distinguish categories □ *, whose objects are n-dimensional cubes, the map a: /" -> I m with □ , being a morphism of category a , provided a = id[n or a is a composition of maps on the other hand this is a morphism in the category if a = id[n of a is a composition of maps elg1.

1.2. D

. A category of cubical objects over the category A is called a category ([I]op,A ) of all contravariant functors defined on □ with values in category A, with their natural transformations as morphisms.

If X e O b ( □ op, A) we write X„ = X ( I n), d^ = Х (г‘’й), s' = Xfa‘ ).

Analogously we define categories (П °Р,А ), ( □ rlop,A ).

1.3. Let A be an abelian category. We define a normalization functor N c: (О 0'’ , A) -> (ch, A) ((ch,A) denotes a category of chain complexes over A):

NAX), = П K e r(4 ’°: X „ - * X „ _ ,), i = 1

N A D . for Х , У е О Ь (П ° г , А ), / е Н о т е орл|(Х ,У ),

<<„ = î ( — l ) i + 1 if,’1 : N AX), ^ N A X ) . - , . i = 1

A functor kc: ( □ op, A) -+ (ch, A) associates to an object X a chain complex with components kc(X)n = X n and differential

d n = Z ( - 1 ) ' w i ° - 4 ‘ )- i — 1

For a cubical object X over category A, the subobject Dn(X) = Im s„_!

of the object X„ is called an n-dimensional degenerate part of the cubical object X, where s„_!: © X j/ ij -> X n is a morphism defined by the formula

j= i

s.-iW j = 4 - , with X ‘J - , = x „ - , , wy X'Ji , ® X f - , . J = t

1.4. T

. Let A be an abelian category and X e O b (Q op,A ), (T„(X): X n- + X n a morphism of a category A defined as follows: -

o a x ) = ( l - s F i d r u i - s ^ ^ - V - a - s j - i C 0).

Then

(i) < t „2(X) = <T„(X),

(ii) Im<7„(X) = NAX).,

(iii) Ker<r„(X) = D,(X),

(iv) Х я = N c(X)n@D„(X), (v) = © N c(X)q,

ry. I n ^ l 4

where rj : /" -> I q runs over epimorphisms in the category □ .

1.5. The inclusion functor □ , .< = □ induces the functor z: ([Ц]ор,А ) A). The latter one has a left-adjoint functor *z: ( □ ° p? A) -> (D 015, A).

Furthermore, we define the functors

J: ( a lop, A ) - > ( a op,A ), N c: (D op, A) ^ (D rlop, A)

as follows:

if X, УеО Ь

A), /: X -> У, then

(J(X))„ = X n, (J(X))(ek°) = 0, (J (X)) (si;1) = 4 1, J(/)„ = /„;

if X, У е О Ь (П ор, A), /: X -► У, then

N C(X)„ = N c(X)n, ( NC(X)) (e^’ 1) = di1, N c(f)n = /nUc(x)„.

T

T he categories

П ор, A ),

, A) are equivalent. T he eguivalence is established by the functors N c and L = *z ■ J.

1.6. D. M. Kan in [4] defines cubical sets with an extension condition indicating their good properties. It is known that every simplicial group object (i.e. and object in ( A op, Gr), where Gr denotes a category groups) satisfies the extension condition. The question arises whether an analogous theorem is valid for cubical group objects. We shall give an example of a cubical group object not satisfying the extension condition.

Let Singc: Top -> ( □ op, Set) denote the known covariant functor as­

sociating to a topological space E a cubical set whose «-dimensional component is a set of «-dimensional singular cubes of the space E, i.e.

(SingC(E))„ = {/ ;/ : / "-> £ }.

Consider the cubical subset H generated in Singc(/) by 1-dimensional singular cubes given by the ordered pairs (f,0 ), ( f , f ) , (f, 1). We denote by G„ the free abelian group generated by the set Hn. The cubical group object G = (G„)„eN d °es not satisfy Kan’s condition.

