Quillen cohomology of $(\infty,2)$-categories

TL;DR

This paper develops a homotopy-theoretic framework for $( abla, 2)$-categories, introducing a twisted 2-cell $ abla$-category and relating parameterized spectrum objects to diagrams indexed by this category, with applications to adjunctions.

Contribution

It constructs the twisted 2-cell $ abla$-category for $( abla, 2)$-categories and establishes an equivalence with parameterized spectrum objects, advancing the understanding of Quillen cohomology in this context.

Findings

Established an equivalence between spectrum objects and diagrams indexed by the twisted 2-cell $ abla$-category.

Identified Quillen cohomology with the two-fold suspension of the inverse limit spectrum.

Provided an obstruction-theoretic proof of the uniqueness of adjunctions at the homotopy $(3, 2)$-category level.

Abstract

In this paper we study the homotopy theory of parameterized spectrum objects in the $\infty$-category of $(\infty, 2)$-categories, as well as the Quillen cohomology of an $(\infty, 2)$-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an $(\infty,2)$-category $\mathbb{C}$, which we call its twisted 2-cell $\infty$-category. We then establish an equivalence between parameterized spectrum objects over $\mathbb{C}$, and diagrams of spectra indexed by the twisted 2-cell $\infty$-category of $\mathbb{C}$. Under this equivalence, the Quillen cohomology of $\mathbb{C}$ with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between $(\infty,1)$-categories are uniquely determined at the level of the homotopy $(3, 2)$-category of $\mathrm{Cat}_{\infty}$.