Sigma limits in 2-categories and flat pseudofunctors

Abstract

In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-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 σ-bicolimit of representable 2-functors, for a suitable choice of Σ, 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 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A⟶Cat is flat if and only if it is a σ-filtered σ-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.