Subject: categories: Re: Comma categories

From: F W Lawvere <[email-masked]>
Date: Tue, 20 Oct 1998 17:11:48 -0400 (EDT)

A crucial point is whether the recipient of the enriching is cartesian or
not. Note that fully internalising always must involve a cartesian aspect
since one must diagonalize on the parametrizers of families of objects 
(at least) in order to explain eg natural transformations, even if the
parametrizers for individual homs are not cartesian (eg linear or metric).

One can envisage replacing individual "comma" categories by families of
categories parametrized by (commutative) coalgebras, which seems just a
way of constructing a cartesian category for the purpose, to which it may
or may not be adequate. 

Symmetric monoidal categories in which the unit object is terminal seem to
have a special role, but that may be illusory.(After all "any" smc is
covered by one with that additional property) . Perhaps the affine modules
( see my paper "Grassmann's dialectics and category theory") constitute a
good test case for proposed constuctions

Bill Lawvere.
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F. William Lawvere			Mathematics Dept. SUNY 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

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On Mon, 19 Oct 1998, Manuel Bullejos wrote:

> 
> Does any body know if comma categories have been defined in
> enriched contexts?
> 
> I have an idea of how they can be defined in some particular
> contexts, such as Cat-categories or Simplicial-categories, but I
> don't know if there is a general definition or even if a
> definition in the above two contexts can be found in the
> literature.
> 
> Thanks
> 
> Manuel Bullejos
> 
> 
>