Centric Maps
and
Realization of Diagrams in the Homotopy Category
W. G. Dwyer and D. M. Kan
University of Notre Dame
Massachusetts Institute of Technology
x1. Introduction
Let D be a small category. Suppose that X is a D-diagram in the
homotopy category (in other words, a functor from D to the homotopy
category of simplicial sets). The question of whether or not X can be
realized by a D-diagram of simplicial sets has been treated by [5]. The
purpose of this note is to study a special situation in which the treat-
ment can be simplified quite a bit. We look at two examples to which
this simplified treatment is applicable; both examples involve homotopy
decomposition diagrams for compact Lie groups. Our results show that
in at least one of these examples ([13])the decomposition diagram is
completely determined by its underlying diagram in the homotopy cat-
egory. It is possible that this "rigidity" result will eventually contribute
to a general homotopy theoretic characterization theorem for classifying
spaces of compact Lie groups (cf. [8]).
Before going any further we have to introduce some notation. The
symbol S will denote the category of simplicial sets and ho S the as-
sociated!homotopy!category obtained by localizing with respect to (i.e.
formally!inverting)!weakequivalences. If C is a category and D is a
small!category,then CD will stand for the categoryof D-diagrams in C;
the!objects!of CD are functors D ! C and the maps of CD are natural
transformations.!There is a projection functor ss: S ! ho S ;we will use
the!same!symbol for induced functors SD ! (ho S )D . Both S and SD
admit!closed simplicial model category structures [19, II] [20, p. 233]
[4,!x2]!, and we will sometimes require without loss of generality that
chosen!objects!in these categories be fibrant.
! Given a small category D and a diagram X in (ho S )D, a realization
of!X! [5, 3.1] is a pair (Y;f) such that Y is an ob ject of SD and f is an
isomorphism!f!: ssY = X in (ho S )D . A weak equivalence between two
!
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2 Dwyer and Kan
such realizations (Y; fY ) and (Z; fZ ) is a weak equivalence h :Y ! Z 2
SD such that fZ ffi(ssh) = fY .
Iff : A ! B is a map of simplicial sets, let Map (A; B)f denote
the component of the simplicial mappingspace Map (A; B) containing
f [17, p. 17]. Call a map f : A ! B between fibrant simplicial
sets centric if precomposition with f induces a homotopy equivalence
Map (A; A)1 ! Map (A; B )f. Here "1" stands for the identitymap of A
and Map (A;A)1, also denoted haut (A)1, is the identity component of
the simplicial monoid of homotopy self-equivalences of A. It is clear that
whether or not a map is centric isa homotopy property. We will say
that a homotopy class of maps OE :A ! B is centric if any representative
is, and that a diagram X in SD (resp. X in (ho S)D ) is centric if X(g)
(resp. X (g)) is centric for each morphism g of D.
Remark: A map i : G ! H of discrete groups induces a centric map
BG ! BH of classifying spaces iff i induces an isomorphism between
the center of G and the centralizer in H of i(G). This explains the choice
of the term "centric".
Given a centric diagram X in (ho S)D construct functors ffiX : Dop !
Ab (whereAb is the category of abelian groups) by setting ffiX (D)=
ssihaut (X (D))1for D an object of D; there is nobasepoint problem with
these homotopy groups because haut(X (D))1is a simple space. For each
map g : D0 ! D1in D the induced map ffiX (g) : ffiX (D1) ! ffiX (D0)
is the composite
# )1
ssihaut (X (D1))1 fl#!ssiMap (X (D0);X (D1))fl(fl! ssihaut (X (D0))1
where fl is a representative of X (g), fl# is induced by postcomposition
with fl and fl# byprecomposition.
Ourmain technical result is the following one.
Theorem 1.1. Suppose that D is a small category and that X 2
(ho S )D is a centric diagram.
(1) If the groups lim i+2 ffiX vanishfor i 1 then at least one real-
izationof X exists.
(2) If in addition the groups lim i+1 ffiX vanish for i 1then any
two realizations of X are in the same weak equivalence class.
Remark: See [3, XI, x6] for a discussion of the right derived functors
lim i of the inverse limit functor lim :Ab D !Ab .
Theorem 1.1 is actually a consequence of a sharper and more geometric
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Centric Maps 3
We study two examples, both involving diagrams which are used to
express the classifying space of a compact Lie group as a homotopy di-
rect limit (or at least as a homotopy direct limit up to Z=p homology).
We first show thata slight modification of the centralizer decomposition
diagram of [12] is centric. We then show that the p-toral decomposi-
tion diagram of [13] is centric, and point out that the results of [13]
imply that this diagram satisfies the hypotheses of parts (1) and (2)
of Theorem 1.1. It follows from 1.1(2), then, that the diagram of [13]
is determined up to weak equivalence by its underlying diagram in the
homotopy category.
Organization of the Paper: Section 2contains a discussion of re-
alization complexes and some calculations in the centric case. Theorem
1.1 is proved in x3. Finally, x 4 deals with the centrializer diagram of
[12] and x5 with the p-toral diagram of[13].