Hi John Interesting result. Especially the way in which the correct 2-structure follows from cartesian closure. An analogous result may be the (exension from posetal domains to categories of) closure under filtered colimits of various shapes. Possibly there is a common generalization to connected limits or colimits ? Bill On Thu 04/16/09 9:22 PM , John Bourke johnb@maths.usyd.edu.au sent:
Dear category theorists, I noticed recently that the category whose objects are categories with pullbacks and whose morphisms are pullback preserving functors is cartesian closed. Given a pair of categories with pullbacks A and B, the internal hom [A,B] has objects: pullback preserving functors from A to B, and morphisms: cartesian natural transformations.I have posted a short paper on the arxiv proving this fact: http://arxiv.org/abs/0904.2486It seems like a fairly natural fact but is not to my knowledge in the literature. I am wondering whether anyone was previously aware of this result, and if so whether it might be mentioned somewhere in the literature?Thanks, John Bourke, University of Sydney.