On a notion of ring groupoid

TL;DR

This paper provides an elementary exposition of the (2,1)-category of ring groupoids, which are animated rings with trivial higher homotopy groups, motivated by their role in prismatic cohomology.

Contribution

It offers a simplified, elementary treatment of ring groupoids without relying on advanced n-category theory, aiding understanding in prismatic cohomology context.

Findings

Elementary description of the (2,1)-category of ring groupoids

Clarification of the role of ring stacks in prismatic cohomology

Simplification of the conceptual framework for animated rings

Abstract

By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.