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:
Set, F is a covariant functor T -> Set and c is a natural transformation from P x F to
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 - 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.