Datum: Mon, 27 Dec 2004 19:06:52 +0100 Betreff: categories: A question for topos theorists. Von: jean benabou <jean.benabou@wanadoo.fr> An: Categories <categories@mta.ca>

I have recently come across a question, which seems to me natural, and before trying to solve it, I wanted to know if an answer was already known.

If E is a category with pullbacks so is the category Cat(E) having as objects the internal categories of E and and as maps the internal functors F: A ---> B. I shall say that such an F is a "Pi-functor" if the pullback functor: Cat(E)/B --->Cat(E)/A has a right adjoint Pi/F: Cat(E)/A---->Cat(E)/B

When E=Set there is a well known Conduché-Giraud condition characterizing such functors. Is there such a characterization when E is an (elementary) topos? If there is, what is the condition and where can it be found?

Best wishes to all, Jean

Dear Jean, Let me draw your attention to the the paper by Phil Heath and myself P.R.Heath, K.H.Kamps Lifting colimits of (topological) groupoids and (topological) categories. Categorical topology and its relations to analysis, algebra and combinatorics : Proceedings, Prague 1988, World Sci. Publishing, Teaneck, NJ, 54-88 (1989). where a related problem on Pi-functors has been dealt with. With best wishes, Heiner. 20-Jan-2005 15:07:43 -0400,1221;000000000001-00000000