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In this paper we introduce sigma limits (which we write $\sigma$-limits), aconcept 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 conesuch that the structural 2-cells corresponding to the arrows of $\Sigma$ areinvertible. 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 theoryof $\sigma$-limits and $\sigma$-bilimits, whose importance relies on thefollowing key fact: any weighted $\sigma$-limit (or $\sigma$-bilimit) can beexpressed as a conical one. From this we obtain, in particular, a canonicalexpression 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 flatpseudofunctors. We define a $\mathcal{C}at$-valued pseudofunctor to be flatwhen its left bi-Kan extension along the Yoneda 2-functor preserves finiteweighted bilimits. We introduce a notion of 2-filteredness of a 2-category withrespect to a class $\Sigma$, which we call $\sigma$-filtered. Our main resultis: A pseudofunctor $\mathcal{A} \rightarrow \mathcal{C}at$ is flat if and onlyif it is a $\sigma$-filtered $\sigma$-bicolimit of representable 2-functors. Inparticular the reader will notice the relevance of this result for thedevelopment of a theory of 2-topoi.

Sigma limits in 2-categories and flat pseudofunctors