II. H O M O T O P Y IN C A T E G O R Y O F C U B IC A L OBJECTS

II. 1.

Let X, У be cubical objects over the category A. The

morphisms X - * Y are called homotopical in (П ор,А ) if there exists

such sequence of maps hn: X „ -► Yn + 1, « ^ 0, in A that

(i) d l i xhn = f * , S = 0,1,

(ii) dïd + l hn = hn_ x<Pn~U6, 1 < i ^ n+ 1, <5 = 0, 1, (iii) sn +1 hn = hn+l sln, 1 < i ^ n+1.

The sequence {hn)„^0 is called a homotopy connecting morphisms f ° and f 1; we write f ° ~ / x.

It is easily seen that in case A is an additive category, the homotopy relation of maps is an equivalence. The functors N c, kc preserve the homotopy.

In the category of cubical sets the relation is identical with the homotopy relation defined according to [4].

III. T H E C A T E G O R Y O F S IM P L 1 C IA L OBJECTS A N D T H E C A T E G O R Y O F C U B IC A L OBJECTS O VE R A

III.1. Let us fix a functor t : □ - > ( A op,Set) ascribing to every cube /" a simplicial set associated to a particular triangulation of /". We define the latter by induction:

the triangulation of the cube /" = I n~ l x l is obtained by triangulizing in a standard manner any prism ax/ , where a is an (n — l)-dimensional simplex in triangulation I n~l .

Note that the vertices of the cube Г are functions a: {1 ,2 ,..., n} {0, 1}.

We introduce the following ordering relation ^ in the set of vertices of the cube /":

a ^ b iff Д a(i) ^ b(i).

1 ' i< n

In terms of this relation the simplexes in t ( I n) can be described as follows:

the sequence of vertices a0, a x,. .. ,a r constitutes an r-dimensional simplex in t ( I n) iff a0 < ax < ... < ar, i.e. iff for any i = l,2 ,...,n the sequence a0( i ) , a r (i) is monotonously non-decreasing and the vertices a0, ax, ..., ar differ from one another. This allows us to identify r-dimensional simplexes in t ( I n) with rectangular matrices with r+ 1 rows and n columns made up from the numbers 0,1, the columns of the matrices being monotonie non-decreasing sequences (the rows are numbered from 0 to r).

There exists a one-to-one correspondence between r-dimensional simplexes a0, a x, .. . ,a r in t ( I n) and functions tp\ {1 ,2 , ..., n} -> {0, 1,2,..., r, r + 1}

that {1, 2,..., r} c= Im (p. The function ç > : {l,2 ,...,n }- > {0 , l , 2 , . . . , r + l } corresponding to the simplex (a0, ax, a r) is defined as follows : cp (/) = number of zeros in the sequence a0 (/), ax (i) , ..., ar(i).

Note that и-dimensional simplexes in t ( I n) are given by all automorphisms of the set {l , 2 , . . . , n }, whereas (n — l)-dimensional simplexes in t ( I n) are determined by the functions ф: {1 ,2 ,..., n} -> {0, 1,2,..., n — 1, n) such that

{1 ,2 ,..., « — 1} cz Im ф.

The functions ф thus satisfy precisely one of the conditions:

(i) V ф{р) = ф { 4) е{ \ , 2 , . . . , П - \ } ,

1 ^ p< q ^ И

(ii) у ф(г) = 0 or ф(г) = «.

1

In case (i) the simplex ф is an (« — l)-dimensional internal simplex of the triangulation t ( I n). It is the common face of two «-dimensional simplexes determined by the functions

<p> <p'- (1 ,2, . . . , « } - * (0, 1 , 2 , . . . , « , « + 1) given by the formulae

<M0, ^{Ф(г) < ф(р),} ф(1) + 1, Ф (0 > Ф(р), ф(р)+ 1, ^{i = р,}

Ф(я), ^{i = q,}

«MO, Ф (0 < Ф(р)>

i^Oj.+ l, ф{1) > ф(р),

Ф(Р), i = р,

Ф ( д ) + 1, ^{i = q,}

«-dimensional simplexes (p, q>' whose common face is an (« —l)-dimensional simplex ф will be denoted as follows:

(p = и„(ф), q>' = v„№) and obviously

dt(p)(<P) = dt(p){<p') = ф.

