PS The Isbell envelope arises from the Isbell conjugate pair which is the adjoint pair of functors connecting set^(T^op) and (set^T)^op. It is thus a special case of the general construction in my thesis (TAC Reprints) of the total category with two descriptions which objectifies the notion of adjointness, in order to free it from dependence on enrichments in small sets. (Of course the example depends on enrichments in small sets.) ************************************************************ F. William Lawvere, Professor emeritus Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Thu, 5 Mar 2009, Andrew Stacey wrote:
Dear Categorists,
I'm interested in looking at the following type of thing:
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.
Thanks,
Andrew Stacey