Variation on a comprehensive theme

TL;DR

This paper establishes a bicategorical factorization system on the bicategory of categories, introducing concepts like ultimate functors and groupoid fibrations, with implications for polynomial functors.

Contribution

It introduces a new factorization system on Cat involving ultimate functors and groupoid fibrations, connecting to polynomial functor theory.

Findings

Every right adjoint functor is an ultimate functor.

Functor factorizations are unique up to isomorphism.

Connections to polynomial functor theory are established.

Abstract

The main result concerns a bicategorical factorization system on the bicategory $\mathrm{Cat}$ of categories and functors. Each functor $A\xra{f} B$ factors up to isomorphism as $A\xra{j}E\xra{p}B$ where $j$ is what we call an ultimate functor and $p$ is what we call a groupoid fibration. Every right adjoint functor is ultimate. Functors whose ultimate factor is a right adjoint are shown to have bearing on the theory of polynomial functors.