Classifying spaces

Classifying spaces

[Peter May]

On colimits of classifying spaces.  This topologist
may be missing something, but the conclusion seems
obviously true, at least in reasonable situations.
With a countable system of inclusions of spaces,
unless the situation is fairly bizarre, the system
will be filtered and we can find a countable cofinal
sequence.  But finite products commute with sequential
colimits in reasonable categories of spaces.  Since
the usual classifying space of G is constructed as
the geometric realization of a simplicial space whose
space of q-simplices is G^q, and since geometric
realization certainly commutes with sequential colimits,
the conclusion seems clear.  It is used all the time
in algebraic topology, in such familiar examples as
BU = colim BU(n), BTop = colim BTop(n), BF = colim BF(n),
etc, the last being a system of monoids rather than groups.
In the standard classical statement that BU classifies
stable complex vector bundles, we are using the first
listed special case.  While the groups are compact in
that case, compactness is not relevant to the argument
and fails for the groups Top(n), for example.  In more
complicated equivariant situations, one has countable
systems that are not naturally sequential, but again the
conclusion is familiar and in common use.

Peter