A Functorial Classification of Toric Stacks

Abstract

Toric stacks enrich the study of toric varieties by accounting for quotients of actions of diagonalizable groups. In practice, toric stacks are constructed and understood via "stacky fans" generalizing the fans appearing in toric geometry. Previous attempts to make this construction into an equivalence of categories have either run into crucial errors or have been restricted to special cases (e.g. only considering separated Deligne-Mumford stacks). In this paper, we solve this problem by introducing a new 2-category $\mathsf{StkFan}$ of stacky fans and constructing an equivalence of 2-categories between $\mathsf{StkFan}$ and a 2-category $\mathsf{TStk}$ of toric stacks. Along the way, we develop a theory of "toric categories" which categorifies the classical theory of toric monoids. We also explain how working with toric categories provides a simple axiomatic framework for Gale duality.

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