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

Abstract: Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $\Sigma \subset \mathcal{A}$ 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 generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of $\sigma$-filtered} pair $(\mathcal{A}, \, \Sigma)$ which generalises the notion of 2-filtered 2-category. We give an explicit construction of $\sigma$-filtered $\sigma$-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 $\sigma$-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.