Constructing Reedy fibrant replacements of projective fibrant simplicial presheaves

Abstract

In this paper, we construct an explicit Reedy fibrant replacement functor for projective fibrant simplicial presheaves \(X : \mathscr {C} \rightarrow {\textbf {sSet}}\), where \(\mathscr {C}\) is a Reedy category. Our approach describes, by hand, all latching maps for the Reedy fibrant replacement by an inductive series of higher homotopies. The concrete nature of the construction means it has application in extending other constructions on Reedy fibrant functors to more general projective fibrant functors, providing an explicit description of the extension. In particular, the author uses the Reedy fibrant replacement functor presented here in his thesis to construct the homotopy bicategory of a projective fibrant 2-fold Segal space by extending a similar construction in the Reedy fibrant case. We illustrate our functor’s behavior by using it to recover the homotopy category of a projective fibrant Segal space from the classical homotopy category construction for Reedy fibrant Segal spaces, which allows us to further recover a standard characterization of completeness for projective fibrant Segal spaces.