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:
Dear category theorists,
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" ?
2) And What kind of topological spaces of the realization of the nerve -Is it a topological monoid , with some extra structure ?
3) And what hapen if M is not strict, or is symmetric, or braided , etc...
Thank you and sorry if these are completely stupid questions