On the squares functor and the Gaitsgory-Rozenblyum conjectures

TL;DR

This paper clarifies the status of eight conjectures in derived algebraic geometry related to $( abla,2)$-categories, proves the last open conjecture, and demonstrates the universal property of the squares functor, a key construction in the field.

Contribution

It provides a proof for the last remaining conjecture and establishes the universal property of the squares functor in the context of $( abla,2)$-categories.

Findings

Proof of the last open conjecture in Gaitsgory-Rozenblyum's list.

Demonstration of the universal property of the squares functor.

Clarification of the status of eight conjectures in derived algebraic geometry.

Abstract

In the seminal work of Gaitsgory and Rozenblyum on derived algebraic geometry, eight conjectures regarding the theory of $(\infty,2)$-categories are stated. This paper aims to clarify the status of these claims, and to provide a proof for the last remaining open one. Along the way, we demonstrate the universal property of the so-called squares functor, a construction that plays an important role in the $(\infty,2)$-categorical foundations of Gaitsgory-Rozenblyum.