From: Tom Leinster <tl@maths.gla.ac.uk>
To: categories@mta.ca
Subject: Re: classifying functor and colimits
Date: Mon, 28 Aug 2006 13:45:07 +0100 [thread overview]
Message-ID: <E1GHksh-00012d-3y@mailserv.mta.ca> (raw)
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>
next reply other threads:[~2006-08-28 12:45 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-08-28 12:45 Tom Leinster [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-08-28 21:45 Stephen Lack
2006-08-24 5:59 David Roberts
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