DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 11

(1.8). Postnikov decompositions, R-localization, and other common tech-

niques of homotopy theory can be developed for the category of admissible G-

invariant Cat (P)-diagrams of G-CW subcomplexes of a G-CW complex.

This is a special case of the conclusions at the bottom of [DF, p. 131] (in this

connection also see [May]).B

The following special case of (1.8) will be particularly important for our pur-

poses:

(1.9). An analog of M0ller's equivariant Barratt-Federer spectral sequence is

valid for the homotopy groups of equivariant Cat (P)-function spaces for Cat(P)-

maps from an admissible Cat(P) finite G-CW subcomplex diagram X to a

Cat(P)-simple G-CW complex diagram Y.

This is true because the treatment of [M0] is basically a formal consequence

of the existence of skeleta for X and Postnikov systems for Y in the appropriate

category. •

There is also a straightforward extension of the Fary spectral sequence of

Theorem 1.5 to diagram categories:

(1.10). An analog of the equivariant Barratt-Federer/Fary spectral sequence

in Theorem 1.5 is valid for the homotopy groups of equivariant Cat (P)-function

spaces for Cat (P)-maps from an admissible Cat(P)-imite G-CW complex diagram

X to a Cat(F)-simple G-CW complex diagram Y.

For the sake of completeness we note that diagram cohomology (of ordinary

spaces) with twisted coefficients is defined in [DK, Section 3]; as usual, one can

define cohomology groups with compact supports by taking colimits of suitable

relative groups.

REMARKS ON THE PROOF OF

(1.10). The setting of [Sc2] shows that a

bifiltration defines a Fary spectral sequence if certain vanishing conditions hold

(see [Sc2, Assumptions A and A', p. 75]). This condition is established for

the mixed skeletal and orbit type bifiltration of a finite simplicial G-complex in

[Sc2, Prop. 4.2, p. 82]; the same argument goes through for diagrams of finite

simplicial G-subcomplexes of a finite simplicial G-complex.•

In [DF] and [DF/Z] the homotopy groups of diagram function spaces are

also analyzed in terms of unstable Ext-type spectral sequences. Although this is

beyond the scope of the present paper, a comparison of the results in this paper

with those of [DF/Z] seems worthwhile and possibly quite illuminating.