10 GIORA DULA AND REINHARD SCHULTZ
(where
Aop
= opposite category of A)
must be restricted in order to guarantee its homotopy representability (compare
[DF, statements 4.15 and 5.8, pp. 119 and 130]). Following [DF, statement
4.15] we can express the appropriate restriction as follows: By [DF, statement
2.1, p. 101] the orbit category of any small category D contains a canonical free
orbit subcategory that is isomorphic to Dop. It follows that the orbit category
Orb(Cat(G) x Cat(P)) ~ Orb
G
x Orb(Cat(P))
equiv
contains a small subcategory of the form Orb^r x Cat(P)op. In analogy with
[DK] we would like coefficients to be given by a contravariant functor A on
this small subcategory, and as in [DF, statement 4.15] we can reconcile the two
settings by taking the coefficient functor M on the entire orbit category to be a
left Kan extension of the functor A on the small subcategory. Frequently it is
convenient to view A a s a covariant functor on Orb£? x Cat(P), and we shall do
so henceforth without further comment. In order to ensure that such cohomology
groups have decent formal properties one must restrict attention to admissible
P-diagrams of (G-invariant) subspaces {Xa}aep of X satisfying the following
conditions:
(i) P has a unique maximal element e, and = X.
(ii) If a (3 G P and jap is the canonical map, then jap is 1-1 and a homeo-
morphism onto its image.
(Hi) If y G X then there is a unique minimal element a(y) G P such that
oc(y) (3 for all (3 such that y G Xp.
These assumptions imply that every Cat(G) x Cat(P)-orbit in X (as in [DF,
final paragraph on p. 97]) lies in the small subcategory Orb ^ x Cat(P)op; as
noted in [DF, statement 4.23, p. 121] the restrictive condition on orbits guar-
antees that certain technical difficulties do not arise.
Having placed ourselves in the setting of [DF], we can use the latter to draw
the following conclusions:
(1.6). For every finite partially ordered set P, every finite group G, and every
covariant functor
A : Orb£? x Cat(P) - Modules
there are G-equivariant Bredon cohomology groups for admissible Cat(P)-dia-
grams of G-invariant subspaces with coefficients in A.
This is discussed in [DF, statements 4.8-4.22, pp. 117-121].•
(1.7). The Bredon diagram cohomology groups are the "correct" domains for
obstructions to extending Cat(P) -maps over Cat(F)-cofibrations.
This is discussed in [DF, statements 5.10-5.11].•
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