tricategories

A tricategory is a weak 3-category in the sense of higher category theory. It has objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, and 3-morphisms between 2-morphisms. The composition laws for 1- and 2-morphisms are not strictly associative or unital; instead, they are equipped with higher-dimensional isomorphisms that satisfy coherence conditions up to further 3-morphisms.

For any pair of objects A and B, there is a hom-bicategory T(A,B). Its objects are 1-morphisms A → B, its 1-morphisms are 2-morphisms between those 1-morphisms, and its 2-morphisms are 3-morphisms between the 2-morphisms. Horizontal composition of 1-morphisms induces a functor between hom-bicategories, and this composition is associative and unital only up to specified 2-isomorphisms, called associators and unitors, which themselves are controlled by 3-morphisms.

The coherence data in a tricategory includes higher morphisms that satisfy pentagon and triangle-like identities, ensuring compatibility of the various ways of composing morphisms up to higher equivalence. In particular, the interchange laws for composing 2- and 3-morphisms hold up to coherent 3-morphisms, and all higher coherence relations are part of the structure.

Examples and relation to other notions: a strict 3-category is a tricategory in which all coherence isomorphisms are equalities. A standard example is the tricategory of bicategories, with pseudofunctors as 1-cells, pseudonatural transformations as 2-cells, and modifications as 3-cells. Tricategories provide a natural framework for higher-dimensional category theory and underpin many constructions in higher algebra and topology. Coherence theorems show that many tricategories are biequivalent to strict 3-categories under suitable conditions.