categorified

Categorified is an adjective used to describe mathematical objects or constructions that have been obtained by categorification. Categorification is a process that replaces set-based structures with category-based analogues, often by moving from sets to categories, functions to functors, equalities to isomorphisms, and elements to objects and morphisms. The result is a higher-dimensional, more structured version of the original object. In many cases, one speaks of a decategorification functor that recovers the original set or numerical invariant by taking isomorphism classes (for example, the decategorification of a category of finite sets under disjoint union yields the natural numbers). A categorified version of an object thus carries more data.

Examples illustrate the idea. Categorified numbers treat natural numbers as isomorphism classes of finite sets, and categorified linear algebra uses categories whose decategorification yields a vector space. A prominent example is Khovanov homology, which categorifies the Jones polynomial by assigning a chain complex to a knot; its graded Euler characteristic recovers the Jones polynomial. Categorified quantum groups, realized by Khovanov–Lauda–Rouquier algebras, provide higher-categorical analogues of quantum groups.

History and scope: The term categorification was popularized by Baez and Dolan in the late 1990s, though related ideas appeared earlier. The concept has since become central to higher category theory, representation theory, and low-dimensional topology, influencing developments across mathematics and mathematical physics.

Impact: Categorification and its categorified objects offer richer invariants and structural frameworks, enabling deeper connections between algebra, topology, geometry, and physics. The phrase “categorified” thus denotes a shift from set-based data to categorically enriched, higher-dimensional structures.