The comprehensive factorization of Burroni's T-functors

TL;DR

This paper extends the comprehensive factorization system to Burroni's category of T-categories under mild conditions, revealing new applications in topology and multicategories.

Contribution

It establishes the existence of the comprehensive factorization system in Burroni's T-category setting, connecting it to topology and multicategory theory.

Findings

Factorization system exists in Burroni's T-categories.

Application to Lambek's multicategories.

Insight into fiberwise compactification in topology.

Abstract

Expanding on the comprehensive factorization of functors internal to a category C, under fairly mild conditions on a monad T on C we establish that this orthogonal factorization system exists even in Burroni's category Cat(T) of (internal) T-categories and their functors. This context provides for some expected applications and some unexpected connections. For example, it lets us deduce that the comprehensive factorization is also available for functors of Lambek's multicategories. In topology, it leads to the insight that the role of discrete cofibrations is played by perfect maps, with the comprehensive factorization of a continuous map given by its fibrewise compactification.