A CONSTRUCTION OF CERTAIN WEAK COLIMITS AND AN EXACTNESS PROPERTY OF THE 2-CATEGORY OF CATEGORIES

作者：Descotte M. Exm^*; Dubuc E. J.; Szyld M.

来源：Theory and Applications of Categories, 2018, 33: 193-215.

• weak colimit
• filtered
• 2-category
• exactness property

Given a 2-category A, a 2-functor A ->(F) Cat and a distinguished 1-subcategory Sigma subset of A containing all the objects, a sigma-cone for F (with respect to Sigma) is a lax cone such that the structural 2-cells corresponding to the arrows of Sigma are invertible. The conical sigma-limit is the universal (up to isomorphism) sigma-cone. The notion of sigma-limit generalizes the well known notions of pseudo and lax limit. We consider the fundamental notion of sigma-filtered pair (A; Sigma) which generalizes the notion of 2-filtered 2-category. We give an explicit construction of sigma-filtered sigma-colimits of categories, a construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that sigma-filtered sigma-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a sigma-filtered sigma-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.

• 出版日期2018