Re: Classifying Space...

[Tom Leinster <tl@maths.gla.ac.uk>]

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.

3. Putting these together, if M is a strict monoidal category then its
classifying space B(M) is a topological monoid.

If M is a non-strict monoidal category then B(M) is not necessarily a
topological monoid in a natural way, but it is a "homotopy topological
monoid" in any of several accepted senses.  For instance, it is a
Delta-space in the sense of Segal, and an A_infinity-space in the sense of
Stasheff (although that doesn't deal satisfactorily with the unit).

Similarly, if M is a symmetric monoidal category then B(M) is a "homotopy
topological commutative monoid", e.g. a Gamma-space or an E_infinity
space.

Best wishes,
Tom


On Thu, 23 Apr 2009, Hugo.BACARD@unice.fr wrote:

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> Dear category theorists,
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> Sorry for my following stupid questions , but i would like:
>
> -Given a monoidal category M , for first assumed to be strict, What kind of
> thing do we obtain when we take it's classifying space ?: we take the nerve of M
> and then realising the simplicial sets obtained
>
> Explicitely there are theses questions :
>
> 1) Does the nerve of M "preserve" (or "reflect")  the monoidal structure of M ?
>
>    -Is it a monoid in the category of simplicial sets ?
>    -If yes , can we have conditions on a monoid of sSet to be the nerve of a
> monoidal category ? I mean, does some kind of "segal condition" ?
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>
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> 2) And What kind of topological spaces of the realization of the nerve
>   -Is it a topological monoid , with some extra structure ?
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> 3) And what hapen if M is not strict, or is symmetric, or braided , etc...
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> Thank you and sorry if these are completely stupid  questions
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