On orthogonal factorization systems and double categories

TL;DR

The paper establishes a precise bridge between orthogonal factorization systems and double $\infty$-categories by constructing a fully faithful functor $\mathop{\mathrm{Fact}}: \mathop{\mathrm{OFS}} \to \operatorname{DCat}$ whose essential image consists of double $\infty$-categories with a unique square for each bottom-left corner. It then provides an inverse on the image via $\mathop{\mathrm{Cnr}}$, proving an equivalence $\mathop{\mathrm{OFS}} \simeq \operatorname{DCat}^{\operatorname{OF}}$, and derives an $(\mathrm{un})$straightening equivalence for fibrations, restricting to op-Gray fibrations and curved orthofibrations. Applications include recovering Barwick's span construction for adequate factorization systems, computing automorphisms of $\operatorname{OFS}^{\bot}$ (giving $\mathbb{Z}/2\mathbb{Z}$), and establishing fibrational correspondences: $\operatorname{Ortho}(\mathcal C^{\dag}) \simeq \mathrm{CoR}(\mathop{\mathrm{Fact}}(\mathcal C^{\dag}))$ and $\operatorname{opGray}(\mathcal C^{\dag}) \simeq \mathrm{CaR}(\mathop{\mathrm{Fact}}(\mathcal C^{\dag}))$. Collectively, these results unify OFS and double-category formalisms in higher category theory and enable fibrational analyses and span-structure interpretations within a common framework.

Abstract

We prove that the $\infty$-category of orthogonal factorization systems embeds fully faithfully into the $\infty$-category of double $\infty$-categories. Moreover, we prove an (un)straightening equivalence for double $\infty$-categories, which restricts to an (un)straightening equivalence for op-Gray fibrations and curved orthofibrations of orthogonal factorization systems.

Figures (1)

• Figure 1: $\mathbb{A}\mathrm{r}\left(\left[ 3 \right]\right)$, the squares are labeled by the the smallest bicosimplicial subspace $P^{i,j}$ in which they are contained

Theorems & Definitions (62)

• Theorem A: \ref{['main-theorem']}
• Theorem B: \ref{['maintheorem']}
• Theorem C: \ref{['Aut(AOFS)']}
• Definition 2.1
• Remark 2.2
• Theorem 2.3: Joyal-Tierney jt07
• Proposition 2.4: bri18
• Definition 2.5
• Remark 2.6
• Definition 2.9