Title: Tower techniques for cofacial resolutions
author: A. Libman
Classification: 55U35,55T15,18A25
Address: Dept. of Math. Sciences, University of Aberdeen, Aberdeen AB24 3UE, UK.
E-mail: assaf@maths.abdn.ac.uk
Let $J$ be a continuous coaugmented functor on spaces.
For every space $X$ one constructs a cofacial resolution
$X \to J^\bullet X$ (namely a cosimplicial resolution without its
codegeneracy maps) in the usual way.
Following Bousfield and Kan, one defines $J_s(X) = tot_s J^\bullet X$.
Suppose $D$ is a small category and that $X$ is a $D$-diagram of
$J$-injective spaces, namely $X(d) \to JX(d)$ admits a left inverse
for every object $d$ in $D$, but in a way which need not be
compatible, namely a map $JX \to X$ cannot be constructed out of this data.
We show that for many free diagrams $F$, the spaces $hom_D(F,X)$ are
$J_s$-injective for $s<\infty$.
Thus, the functors $\mathbb{Z}_s$ of Bousfield and Kan capture a large class
of polyGEMs as their injective spaces.
This generalises earlier results by the author.
Our methods use pro-object arguments, which are originally due to Farjoun.