ONTHE

HOMOTOPY

RELATION

FORc.s.s.. MAPS...

By Danien M. Kan

1. Introduction
A c.s.s. complex(see [2]) mayCo|orRA

be considered as a collection of sets together

with a collection of maps between them satisfying certain identities. Similarlya ¢.8.8. group may be considered as a collection of groups together with a col-lection of homomorphisms between them satisfying the same identities. Thissuggests the notion of a c.s.s. object over an arbitrary category C.

Let © be a category and let @” denote the category of a c.s.s. objects over €.

Then it will be shown that if in € a notion of sum is defined, it is possible to introduce in @" a homotopy relation in a rather natural way.

aLet € and © be such categories with sums. Then we will show that undef

certain conditions a functor T:€" — D” preserves homotopies.
This generalizes

a result of A. Dold ([1]).

aAs an application we prove an analogue for c.s.s. groups of :4 theorem of: J.
H. C. Whitehead.

:

2. C.s.s. categories.

For every integer n = 0 let [n] denote the ordered set (0, ---, n). By a map

a:[m] — [n] we mean a monotone function, i.e., a(t) < a(j) for0 £ ¢ £ j= m.
Clearly the sets [n] and the maps a:[m] — [rn] form a category. This category
will be denoted by 0.
||i
DeriniTioN (2.1). Let @ be a category. The function category €' (see 31)
will be called the c.s.s. category over €; its objects and maps will be called c.s.s.
objects and c.s.s. maps over @. We recall that an object of @ is any contravariant
functor K:U — € and that for two objects K, L e@" a map f:K — L is a. natural transformation. Instead of Kn}, Ka and f[n] we usually write K,, K.

and f,, .
ExaMpLEs||(2.2). (a) Let 9M be the category of sets. Then on” is the category -of c.s.s. complexes ([2]).
(b) Let 9T be the category of sets with a distinguished element. Then 9" isthe category of c.s.s. complexes with a base point ([4], §2).

(c) Let £ be the category of modules (over a ring A). Then £&” is the category of c.s.s. modules over A ([1]). In particular if A = Z, then £" is the. category of c.s.s. abelian groups.

(d) Let G be the category of groups. Then G" is the category of c.s.s. groups(14).

1 The author was partially supported by Air Force Contract AF 18 (600)-1494 during theperiod when the work on this paper was being done at Princeton University.
75

76

DANIEL M. EAN
3. Categories with sums

DEriniTION (3.1). Let © be a category and let M be a set. Let C' ¢@ be anobject and for every element ue M let be given an object C, ¢@ and a map
Ju:Cy — C. Then C is called the sum of the objects C, under the maps j, if forevery object D eC and every set of maps d,:C, — D, u ¢ M there is a unique
map d:C — D such that for every u e M commutativity holds in the diagramc, —*,¢NYNyDWe then write C = 2 ux 5,0, of C = ju, Coy + §ugChy + +This definition is a special case of the definition of direct limit of [3], chapter
II.
DEFINITION (3.2). A category € is called a category with sums if for every set
IM and function I' which assigns to the elements of M an object of ©, there are
given an object M-T eC and maps pu -I'(n): T(x) — M-T (up eM) such that
M-T = per)(u T())T(n).
ExampLEs (3.3). All categories in example (2.2) are categories with sums.
Using the same notation we have(a) The sum of a collection of objects of 9M is what is usually called theirunion.
(b) The sum of a collection of objects of IT is their union with identificationof all the distinguished elements.
|(¢) The sum of a collection of objects of £ is their direct sum.
(d) The sum of a collection of objects of G is their free product.
DerFINITION (3.4). Let € be a category with sums. Then we define a functor® : 9M, @ — C as follows. Let M 9 and C €C be objects and let T be the funection given by T(x) = C for all u e M. We then define i ® C byM®C=M-T(ie., M ® C is the sum of as many copies of C as there are elements in M).
For maps g:M —- N eM and f:C —DeClet g ® fr: M ® C — N ® D be the(unique) map such that for every u e M commutativity holds in the diagram¢c *C,Ifpmec
7 Q fDN® D

It is readily verified that the function ® so defined is a covariant functor.

HOMOTOPY

RELATION FOR C.S.S. MAPS

71

It should be noted that definition (3.2) and hence definition (3.4) contains anelement of choice. However it follows from [3], chapter II, that if in definition(3.2) the given object C ¢@ and maps 7,:Cy — C are changed, then the functor® gets changed by a unique natural equivalence.

