Discrete and Conservative Factorizations in Fib(B)

AbstractWe focus on the transfer of some known orthogonal factorization systems from$$\mathsf {Cat}$$Catto the 2-category$${\mathsf {Fib}}(B)$$Fib(B)of fibrations over a fixed base categoryB: the internal version of thecomprehensive 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$${\mathsf {Fib}}(B)$$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$$\mathsf {Cat}$$Cat, providing suitable conditions on a 2-category$${\mathcal {C}}$$C, that allow the transfer of the construction of coinverters and coidentifiers from$${\mathcal {C}}$$C to$${\mathsf {Fib}}_{{\mathcal {C}}}(B)$$FibC(B).