Re: Bi-presheaves

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

Ross,

Thanks for the information.  I'm not surprised to hear the name 'Isbell'
connected with this (nor Lawvere) as there are hints of this idea in 'Taking
Categories Seriously' where Lawvere talks about 'Isbell conjugation' (though
in Isbell conjugation one considers the categories of covariant and
contravariant functors as separate).  Looking up 'Isbell envelope' on
MathSciNet came up with nothing whilst 'Isbell conjugation' only came up with
two papers (one reviewed by you, I think!).  One is about 'total categories',
though I've yet to look at it to see if it is relevant.

When you say that you call it 'the Isbell envelope', what do you mean?  Is the
category the 'Isbell envelope of/on the original category' or are the objects
'Isbell envelopes' and we have the category of Isbell envelopes (in/on/of the
original category)?

Thanks for the quick reply,

Andrew

On Fri, Mar 06, 2009 at 04:13:11PM +1100, Ross Street wrote:
> 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.
>>