Concentration structures on categories and horizontal categorification

We introduce a theory for encoding and manipulating algebraic data on categories via $\mathit{\text{concentration}}\mathit{\text{structures}}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\mathit{\text{concentration}}\mathit{\text{monoid}}$, 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.