In case (ii) the simplex ф lies on the face tr = 1 when ф(г) = 0 or on the face tr = 0 when ф{г) = n. This is the face of an «-dimensional simplex q> = w„(^) given by

q>{i) = ф(г), ф (r) = «, Ф (0 + 1, Ф(г) = o.

The maps *7n are simplicial maps with respect to the triangulation described above.

According to Theorem 1.3 of part II [2], the functor t determines a pair of adjoint functors

Eh(t): ( □ op, Set) -> (Д ор, Set), т: (Л ор, Set) -> ( □ op, Set).

The functor Eh(t) is Kan’s left extension of the functors t along Yoneda’s functor h: □ -» ( □ op, Set).

Before giving a description of the functor Eh(t) we introduce the following notation:

if X e O b (D op, Set), then by П/Х we denote the category whose objects

are the pairs (/", u: HomD ( - , / " ) - > X), the map a: (/", u) -> (/m, y) is a mor­

phism in П/Х provided a: /" -> I m is a map in □ and following diagram commutes

Hont, ( - , / " )

Н о ш а ( - , П

Since the maps of cubical sets /x: HomD (-,/ ")-*• X are in one-to-one correspondence with elements of the set X„, the objects of category П/Х can be writen in the form (Г , х п), x „ e X„.

Functor tx : П/Х -* ( A op, Set) is determined as follows:

tx ((/", *«)) = t ( I n), tx (a) = t ( a) for a: (/” , x„) ( Г , x j .

Further, if /: X -♦ У is a map of cubical sets, we obtain a naturally induced functor □//: □ /X -► Щ/У.

The functor Eh(t) is given by formulae

Eh(t) (X) = lim tx for X e Ob (C]op, Set), nix

Eh( t ) ( f ) = lhn □// for /: X -> F e ( D op,Set).

By w~ = w ~ : t ( I n)-+ lim tx we shall denote the structural map of the

.. . n (' *

limit.

Using the above information, one can find for a cubical set the n-dimensional component of a simplicial set

(£»(f)(X))„ =

P

where Bl is the set of n-dimensional internal simplexes in t ( P ) or degenerations of internal simplexes.

The functor t , right-adjoint to Eh(t), is given by the formula

t (X)„ = Hom(AoPjSet) (r (/"), X) for X e Ob ( Д ор, Set).

Ш.2. Let A = RMod be a category of Я-modules. A pair of adjoint functors Set £ KMod (3 — forgetful functor, F — functor of free module)

3 Fi

produces a pair of adjoint functor ([Цор, Set) ( □ op, RMod) (respectively

( Л ор, Set) Ц ( A op, „Mod)).

We have the diagram

(□ ° p, RMod)

The functors

Eh (t) = F 2 ■ Eh (t) ■ 31 : ( С Г , « Mod) - ( Л ор, RMod), f = F t ■ г • з2 : ( A op, RMod) - (П ор, RMod),

also form a pair of adjoint functors.

The question arises whether the diagram

(□ »', „Mod) 5^2 (Д ор, „Mod) (Д ор, „Mod) -Д „Mod)

(ch,RMod) (ch,RMod)

commutes up to a homotopy.

One of the relations between functors in diagram (*) is a transformation of functors A: N c -> к ■ Eh(t), whose description is given below.

We have to define the maps A„(X): N c (X)„ -* (k • Eh (t)) (X)n natural with respect to X.

