Constructing Reedy Fibrant Replacements of Projective Fibrant Simplicial Presheaves

TL;DR

The paper constructs an explicit Reedy fibrant replacement functor $R$ for projective fibrant simplicial presheaves over a Reedy category, using an inductive, higher-homotopy approach to define latching maps. With a natural transformation $\kappa_X:X\to R(X)$, it shows that $R(X)$ is Reedy fibrant, $\kappa_X$ is a levelwise weak equivalence (trivial cofibration in elegant Reedy contexts), and $R$ preserves levelwise weak equivalences and finite products. This framework yields concrete models for homotopy pullbacks and homotopy limits, enabling explicit fibrant replacements without the small-object argument and offering potential applications to $\Theta_n$-spaces and $(\infty,n)$-categories. Overall, the work provides a hands-on, explicit mechanism to pass from projective fibrant objects to Reedy fibrant ones, with direct implications for constructing homotopy bicategories and related higher-categorical structures.

Abstract

In this paper, we construct an explicit Reedy fibrant replacement functor for projective fibrant simplicial presheaves $X : \mathscr{C}^{op} \rightarrow \textbf{sSet}$, 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. We explore the nature of our functor by using it to recover some standard homotopy limit constructions.

Theorems & Definitions (64)

• Theorem 1.1
• Definition 3.1: dwyerSpalinskiHomotopyTheoriesModel1995
• Definition 3.2: dwyerSpalinskiHomotopyTheoriesModel1995
• Definition 3.3
• Definition 3.4: may1999AConciseCourse
• Definition 3.5
• Definition 3.6
• Proposition 3.7
• proof
• Proposition 3.8