Vertical categorification replaces set-theoretic structures by higher categorical analogues, while horizontal categorification, replaces one-object categories by multi-object categories. Unlike vertical categorification, horizontal categorification and decategorification are still not systematically developed.

In joint work with Yangxiao Luo, we introduce concentration structures on categories: equivalence relations on morphisms satisfying natural axioms. Each such structure gives rise to an associated monoid, which may be viewed as a horizontal decategorification of the category. This talk will present the basic definitions, properties, and examples, with possible connections to equivariant direct limits, real braid groups, and fundamental groupoids.