First we define the maps I „ ( X ) : kc(X)n -> (k- Eh(t))(X)n. Let

This element determines precisely one map of cubical modules Jc„: /?(HomD ( — , /")) - *‘X such that ( a „)„(1z„) = xn.

The map xn induces the map of simplicial objects

Eh (t) (x„) : E„ (t) (R (HomD ( - , /"))) Eh(t){X) = lim tx n/x

i.e. Eh(t)(x„) = w-Xn.

Requesting that the family X{X) establishes a natural transformation of functors, the following diagram should commute

<*n>n

(R (Н ош ( —, /")))„

l„(X)

I„(R(HomD (-,/"))) (t d %

Hence

I „ ( X ) W = (Я„(Х)-(5с„)„)(1;„)

= [> ;„ )„ • I„ (R (Horn ( - , /”)))] (1,„).

Thus it suffices to determine the maps A„(R(Horn ( —, /"))). We assume Я„(R(Horn ( — , 7")))(l/lt) = Y j (sgn a)a, where A„ denotes the set of n-dimen-

аеЛ„

sional simplexes in triangulation Г and sgn a stands for the sign of the permutation of the set {1 ,2 , determined by a (cf. III.2).

If у is a degenerate element in R(Hom ( —, /")), we assume I„(R (H om (-,_/"))) (y) = 0.

The map Xn{X) is thus given by the formula

X„(X)(x„) = (win)„ X (sgn а) a,

аЫ „

where 2„(X)(x„) = (X„ (X) • w„ (X)) (x„) and w„(X): N C(X)„ cz_, X„.

The maps X„(X) are maps of chain complexes natural with respect to X.

Having defined X: N c -> к ■ Eh(t) and making use of the canonical trans­

formation ft\ Eh(t) • t -> 1 we obtain a natural transformation of functors у : iVc т -> к given by the family of maps

(JV ct )(X) ^ ( f c - ^ W - ï ) ( X ) ^ f c ( X ) ,

where X 6 Ob (Д ор, RMod).

It is not known whether A establishes a homotopy equivalence of functors N c and к ■ Eh(t).

III.3. The functors corresponding to т and Eh(t) can be also defined in case when the category RMod is replaced by an arbitrary abelian category.

De f i n i t i o n.

The functor

, A) ->

, A) is defined as follows: if X , Y Ob (Д ор, A), /: X - * Y , then

т(Х)„ = Ker ( П К 4 П

aeAn P eBn

— 1

where XI = X n, X ^ 1 = A „ _ l9 An is the set of n-simplexes in triangulation of denotes the set of (n — l)-dimensional internal non-degenerate simplexes in triangulation of Г and the morphisms u*, v* are given by the formulae

щи* = di{p)KUnm, npv* = énip)TiVnm. ' The я-dimensional component of morphism x (/) is defined by

in{Y)(x{f ))n = ( П fn)in(X),

xeAn

where i„(X): т(Х)„е ^ f ] XI is an embedding morphism, /„* = /„.

*eA n

The cubical structure of the object f(A ) can be also given directly.

The functor

sends homotopic maps into homotopic maps. The functor Eh(t): (G op, A ) (Д ор, A) assumes the value (Eh(r)(X))„ = 0 © ( N C(X))P on

p Рев?

the cubical object X.

References

[1 ] S. B a lc e r z y k , P h a n H u u C h a n and R. K ie lp in s k i, On three typ es o f simplicial objects, Fund. Math. 91 (1975), p. 145-160.

[2 ] P. G a b r ie l and M. Z ism a n , Calculus o f fra ctio n s and hom otopy theory, Berlin 1967.

[3 ] P. J. H ilt o n and S. W y le e , H o m o lo g y theory, Cambridge 1960.

[4 ] D. M. K an , A b s tra ct h o m o to p y I, Proceedings Nat. Acad, of Sci. 41 (1955), p. 1092-1096.

[5 ] A. S w i^ te k , K a t e g o r i e obiektôw kostkow ych , Preprint N o 15, Torun 1975.

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