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£ = 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].•