Discrete 2-fibrations (Q6617077)

This paper is a study of 2-dimensional discrete fibrations, which are a specialization of 2-fibrations, just as fibrations generalizes discrete fibrations.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] defines a discrete 2-fibration (Definition 2.4), giving an elements construction (Construction 2.1). Examples are given, particularly that of\N\[\N\mathrm{cod}:\boldsymbol{DFib}\rightarrow\boldsymbol{Cat}\N\]\Nas a discrete 2-fibration.\N\N\item[\S 3] justifies the above definition by exhibiting the pseudo-inverse (Construction 3.2), which leads to representation theorem (Theorem 3.7). It is finally shown (Theorem 3.23) how the equivalence restricts the established equivalence for 2-fibrations in [\textit{M. Buckley}, J. Pure Appl. Algebra 218, No. 6, 1034--1074 (2014; Zbl 1296.18006)].\N\N\item[\S 4] gives several monadicity results. It is shown (Theorem 4.13) that (discrete) 2-fibrations are monadic, the monad being given by an action of Bénabou's cylinder construction [\textit{J. Bénabou}, Lect. Notes Math. 47, 1--77 (1967; Zbl 1375.18001), \S 8.2]. A 3-categorical setting is developed to describe universal constructions arising in the description of (discrete) 2-fibrations as algebras.\N\N\item[\S 5] is a prospectus.\N\end{itemize}