CONICET | Buscador de Institutos y Recursos Humanos

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 $cc{A}$, a $sigma$-cone for a $2$-functor $cc{A} mr{F} cc{B}$ is a lax cone such that the structural 2-cells corresponding to the arrows of $Sigma$ are invertible. The conical emph{$sigma$-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: emph{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 $Cat$-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 $Cat$-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda mbox{$2$-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 emph{$sigma$-filtered}. Our main result is: emph{A pseudofunctor $cc{A} mr{} cc{C}at$ is flat if and only if it is a $sigma$-filtered mbox{$sigma$-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.