Given a 2-category $A$, a 2-functor $F : A \to Cat$ and a distinguished 1-subcategory $\Sigma \subset A$ 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.
Keywords: weak colimit, filtered, 2-category, exactness property
2010 MSC: Primary: 18D05. Secondary: 18A30
Theory and Applications of Categories, Vol. 33, 2018, No. 8, pp 192-215.