A construction of certain weak colimits and an exactness property of the 2-category of categories

DESCOTTE, M. EMILIA; DUBUC, EDUARDO J.; SZYLD, MARTIN

THEORY AND APPLICATIONS OF CATEGORIES

Mount Allison University

Año: 2017

1201-561X

Given a 2-category , a 2-functor ⟶Fat and a distinguished 1-subcategory Σ⊂ containing all the objects, a σ-cone for F (with respect to Σ) is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit is the universal (up to isomorphism) σ-cone. The notion of σ-limit generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of σ-filtered} pair (,Σ) which generalises the notion of 2-filtered 2-category. We give an explicit construction of σ-filtered σ-colimits of categories, construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that σ-filtered σ-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a σ-filtered σ-colimit of exact category valued 2-functors is exact. This corollary is essential in the 2-dimensional theory of flat and pro-representable 2-functors, that we develop elsewhere.