On flat functors

DUBUC, EDUARDO J; SZYLD, MARTÍN; DESCOTTE, MARIA EMILIA

Norman

Seminario; OU Algebra and Representation Theory Seminar; 2016

The University of Oklahoma

The notion of flat module has a classic generalization to set-valued functorsF. The main theorem of that theory expresses the equivalencesi) F is flat.ii) F is a filtered colimit of representable functors.iii) The diagram of F is a filtered category.For an arbitrary base category V instead of Ens, Kelly has developed atheory of flat V-enriched functors into V, but there is no known generalizationof the theorem above for any V other than Ens.We have established a 2-dimensional version of this theorem, i.e. for a 2-functor F: C --> Cat, where C is a 2-category and Cat is the 2-category of categories.As it is usually the case for 2-categories, the Cat-enriched notion of limitisn?t adequate for most purposes and the relaxed bi and pseudo notions are theimportant ones. We will explain these concepts, review the theories mentionedabove and present our theorem.