Effectivity of Generalized Double $\infty$-Categories
Félix Loubaton
TL;DR
This work provides an $ extit{(∞,n)}$-categorical generalization of effectiveness for internal structures by establishing an adjunction between $(n,m)$-double $ extinfty$-categories with enough companions and filtrations $A_0 o brace o A_m$, characterized by a realization and a Čech nerve built from the Gray tensor product. When the necessary companion conditions hold and the maps are essentially surjective on lower-dimensional cells, this adjunction becomes an equivalence, generalizing the Rezk–Lurie internal-groupoid result and recovering Ayala–Francis for $n=0$ and certain square-functor conjectures for $n=1$. The construction relies on an intricate framework of Gray operations, marked/bi-marked structures, two-sided fibrations, and directed Čech nerves, providing a unified approach to higher double-categorical effectiveness. These results offer a robust foundation for higher topos-like phenomena in $( abla n)$-type contexts and connect to key conjectures in DAG via a general, operadic viewpoint.
Abstract
We construct an adjunction between $m$-categories internal to $(\infty,n)$-categories, called $(n,m)$-double $\infty$-categories, and filtrations $A_0\to \dots\to A_m$ where for all $i<m$, $A_i$ is a $(n+i)$-category. We show that this adjunction induces an equivalence between $(n,m)$-double $\infty$-categories admitting enough companions and filtrations such that each morphism $A_i\to A_{i+1}$ is essentially surjective on cells of dimension lower than or equal to $i$. This result can be seen as a $(\infty,n)$-categorical generalization of the equivalence between internal groupoids and effective epimorphisms in the category of $\infty$-groupoids proven by Rezk and Lurie. In the case $n=0$, this recovers the characterization of flagged $m$-categories given by Ayala-Francis, and in the case $n=1$, it allows us to prove some conjectures concerning the square functor and its variants, stated by Gaitsgory-Rozenblyum in the appendix of their book on Derived Algebraic Geometry.