Highercategorical

Highercategorical refers to the field of higher category theory, which generalizes ordinary category theory by allowing morphisms between morphisms at all levels. In this setting, objects have 1-morphisms between them, 2-morphisms between 1-morphisms, and so on, with composition defined up to higher cells. The primary objects of study include n-categories and ∞-categories, especially (∞,1)-categories where higher morphisms are invertible.

Historically, higher category theory grew from Grothendieck’s ideas and the need to model homotopy types categorically. Multiple formalisms have been developed, including quasi-categories (∞-categories) via simplicial sets, complete Segal spaces, simplicial categories, and ω-categories. Foundational contributions came from Joyal, Lurie, Rezk, Dwyer–Kan, among others.

Key concepts include higher morphisms with coherence data, higher limits and colimits, adjunctions, equivalences, and monoidal structures. Different models are linked by comparison theorems, and the theory emphasizes working up to coherent homotopy rather than strict equalities.

Variants include (∞,n)-categories, where morphisms above level n are invertible, and ω-categories, with morphisms in all levels. The framework provides a natural language for spaces and homotopy types and underpins areas such as derived algebraic geometry, higher topos theory, and topological quantum field theory.

Impact: highercategorical methods unify and extend many constructions in topology and algebra, offer tools for organizing complex algebraic structures, and remain an active area of foundational research with ongoing developments in models and applications.