Re: Classifying Space...

[Robert L Knighten <RLK@knighten.org>]

Tom Leinster writes:
 > Dear Hugo,
 >
 > Your question involves the functors
 >
 >        N           | |
 > Cat -----> SSet ------> Top
 >
 > (nerve and geometric realization) and their composite, the classifying
 > space functor B.
 >
 > 1. The nerve functor N has a left adjoint, so in particular it preserves
 > finite products.  Hence if M is a monoid in Cat (i.e. a strict monoidal
 > category) then N(M) is, in a natural way, a monoid in SSet.
 >
 > 2. It's also true, though not totally obvious, that the geometric
 > realization functor | | preserves finite products.  So if X is a monoid
 > in SSet then |X| is a topological monoid.

A small issue - this works provided the destination of the geometric
realization is the category of compactly generated Hausdorff spaces.
Otherwise, as in Milnor's original paper, there are limitation on the
simplicial sets involved.

-- Bob

-- 
Robert L. Knighten
RLK@knighten.org