Discrete and conservative reflections of fibrations

TL;DR

This paper investigates two specific factorization systems for opfibrations in the 2-category of fibrations over a fixed base, using 2-colimits and extending results from Cat to Fib(B).

Contribution

It introduces two factorization systems for opfibrations in Fib(B) and shows how to obtain them via 2-colimits, extending known results from Cat to Fib(B).

Findings

Factorizations obtained by 2-colimits as coidentifiers and coinverters.

Conditions on 2-category C enable transfer of constructions from Cat to Fib(B).

Results unify and extend fibrational factorizations in a 2-categorical context.

Abstract

We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class is given by internal discrete opfibrations. The second one has as its right orthogonal class internal opfibrations in groupoids, i.e. with groupoidal fibres. These factorizations can be obtained by means of a single step 2-colimit. Namely, their left orthogonal parts are nothing but suitable coidentifiers and coinverters respectively. We will show how these results follow from their analogues in Cat. To this end, we first provide suitable conditions on a 2-category C, allowing the transfer of the construction of coinverters and coidentifiers from C to Fib(B).