A Remark on Lurie's Representability Theorem

TL;DR

This paper revisits Lurie's representability theorem for geometric stacks, showing a mild relaxation of one condition, and explores variants and the connection between deformation theory and homogeneity conditions.

Contribution

It provides a relaxed condition for Lurie's theorem and clarifies the relationship between deformation theory and homogeneity in prestacks.

Findings

Relaxation of a key condition in Lurie's theorem

Introduction of variants of the main theorem

Analysis of deformation theory and homogeneity

Abstract

In this note we revisit Lurie's representability theorem for geometric stacks and prove that one of the conditions can be mildly relaxed. The proof uses ideas from Hall--Rydh's work on the (classical) Artin's representability theorem. We also spell out a couple of variants of the main theorem and review the relation between deformation theory and homogeneity conditions on prestacks.