Dear Andrew Stacey, When John Isbell introduced this construction in the early 1960's, he called it the 'double envelope', so I often call it the Isbell envelope. You just re-discovered it! Bill Lawvere ************************************************************ 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