On flat 2-functors

MARÍA EMILIA DESCOTTE; MARTÍN SZYLD; EDUARDO JULIO DUBUC

Louvain la Neuve

Workshop; 5th Workshop on Categorical Methods in Non-Abelian Algebra; 2017

Université catholique de Louvain

The notion of flat module has a classical generalization to set-valued functorsP: C−→ Ens ([1], [4]). The main theorem of that theory expresses the equivalencesi) P is flat.ii) P is a filtered colimit of representable functors.iii) The diagram of P is a filtered category.For an arbitrary base category V instead of Ens, Kelly [3] has developed atheory of flat V-enriched functors P: C−→ V, but there is no known generalizationof the theorem above for any V other than Ens.In [2] we have established a 2-dimensional version of this theorem, i.e. fora 2-functor P: C−→ Cat, where C is a 2-category and Cat is the 2-category ofcategories. As it is usually the case for 2-categories, the Cat-enriched notion oflimit isn?t adequate for most purposes and the relaxed bi and pseudo notionsare the important ones.References[1] Artin M., Grothendieck A., Verdier J., SGA 4, Ch IV, Springer LectureNotes in Mathematics 269 (1972).[2] Descotte M.E., Dubuc E., Szyld M.,On the notion of flat 2-functor, submitted,arXiv:1610.09429v2 (2016).[3] Kelly G. M.,Structures defined by finite limits in the enriched context I,Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques 23 (1982).[4] Mac Lane S., Moerdijk I, Sheaves in Geometry and Logic: a First Introductionto Topos Theory, Springer, New York (1992).