Some properties of Fib as a fibred \(2\)-category
Not only does this paper analyse the particular example \({\mathcal F}ib\) as the title implies, but it also defines (for the first time, I believe) what it means for a 2-functor to be a fibration. Here \({\mathcal F}ib\) denotes the 2-category whose objects are functors \(E\to B\) which are fibrations. The 2-functor \({\mathcal F}ib \to{\mathcal C}at\), taking each fibration to its codomain, is a fibration. Writing \({\mathcal K}_{\text{ladj}}\) for the sub-2-category of the 2-category \({\mathcal K}\) obtained by restricting to the 1-cells with right adjoints, the author proves that the 2-functor \({\mathcal F}ib_{\text{ladj}}\to{\mathcal C}at_{\text{ladj}}\) is a subfibration of \({\mathcal F}ib \to{\mathcal C}at\). Results about the completeness and Cartesian closedness of \(E\), for an object \(E\to B\) of \({\mathcal F}ib\), are simply deduced. The paper also discusses comma objects in \({\mathcal F}ib\) and \({\mathcal F}ib(B)\), fibrations in \({\mathcal F}ib(B)\), and the Kleisli construction for comonads in \({\mathcal F}ib\).
scientific article; zbMATH DE number 1308286