Abstract
We focus on the transfer of some known orthogonal factorization systems from mathsf {Cat} to the 2-category {mathsf {Fib}}(B) of fibrations over a fixed base category B: the internal version of the comprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in {mathsf {Fib}}(B), the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in mathsf {Cat}, providing suitable conditions on a 2-category {mathcal {C}}, that allow the transfer of the construction of coinverters and coidentifiers from {mathcal {C}} to {mathsf {Fib}}_{{mathcal {C}}}(B).
Highlights
The crucial point of the work [2] was to recover Yoneda’s Classification Theorem in [17, §3.2] as a result of the factorization of a regular span S : X → B × A through an internal discrete opfibration, in the 2-category Fib(B) of fibrations over B
The first aim of the present work is to show that the above mentioned result, obtained in [2] via an ad hoc construction, is an application of an orthogonal factorization system in Fib(B), whose right class is given by internal discrete opfibrations
This is explained in Theorem 4.10, which generalizes to Fib(B) the well known comprehensive factorization system of Cat [15]
Summary
The crucial point of the work [2] was to recover Yoneda’s Classification Theorem in [17, §3.2] as a result of the factorization of a regular span S : X → B × A through an internal discrete opfibration, in the 2-category Fib(B) of fibrations over B. The first aim of the present work is to show that the above mentioned result, obtained in [2] via an ad hoc construction, is an application of an orthogonal factorization system in Fib(B), whose right class is given by internal discrete opfibrations. This is explained in Theorem 4.10, which generalizes to Fib(B) the well known comprehensive factorization system of Cat [15]. Throughout the paper, 2-limits and 2-colimits are to be understood in a strict sense
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