On flat 2-functors

EDUARDO JULIO DUBUC; MARTÍN SZYLD; MARÍA EMILIA DESCOTTE

Vancouver

Conferencia; CT2017 - International Category Theory Conference; 2017

University of British Columbia

The main theorem of the theory of flat functors ([1], [4]) states that P from A to Ens is flat if and only if P is a filtered colimit of representable functors, i.e. there is a filtered category I and a diagram X from I to A such that P is the colimit of the composition hX where h is the Yoneda embedding. For an arbitrary base category V instead of Ens, Kelly ([3]) has developed a theory of at V-enriched functors P from A to V, but there is no known generalization of the theorem above for any V other than Ens.In [2] we have established a 2-dimensional version of this theorem, i.e. for a2-functor P from A to Cat, where A is a 2-category and Cat is the 2-category of categories.As it is usually the case for 2-categories, the Cat-enriched notions are not adequate for most purposes and the relaxed bi and pseudo notions are the important ones.We define a 2-functor P from A to Cat to be flat when its left bi-Kan extensionP^* along the Yoneda 2-functor h from A to Homs(A^op; Cat) is leftexact. Homs(A^op; Cat) denotes the 2-category of 2-functors, 2-natural transformations and modications. By left bi-Kan extension we understand the bi-universal pseudonatural transformation P =) P^*h, and by left exact we understand preservation of finite weighted bilimits. Let (A; \Sigma) be a pair where A is a 2-category and \Sigma a distinguished 1-subcategory. A \sigma-cone for a 2-functor F from A to B is a lax cone such that the 2-cells corresponding to the distinguished arrows are invertible. The \sigma-limit ofF is a universal \sigma-cone (characterized up to isomorphism). We introduce a notion of 2-filteredness of A with respect to \Sigma, which we call \sigma-filtered. Our main result states the following:A 2-functor P from A to Cat is flat if and only if there is a \sigma-filtered pair (I^op; \Sigma) and a 2-diagram X such that P is pseudo-equivalent to the sigma-bicolimit of the composition hX. As in the 1-dimensional case, X can bechosen as the 2-fibration associated to P.References:[1] Artin M., Grothendieck A., Verdier J., SGA 4, Springer Lecture Notes in Math-ematics 269 (1972) Ch IV.[2] Descotte M.E., Dubuc E., Szyld M., On the notion of at 2-functor, submitted,arXiv:1610.09429v2 (2016).[3] Kelly G. M., Structures dened by finite limits in the enriched context I, Cahiersde Topologie et Geometrie Diferentielle Categoriques 23 (1982).[4] Mac Lane S., Moerdijk I, Sheaves in Geometry and Logic: a First Introductionto Topos Theory, Springer (1992).