Sigma limits in 2-categories and flat pseudofunctors

In this paper we introduce sigma limits (which we write $\sigma$-limits), a concept that interpolates between lax and pseudolimits: for a fixed family $\Sigma$ of arrows of a 2-category $\mathcal{A}$, a $\sigma$-cone for a $2$-functor $\mathcal{A} \stackrel{F}{\rightarrow} \mathcal{B}$ is a lax cone such that the structural 2-cells corresponding to the arrows of $\Sigma$ are invertible. The conical $\sigma$-limit of $F$ is the universal $\sigma$-cone. Similary we define $\sigma$-natural transformations and weighted $\sigma$-limits. We consider also the case of bilimits. We develop the theory of $\sigma$-limits and $\sigma$-bilimits, whose importance relies on the following key fact: any weighted $\sigma$-limit (or $\sigma$-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary $\mathcal{C}at$-valued 2-functor as a conical $\sigma$-bicolimit of representable 2-functors, for a suitable choice of $\Sigma$, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a $\mathcal{C}at$-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class $\Sigma$, which we call $\sigma$-filtered. Our main result is: A pseudofunctor $\mathcal{A} \rightarrow \mathcal{C}at$ is flat if and only if it is a $\sigma$-filtered $\sigma$-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.