DermviTioN (3.5). Let @ be a category with sums, Then a covariant functor®:9m”, ¢” — e¥ may be defined by(K®

4),(K®

A)a=K.®

(gfor every object K eM"

®

=fa

=K,

gn

®

Qand 4 ec” and map

A,A.
fng eM"and fe@’.

It is clear that the use of the symbol ® for two different functors will notcause any trouble. In both cases we often write ¢ ® A and K ® f instead ofgd

14andExampresix®

f.

(3.6). (a) Let K, Le’.

Then K ® L is usually called theircartesian product.
(b) Let K ean’, A e Gg". The productK ® A then is as in [5], §3.

4. The homotopy relation
DermviTiON (4.1). Let © be a category with sums. Let the standard simplices
P = A[0] and I = A[l] and the c.s.s. maps Ae*:P — I (z = 0, 1) be as in [5],§2. Then two maps fo, fi:A — B eG’ are called homotopic (over @) if there exists

a map fri] ® A — Bee” (called homotopy) such that commutativity holds
in the diagram
01PRAAe®4~ |i|4aI®AAe®A4Js—h ,BpPRAmiBy

where 7:P ® A ~~ A is the natural isomorphism. Notation fr:fo ~ fi (over @)or fo ~ fi (over @).
||A map f:4 — Be ec” is called a homotopy equivalence (over @) if there exists

a map g:B— A ec’ such that the composite maps go f and fo g are homo-topic (over @) to the identity maps of A and B. The objects 4 and B then aresaid to have the same homotopy type (over @).

ExampLES (4.2). Using the notation of example (2.2) we have:(a) Two maps of 9M” are homotopic over 9M if and only if they are homotopic in the usual sense ([4] §2).

78

DANIEL

M. KAN(b) Two maps of %” are homotopic over 9 if and only if they are homotopic rel. the base point ([4], §2).
(¢)

Two maps of £" are homotopic over £ if and only if they are homotopic

in the sense of [1].
|(d) Two maps of G” are homotopic over G if and only if they are loop homotopic in the sense of [5].
ExamprEs (4.3). Sometimes several homotopy relations may be defined overdifferent categories.

(a) As a c.s.s. group also may be regarded as a c.s.s. complex it follows thaton G' there is a homotopy relation over G and one over 91. Clearly two maps
homotopic over G are also homotopic over 9, but the converse need not betrue. This may be seen from the following example: Let z: K(w, n) — K(¢, q)
be a map representing a non-zero element of H(w, n; ¢) which suspends intozero. Then (see [5]) the c.s.s. homomorphism Gz:G(K(w, n)) — G(K(¢, ¢)) ishomotopic over 9 to the trivial map, but by [5], §11 this is not the case over G.
(b) Let £ momentarily denote the category of abelian groups. As a c.s.s.
abelian group may also be regarded as a ¢.s.s. group or a c.s.s. complex there

corresponds for the category £' three homotopy relations (over §, £ and IN).
It 1s however readily verified that the homotopy relations over G and £ areequivalent. Clearly maps of £° homotopic over £ are also homotopic over IN,
but the converse need not be true.

REMARK(4.4). It should be noted that the homotopy

relation defined in(4.1) need nof be an equivalence relation. For c.s.s. complexes counter examples

can easily be found. However the homotopy relation always has the following
property.

PROPOSITION (4.5). Let @ be a category with sums. Let f:A — B, go, g1:B — Cand h:C — D be maps of C' and let go ~ gy over C. Then hogoof ~ hogiof
over C.

Proor. Let g;:g0 ~ g1 over @. Then it follows immediately from the definitions that the composite map

reaL185,198

9%,

¢ 2,p

is the desired homotopy.
Special cases of Proposition (4.5) are [5], Proposition (2.5) and (3.4).
5. C.s.s. functors

We now define a class of functors involving c.s.s. categories (roughly speaking: functors such that “dimension n of the range’’ only depends on ‘‘dimensionn of the domain”) and show that these functors map homotopic maps into
homotopic maps. This generalizes a result of A. Dold ([1]).

