Subject: A question about homotopy limits
From: Johannes Ebert
Date: Tue, 10 Jul 2007 18:58:51 +0200 (CEST)
I have the following question for the list.
I have a small category C and a functor F: C \to Top
(topological spaces). I want to know whether the natural map for c \in
Ob(C)
f_c:holim F \to F(c)
is a homotopy equivalence. In the example I am interested in, the
classifying space BC is contractible and F maps all arrows in C to weak
homotopy equivalences. Apart from that, I can assume that all F(c) are
of the homotopy type of a CW, but nothing else. It sounds reasonable that
f_c is a (weak) homotopy equivalence, but is it true?
I understand that f_c is a homotopy-equivalence if c is a terminal object
by a special case of the "cofinality theorem" of Bousfield-Kan, LNM 304
("BK" in the
sequel), p.317 (the assumption that F maps arrows to w.h.e.s is
unnecessary in this case. But my category C has no initial object.
There is a spectral sequence for the homotopy groups of holim F (BK,p.
309) starting with E_2 = lim^s \pi_t, which is "closely related to
\pi_{t-s}".
This spectral sequence led me to the following fragmentary argument, and
my question is of course
whether this is correct or nonsense:
1. Because \pi_0 (holim F) = lim \pi_0 (F), (?) I can assume without loss
of generality that all F(c) are connected.
2. If the spectral sequence converges (this is NOT asserted in BK), then
by naturality it suffices to show that lim^s \pi_t (F) vainshes for s > 0.
There is a comparison map from the spectral sequence for F and the
spectral sequence for the restriction of F to the trivial category \{
c\}, which is an isomorphism for s=0.
3. It remains to show that in the present situation, the higher derived
limits are trivial. At least for t > 1 (pi_t abelian), this is true:
Any functor M : C \to Ab defines a sheaf sh(M) on BC. The functor Ab^C \to
Sh(BC) is exact. Moreover, the inverse limit of M is the same as the
global sections \Gamma (BC; sh(M)) of the sheaf sh(M), in other words, the
inverse limit is the composite of the exact functor sh with the left exact
\Gamma. By the Grothendieck spectral
sequence for derivatives of composite functors, it follows that H^s (BC;
sh(M)) \cong lim^s M.
If M sends all arrows in C to isomorphisms, then sh(M) is a locally
constant sheaf, in other words, a local coefficient system. Because BC is
contractible, the cohomology and hence the derived limits vanish.
There are several problems with this "argument"
1. BK assume that F preserves basepoints - I do not have
basepoints in my example and there is no way to introduce them
(coherently).
2. What are the derived inverse limits for diagrams with values in
nonabelian groups, or better in groupoids, like c \mapsto (fundamental
groupoid of F(c))?
3. Can one remove the condition that the E_2 term only exists for 0 \leq s
\leq t ? (BK,p.309) And can one achieve that the spectral sequence
converges?
Is there a more comprehensive reference on homotopy limits than BK?
Thanks in advance,
Johannes Ebert