Re: Bi-presheaves

[Ross Street <street@ics.mq.edu.au>]

Dear Andrew

This is what Lawvere told me about once, long ago.
I think he called it the Isbell envelope; that is what I've called it
ever since. It has nice properties. Lawvere explained that, applied
to finite dimensional vector spaces, it fully contains the category
of banach spaces and bounded linear maps. (I think I've got that
right; it's awhile since I checked it.)

Ross

On 06/03/2009, at 2:34 AM, Andrew Stacey wrote:

> Start with an essentially small category, T, and look at the
> category whose
> objects are triples (P,F,c) where: P is a contravariant functor T -
> > Set, F is
> a covariant functor T -> Set and c is a natural transformation from
> P x F to
> the Hom bi-functor.  Morphisms are pairs of natural transformations
> P_1 -> P_2
> and F_2 -> F_1 that intertwine the natural transformations c_1 and
> c_2.
>
> One could also enrich the whole structure.
>
> Has this cropped up anywhere before?  If so, what is it called and
> where can
> I learn about it?  If not, what shall I call it?
>
> If this is something standard then please pardon my ignorance.  I'm
> fairly new
> to _real_ category theory and am still just learning the basics.
>