HOMOTOPY

RELATION FOR C.S.S. MAPS79Let © and © be categories. A covariant functor T':€ — ©" induces a functor
D(I):e" — D" given by~(D(T)A)s = (TAz)n(D(T)A)e=(T4a)e(D(T)g)n = (T'gn)nfor every object A ¢@” and map g e@’. Denote by Z:¢ — €" the constantfunctor, i.e. for every object C eC and map c eC(ZC)(EC)

a=(C=ic(Ze), = c.

Then with every covariant functor 6:€7 — D" one may associate the compositefunctor © 0 5:@ — D" and it is readily verified that|Proposition (5.1). There exists a natural equivalence n:T' — D(T') o E.
However in general the functors © and D(O o =) do not differ by a naturalequivalence. We therefore define

DerFiNtTION (5.2). A covariant functor 0:7 — DV is called a c.s.s. functor
if there exists a natural equivalence £:0 — D(0 o E)
The theorem now may be stated as follows

THEOREM (5.3). Let @ and D be categories with sums and let 0:€°7 — DY beac.s.8. functor. Then © maps maps homotopic over © into maps homotopic over D.

Proor. It clearly suffices to show that for every object 4¢@" there exists amap a:] ® 604 —6(I ® A) ¢ D” such that commutativity holds in the diagram
0Pood(54)—2®0% |

=~ |!

op»

—28 84)

where 7:P

® 04=~ 6(Pn=0let

al,® OA), —®

1OBA

OIholds in the diagram

1 o'ea), =,~—CBwAD,Oo

For each integer:® A)). ¢D be the (unique) map such that.

.

(0°24),

gpg yA) is the natural isomorphism.

04), 2,pt

pges= Isgrg4

for every element pu eI, commutativityR~2%%

lo04)

1

1904

=

‘4,(0 ® A).

SUCIn/(g.m)1, © 4).
ye80DANIEL

M. KANIt then follows from the naturality of all functions involved that the functiona:] ® 64 —-6(I ® A) so defined is a map in D' and is such that commutativity holds in the diagram (6.4).
REMARK(5.5). Theabove

result also holds for categories with finite SUMS

1.e, if in definition (3.2) M is restricted to be finite.
6. An application
We now prove the following analogue for free c.s.s. groups (see [5], §4) of atheorem of J. H. C. Whitehead.
TaEOREM (6.1). Let A and A’ be connected free c.8.s. groups and let fi 4 — A’
be a c.s.s. homomorphism. Let B = A/[A, A] and B' = A’'/[A’, A'] be their abeltantzations and let g:B — B’ be the map induced by f. Then f is a homotopy equivalence (over GQ) if and only if g is so.

Proor. If f is a homotopy equivalence, then by Theorem (5.3) so is g. Inorder to prove the converse consider the commutative diagramawa

20,4| 4| q

GWWA/IGTWA, GWA] —2

B = AJA, A]

where o'(2) is as in [5], Theorem (11.3) and p and ¢ are the projections. By [5],Theorem (11.3) o/(2) is a homotopy equivalence over G and hence by Theorem(5.3) so is b. Repeating this for A’ we get (see [4], §15) a commutative diagram—

H,(WA)—_Tn_1(b)—Ta—1(b")= m,2(GWA/IGWA, GWA]——

H,(WA") = 7,_1(GWA'/[GWA’, GWA")‘=~~~maa(B)Ta1(B’)

If g is a homotopy equivalence, then clearly (Wf)x:H.(WA) — H,(WA’) is an‘isomorphism for all n. The connectedness of A and A’ implies the simply connectedness of W4 and WA’. Hence the theorem of J. H. C. Whitehead ([6],Theorem 3) yields that Wf:WA — WA’ is a homotopy equivalence over IN.
That f is a homotopy equivalence over § now follows from [5], §11.
HEBREW

UNIVERSITY, JERUSALEM,ISRAELHOMOTOPY

RELATION

FORC.S.S. MAPS

81

Bi1BLIOGRAPHY
1. A. Dorp, Ann. of Math., to appear.
2. S. EILENBERG and J. A. Z1LBER, Semi-simplicial complexes and singular homology, Ann,of Math., 51 (1950), 499-513.
3. D. M. KaN, Adjoint functors, Trans. Amer. Math. Soc., to appear.
4. D. M. XaN, A combinatorial definition of homotopy groups, Ann. of Math., to appear.
5. D. M. KAN, On homotopy theory and c.s.s. groups, Ann. of Math., to appear.
6. J. H. C. WHITEHEAD, Combinatorial homotopy I1., Bull. Amer. Math. Soc., 55 (1949), 213245.