highercategory

Highercategory, or higher category theory, is the study of categories that include not only objects and arrows between objects but also higher levels of morphisms between arrows, and so on to a chosen finite or infinite height. An n-category has objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, up to n-morphisms, with composition operations and unit laws whose coherence is governed by higher-dimensional equivalences. In strict n-categories these laws hold exactly, while in weak (or otherwise “up to isomorphism”) n-categories they hold only up to specified higher morphisms that satisfy their own coherence conditions.

A basic example is a 2-category, where one has objects, 1-morphisms between objects, and 2-morphisms between 1-morphisms. The 2-category Cat consists of categories as objects, functors as 1-morphisms, and natural transformations as 2-morphisms. A related notion is a bicategory, which weakens associativity of composition of 1-morphisms to an isomorphism rather than strict equality.

Beyond finite height, many articles study infinity categories (∞-categories), which have morphisms in all higher dimensions and where morphisms above dimension 1 are not necessarily invertible or are inverted in a controlled way depending on the model. Important models include quasi-categories (simplicial sets satisfying inner horn-filling conditions), complete Segal spaces, Segal categories, and simplicial categories.

Higher category theory provides a framework for organizing structures in homotopy theory, higher algebra, and derived geometry, and it underpins approaches to topological quantum field theory, algebraic topology, and modern formulations of mathematical foundations.