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.