On the notion of flat 2-functors

DUBUC, EDUARDO J.

Montreal

Seminario; McGill Logic, Category Theory, and Computation Seminar; 2016

Let (I; ) be a pair where I is a 2-category and a distinguished 1-subcategory. A -conefor a 2-functor I 􀀀! C is a lax cone such that the 2-cells corresponding to the distinguishedarrows are invertible, -limits are as usual the universal -cones. Similarly we dene -diconesand -ends for 2-functors Aop A 􀀀! B.The beautiful thing is that -ends can be expressed as -limits. This is an important factof limits and ends in category theory that was not available in 2-dimensional category theory.We introduce the notion of -ltered pair which generalises 2-lterness, and we develop asuccessful theory of at 2-functors. We dene a 2-functor A P 􀀀! Cat to be at when its leftbi-Kan extension Homs(Aop; Cat) P􀀀! Cat along the Yoneda 2-functor A h 􀀀! Homs(Aop; Cat)is left exact (where Homs(Aop; Cat) denotes the 2-category of 2-functors, 2-natural transfor-mations and modications, and by left exact we understand preservation of nite weightedbilimits). The main result is:A 2-functor A P 􀀀! Cat is at if and only if there is a -ltered pair (Iop; ), a2-diagram I X 􀀀! A, and P is pseudo-equivalent to the -bicolimit of the compositionIop X 􀀀! Aop h 􀀀! Homs(A; Cat).This establishes that the 2-category of points of the "2-topos" of 2-presheavesHoms(Aop; Cat) (A small), is equivalent to the 2-category -Pro(A) of -pro-objects of A.In other words, the 2-category of models is equivalent to the 2-category -Ind(Aop) of -ind-objects of Aop.