Subject: Re: followup & conference Date: Sat, 6 Apr 2002 17:00:22 -0500 From: Tom Goodwillie >Recall that the question was whether the functor F(?,X) converts >"finite" holimits to homotopy hocolimits in the category of spectra. > >Or, more or less equivalently, when are "finite" holimits hocolimits? > >However it is unclear is what "finite" means. I thought at first that >the indexing category should be finite with finitely many morphisms. >However it does not seem to be finite enough as actions of finite >groups make clear. > >I now believe that the correct notion of finiteness here is that the >indexing category I is very small in the sense of Dwyer-Spalinsky. >That is, not only the set of morphisms and objects is finite, but >also the all morphisms go in the same direction, that is a long >enough string of morphisms must contain an identity morphism. > Yes, that kind if finiteness (in other words, the nerve of I is a finite simplicial set -- finitely many nondegenerate simplices) is enough. I wonder what other small categories have this property regarding holim and hocolim. Although finite groups do not, groups G such that BG has the homotopy type of a finite complex do. It seems to me that one can generalize this last class of examples and talk about small categories I such that the constant functor S:I -> Spectra given by the sphere spectrum has what you might call a finite projective resolution. These also have that property. But that class of examples doesn't include those "very small" categories. I wonder if there is a common generalization. By the way, the nerve of I must be at least stably homotopy equivalent to a finite complex if hocolim F(S,X) is going to coincide with F(holim S,X) where S is that constant functor again. I don't know any references. Which reminds me, does anybody have a reference for the fact that holim over a very small category commutes (up to weak equivalence) with hocolim over a filtering category (or a directed set)? Tom Goodwillie Subject: Re: followup & conference Date: Sun, 7 Apr 2002 15:30:34 -0500 From: Tom Goodwillie A further thought: If J -> I is a functor satisfying Bousfield and Kan's "left cofinality" condition, which insures that any holim over I is equivalent to the associated holim over J, then (since the associated functor J^{op} -> I^{op} satisfies right cofinality, which implies the same thing for hocolim) we can say that if J has the property under discussion (hocolim of dual is dual of holim), then so does I. So this gives more examples. And in fact that class of examples I mentioned (groups with homotopy- finite classifying spaces) can be obtained in that way from the other class of examples (very small categories). If X is a finite connected simplicial complex with contractible universal covering space, then the barycentric subdivision of X yields a poset that has a left cofinal functor to the fundamental group of X. Tom Goodwillie