The classifying space of an \(\infty\)-category is the homotopy type obtained by universally inverting all of its morphisms. Many interesting homotopy types arise in this way, for example, the classifying space of the \(\infty\)-category of \(k\)-dimensional bordisms embedded in \(\mathbb{R}^n \times [0,1]\) yields the \(n\)-fold loop space of the space of \(k\)-planes in \(\mathbb{R}^{n+1}\) (including the “empty plane”).
Similarly to the situation for topological spaces, it is useful for computations to know which commutative squares of \(\infty\)-categories are homotopy pullbacks, i.e., produce pullback squares of classifying spaces. A functor of \(\infty\)-categories is called a realisation fibration if any pullback along it is a homotopy pullback in this sense. The main theorem of the paper under review is:
Theorem: Any functor which is both a locally Cartesian fibration and a locally coCartesian fibration is a realisation fibration.
Thus, the main theorem generalises a result of W. Steimle [Algebr. Geom. Topol. 21, No. 2, 601–646 (2021; Zbl 1475.57046)] stating that functors which are both Cartesian fibrations and coCartesian fibrations are realisation fibrations.
With a view towards applications, the author characterises such functors in three different models: quasi-categories, weakly unital Segal spaces, and weakly unital topological categories. In one such application, functors between several variants of weakly unital topological categories of cospans of finite categories are shown to be realisation fibrations, allowing for explicit computations of their fibres.
The paper is clearly structured, recalls all relevant definitions (thus making it largely self-contained), and carefully carries out all arguments in all three models.