Topological and Smooth Stacks

TL;DR

This paper reviews the foundational concepts of stacks, focusing on topological and smooth contexts, and clarifies subtle distinctions such as stacks over stacks and the difference between small and large stacks.

Contribution

It provides a detailed clarification of the definitions and distinctions essential for understanding topological and smooth stacks, addressing subtle theoretical issues.

Findings

Clarified the definition of stacks over stacks.

Distinguished between small algebraic stacks and large generalized space stacks.

Provided foundational insights for further research in topological and smooth stacks.

Abstract

We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).