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 = . 2 = . --> . 3 = . --> . --> . 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,
------------------------------------------------------------------------ -- 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
-- Tom Leinster <tl@maths.gla.ac.uk>