Sigma limits in 2-categories and flat pseudofunctors

DESCOTTE, M. EMILIA; DUBUC, EDUARDO J.; SZYLD, MARTIN

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2018 vol. 333 p. 266 - 313

0001-8708

In this paper we introduce sigma limits (abbreviated $sigma$-limits), a concept which interpolates between lax and pseudo limits: 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 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 $sigma$-limit ($sigma$-bilimit)}. 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, 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. end{abstract}