Jack Romö | Searcharxiv

arXiv2025

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 $X : \mathscr{C}^{op} \rightarrow \textbf{sSet}$, 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.

math.AT↗

arXiv2025

Homotopy Bicategories of 2-fold Segal Spaces

In this paper, we address the construction of homotopy bicategories of $(\infty,2)$-categories, which we take as being modeled by 2-fold Segal spaces. Our main result is the concrete construction of a functor $h_2$ from the category of Reedy fibrant $2$-fold Segal spaces (each decorated with chosen sections of the Segal maps) to the category of unbiased bicategories and pseudofunctors between them. Though our construction depends on the choice of sections, we show that for a given $2$-fold Segal space, all possible choices yield the same unbiased bicategory up to an equivalence that acts as the identity on objects, morphisms and $2$-morphisms. We illustrate our results with the example of the fundamental bigroupoid of a topological space.

math.CT↗