Straightening for lax transformations and adjunctions of $(\infty,2)$-categories

TL;DR

This paper establishes a unstraightening theorem for lax transformations in $( abla,2)$-categories, enabling new fibrational descriptions of (co)limits and characterizations of adjoints and mate correspondences in higher category theory.

Contribution

It introduces an unstraightening result for lax transformations in $( abla,2)$-categories, advancing the understanding of (op)lax limits, adjunctions, and mate correspondences in higher categories.

Findings

Fibrational descriptions of (co)limits in $( abla,2)$-categories

Characterization of adjoints in $( abla,2)$-categories of functors

Mate correspondence between lax and oplax transformations

Abstract

We prove an unstraightening result for lax transformations between functors from an arbitrary $(\infty,2)$-category to that of $(\infty,2)$-categories. We apply this to study partially (op)lax and weighted (co)limits, giving fibrational descriptions of such (co)limits for diagrams valued in $(\infty,2)$-categories, to characterize adjoints in $(\infty,2)$-categories of functors and (op)lax transformations, and to prove a mate correspondence between lax transformations that are componentwise right adjoints and oplax transformations that are componentwise left adjoints, for such transformations among functors between arbitrary $(\infty,2)$-categories.