Dear David, The classifying space functor is the composite of the nerve functor N:Cat-->SSet and the geometric realization functor SSet-->Top, and it makes sense to consider them separately. The nerve functor preserves all limits (since it is a right adjoint) but not all colimits. It does preserve=20 sequential colimits, since it is given by homming out of the finite ordinals, which are finitely presentable.=20 More explicitly, the nerve NC of a category C is the simplicial set whose set (NC)_n of n-simplices is the set of all functors from [n] to C, where [n] is the category {0<1<...<n}. Now each of these [n] is finitely presentable, meaning that homming out of it preserves filtered colimits, so N itself preserves filtered colimits. Similarly, N does preserve coproducts, since the [n] are all connected. (To preserve coproducts and filtered colimits is to preserve what=20 Mac Lane calls pseudofiltered colimits.) The geometric realization functor is a left adjoint (it is the left Kan extension along Yoneda of the standard map Delta-->Top) and=20 so preserves all colimits, but relatively few limits. It does, as Tom Leinster observes, preserve finite products. Putting these facts together, one sees that the classifying=20 space functor preserves coproducts, filtered colimits, and finite = products. Steve Lack. -----Original Message----- From: cat-dist@mta.ca on behalf of Tom Leinster Sent: Mon 8/28/2006 10:45 PM To: categories@mta.ca Subject: categories: Re: classifying functor and colimits =20 Dear David,
I have been plagued by the following question: does the classifying space functor commute with (co)limits?
The classifying space functor (from Cat to Top) does preserve finite products. It doesn't preserve all infinite products, e.g. let A be the discrete category with two objects and consider the product of infinitely many copies of A. Nor does it preserve all colimits, as the following example shows. Let 1 be the terminal category, 2 the category consisting of a single arrow, and 3 the category consisting of a commutative triangle: 1 =3D . 2 =3D . --> . 3 =3D . --> . --> . Take the two different functors from 1 to 2. The pushout of the diagram in Cat formed by these functors is 3, and B3 is Delta^2, the standard topological 2-simplex. However, B1 is the one-point space and B2 is the unit interval, so the pushout of B1 and B2 is an interval of length 2, which is not homeomorphic to Delta^2. This doesn't answer your question about sequential colimits, but maybe it gives some helpful context. Best wishes, Tom
In particular, I have a system of compact topological groups G_i indexed by the natural numbers, and a whole lot of inclusions.
Is B colim G_i homotopic to colim BG_i ?
I have a hint that this should be so in my particular situation (in a letter of Serre to Grothendieck), but I'd like to know how the general case goes.
Cheers,
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-- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 =
-- droberts@maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts www.trf.org.au
--=20 Tom Leinster <tl@maths.gla.ac.uk>