Concentration structures on categories and horizontal categorification
Abstract
We introduce a theory for encoding and manipulating algebraic data on categories via concentration structures, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a 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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