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arXiv · 2510.07553

Concentration structures on categories and horizontal categorification

Yangxiao Luo·Shunyu Wan
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Abstract

We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.

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Yangxiao Luo, Shunyu Wan. 2025-10-08. Concentration structures on categories and horizontal categorification. https://arxiv.org/abs/2510.07553

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