Homotopy Bicategories of 2-fold Segal Spaces

TL;DR

The paper develops a concrete bridge from homotopy-theoretic $( extbf{∞},2)$-categories modeled by $2$-fold Segal spaces to algebraic, unbiased bicategories. It constructs a functor $h_2$ from Reedy fibrant $2$-fold Segal spaces with chosen Segal sections to unbiased bicategories, using left homotopies and cosimplicial resolutions to encode coherence and associativity. It proves that choices of Segal map sections yield equivalent bicategories up to identity on objects, morphisms, and $2$-morphisms, and it provides a robust coherence framework via globular $2$-homotopies. The framework culminates in a fundamental bigroupoid interpretation for topological spaces and a functorial path from Top to unbiased bicategories, illustrating a concrete path between homotopy-theoretic models and algebraic higher-category structures with potential applications to extended TQFTs and cobordism hypotheses.

Abstract

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.

Theorems & Definitions (148)

• Definition 2.1
• Definition 2.2: heutsMoerdijkSimplicialDendroidalHomotopy2022
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.1
• Proposition 2.1
• proof
• Proposition 2.2