Categorical notions of fibration (Q2221495)

Fibrations in category theory, due to Grothendieck, were developed in [\textit{J. W. Gray}, in: Proc. Conf. Categor. Algebra, La Jolla 1965, 21--83 (1966; Zbl 0192.10701)]. Definitions of fibrations internal to \(2\)-categories were given in [\textit{R. Street}, Lect. Notes Math. 420, 104--133 (1974; Zbl 0327.18006)], and those internal to bicategories were presented in [\textit{R. Street}, Cah. Topologie Géom. Différ. Catégoriques 21, 111--159 (1980; Zbl 0436.18005)]. This expository paper tours the various categorical notions of fibration in order of increasing complexity. A synopsis of the paper goes as follows. \begin{itemize} \item \S 2 deals with the classical definitions of fibrations and discrete ones in ordinary \(1\)-category theory. \item The internalization in a \(2\)-category and generalization in a bicategory are given in \S 3 and \S 4. \item The real goal, pursued in parallel, is to define two-sided discrete fibrations in \(\boldsymbol{Cat}\), where two-sided discrete fibrations encode functors \[ B^{\mathrm{op}}\times A\rightarrow\boldsymbol{Set} \] known as \textit{profunctors} from \(A\)\ to \(B\), while in \(\mathcal{V} \)-\(\boldsymbol{Cat}\) the dual two-sided codiscrete cofibrations encode \(\mathcal{V}\)-profunctors \[ B^{\mathrm{op}}\otimes A\rightarrow\mathcal{V} \] \item This paper concludes with a construction of a bicategory, defined internally to \(\mathcal{V}\)-\(\boldsymbol{Cat}\), whose \(1\)-cells are two-sided codiscrete cofibrations. \end{itemize} This theory has been extended to \(\left( \infty,1\right) \)-categories modeled as quasi-categories by \textit{J. Lurie} [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)], where the equivalence between fibrations and pseudofunctors is implemented by \textit{straightening} and \textit{unstraightening} constructions.