Constructing Reedy Fibrant Replacements of Projective Fibrant Simplicial Presheaves

In this paper, we construct an explicit Reedy fibrant replacement functor for projective fibrant simplicial presheaves $X:{\mathcal{C}}^{op}\to \mathbf{\text{sSet}}$, where $\mathcal{C}$ 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.