TL;DR
This paper develops the algebraic geometry of fans, a category related to monoids, by studying sheaves, maps, and properties, highlighting its foundational role across various geometric frameworks.
Contribution
It introduces the basic algebraic geometry of fans, connecting them to other geometric categories and exploring their structural properties.
Findings
Fans are analogous to schemes but built from monoids.
The study covers sheaves, maps, and geometric properties of fans.
Fans serve as a unifying framework over multiple geometric categories.
Abstract
The category of (abstract) fans is to the category of monoids what the category of schemes is to the category of rings: a fan is obtained by gluing spectra of monoids along open embeddings. Here we study the basic algebraic geometry of fans: coherent sheaves, group fans, affine maps, flat maps, proper maps, and so forth. This study is motivated by the fact that the category of fans lies over all other reasonably geometric categories (schemes, differentiable spaces, analytic spaces, etc) as well as the "log" versions of all such categories.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory