Re: Classifying Space...

Re: Classifying Space...

[F William Lawvere]

The small issue cited by Bob has further ramifications.

Actually the geometric realization  functor  "should"
preserve all finite limits. That is probably a main reason
(along with the lack of function spaces) for the gradual demise 
of Top (and its relatives like Locales) as the default model 
of cohesion.

A functor with a right adjoint and preserving finite limits
is the inverse image of a geometric morphism of toposes,
provided it is between toposes. Can Top be reasonably 
replaced by a topos that will serve all purposes of
algebraic topology, functional analysis, etc ? Yes, as 
Peter Johnstone showed some years ago, using in fact 
a well-known monoid as site.

The issue for simplicial realization in particular is whether
- the unit interval is totally ordered or not and whether
-its endpoints are distinct. 
That is because simplicial sets is precisely the classifying
 topos for such structures, as was pointed out by Joyal and
explained well  both in Johnstone and in Mac Lane & Moerdijk.
(The second condition justifies the omission of 0 from the Delta site).

Johnstone used the monoid of continuous endomaps of the
generic convergent sequence to achieve the total order of the unit
interval; the internal meaning of the latter is that inside the square
there is no subobject containing both solid triangles. No sheaf, that is.

But Peter achieved that solution after detailed study led him  to reject 
another  model, proposed by several people whose 
geometric intuition does not include Peano curves, undecidable
 statements, etc (e.g., me). That proposal, namely that the basic
figures of topology are continuous curves, had to be rejected 
at the then-current level of knowledge because the usual model
for the shape of these figures, the unit interval as constructed in 
traditional set theory, admits far too many endomaps, along which 
coverings must be stable by pullback; thus too few coverings , too
many sheaves.  This leads to the reasonable demand for a submonoid
of those continuous reparametrizers, containing polynomials and lattice
operations but not containing, for example, the coordinates of a 
Peano curve  in the square.

That demand is very similar to the one put forth by Grothendieck in 
his proposal for TAME TOPOLOGY.  I pointed out several years ago
that Grothendieck's demand is related to the achievements
of mathematicians working on so-called O-minimal models, and  some 
of the of them in fact mention Grothendieck's slogan in their discussions.
But I am not aware of any publications addressing the issue exposed by Peter.

Can we now understand by example the kind of monoid desired ?

Best wishes
Bill

On Sat 04/25/09  2:54 AM , Robert L Knighten RLK@knighten.org sent:
> 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
> 
> 
> 
> 
> 

Re: Classifying Space...

[Robert L Knighten]

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

Re: Classifying Space...

[Tom Leinster]

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
>
>
>
>
>
>
>
>
>

Classifying Space...

[Hugo.BACARD]

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