, *
*we
define the composite (fi, t) O (ff, q) to be (fi O t*ff, t O q).
We define the category of G*-categories as a certain category built out of th*
*e category
of functors from G into the category of small categories. In order to avoid po*
*ssible
confusion as to the meaning of "functor" and "natural transformation" where they
occur below, we define G*-objects in any category C with a final object. We wri*
*te * for
a chosen final object in this category, and C* for the category of objects of C*
* equipped
with a structure map from *. We think of C* as the category of based objects i*
*n C.
However, in our applications C is always Cat . Note that for us, a based categ*
*ory
therefore consists merely of a category with a selected base object.
Definition 5.2. A G*-object in C consists of a functor F :G ! C together with a
map * ! F () such that F (m1, . .,.mr) = * whenever any mi = 0 and such that the
following diagram commutes,
*___=__//F (0)
| |
| |
fflffl| fflffl|
F ()____//_F (1),
where the left hand map is the given map, the top map is the unique map, the ri*
*ght
hand map is induced by the unique map (0) ! (1) in G, and the bottom map is
induced by the map () ! (1) in G from the unique map 0_! 1_in Inj and the ident*
*ity
map on 1 in F. A map of G*-objects F ! G is a natural transformation f :F ! G
making the following diagram commute:
*C
__ CCC
___ CCC
""___ !!C
F ()____f()____//G().
We denote the category of G*-objects as G*-C.
We remark that for a G*-object F , the objects F ~~ = (S1, . .,.Sk) of subsets Si {1, . .,.ni}. In ord*
*er to
explain the properties we require for these systems, we need some notation: gi*
*ven
a subset T {1, . .,.ni} for some 1 i k, we write ~~~~, ae~~~~;i,T,U}, where
(1) ~~~~ = (S1, . .,.Sk) runs through all k-tuples of subsets Si {1, . .,.ni*
*},
(2) For ae~~~~;i,T,U, i runs through 1, . .,.k, and T, U run through the subs*
*ets of Si
with T \ U = ; and T [ U = Si,
(3) The C~~~~are objects of C, and
(4) The ae~~~~;i,T,Uare morphisms C~~~~in C,
such that
(1) C~~~~= 0 if Si= ; for any i,
(2) ae~~~~;i,T,U= idif any of the Sj (for any j), T , or U are empty,
(3) For all ae~~~~;i,T,Uthe following diagram commutes:
ae~~~~;i,T,U
C~~~~
| ||
fl|| ||||
| ||
fflffl| ||
C~~~~;i,U,T//_C~~~~,
MULTIPLICATIVE STRUCTURE IN INFINITE LOOP SPACE THEORY 39
(4) For all ~~~~, i, and T, U, V {1, . .,.ni} with T [ U [ V = Si and T , U*
*, and V
mutually disjoint, the following diagram commutes:
ae~~~~;i,T[U,V|
| |
fflffl| fflffl|
C~~~~;i,T,U[V______//C~~~~,
(5) For all ae~~~~;i,T,Uand ae~~~~;j,V,Wwith i 6= j, the following diagram com*
*mutes:
C~~~~;j,V,W33
C~~~~.
| fEEfff
| ffff
fflffl| ffff
C~~~~;i,T,Ufffff
WWWWW ffff
WWWWWW ffff
(ae~~~~, ae~~~~;i,T,U} ! {C0~~~~, ae0~~~~;i,T,U} consists of morphisms*
* fS :C~~~~!
C0~~~~in C for all ~~~~ such that f~~~~is the identity id0when Si = ; for any i, a*
*nd the
following diagram commutes for all ae~~~~;i,T,U:
ae~~~~;i,T,U
C~~~~
f~~~~|
fflffl| fflffl|
C0~~~~.
ae0~~~~;i,T,U
If any of the ni = 0 in the definition above, then JC~~~~, ae~~~~;i,T,U} := {Cff~~~~, aeff~~~~;i,T,U},
where
Cff~~~~= C(ff-11S1,...,ff-1kSk)
and
aeff~~~~;i,T,U= ae~~~~, ae~~~~;i,T,*
*U} is sent
to the object {Coe~~~~. The morphism {f~~~~}*
* is sent to
the morphism {foe~~~~, ae~~~~;i,T,U} is sent to the object {Ce~~~~,aee(S1,...,Sk,{1});i,T,U= ae~~~~;i,T,Ufori < k + 1,
Ce(S1,...,Sk,;)= 0, aee(S1,...,Sk,;);i,T,U= id,aee(S1,...,Sk,{1});k+1,T,*
*U= id.
The morphism {f~~~~} is sent to the morphism {fe~~~~, fe(S1,...,Sk,;)= id.
This description of the components of the objects and morphisms is complete sin*
*ce
the only two subsets of {1} are {1} and ;. The inverse of this isomorphism is i*
*nduced
by dropping the {1} from (k + 1)-tuples of the form (S1, . .,.Sk, {1}). This de*
*scribes
image functors for a generating set of morphisms of G, and since JC~~~~by defining
C~~~~:= F ~~~~.
We also need to produce the structure maps in the system, so suppose given subs*
*ets
T and U of {1, . .,.mi} with T \ U = ; and T [ U = Si. We define the associated
structure map ae~~~~;i,T,Uto be the image under F of the 2-morphism in Emi given*
* by
(T, U) ! T [ U = Si,
together with the objects in the other slots in ~~~~. Now the coherence properti*
*es (1)
and (2) follow from F being a based multifunctor, (3) follows from the commutat*
*ive
diagram
(T, U)_____//T [ U
| ||
| |=
fflffl| fflffl|
(U, T )____//T [ U
in Emi, (4) follows from the commutative diagram
(T, U, V )____//_(T, U [ V )
| |
| |
fflffl| fflffl|
(T [ U, V )____//T [ U [ V
in Emi, and (5) follows from bilinearity.
The reverse direction is the most significant part of the proof: given an ob*
*ject
(C~~~~, ae~~~~;i,T,U) of JC~~~~ := C~~~~. Now suppose given an n-morphism in Em*
*i,
say (T1, . .,.Tn) ! Si, so Tr \ Ts = ; unless r = s, and T1 [ . .[.Tn = Si. We *
*need
to construct the image n-morphism in UC under our multifunctor F , which will b*
*e a
morphism in C
C~~~~.
We define this inductively, requiring the morphism to be id0if n = 0 and idC~~~~*
*if n = 1.
For larger n's, we define the image n-morphism by induction to be the composite
ae~~~~;i,Si\Tn,Tn
C~~~~,
where the first map is given by induction on the first n - 1 terms, and the sec*
*ond is
the structure map given by the object of JC~~