tricategory

A tricategory is a weak 3-category. It has objects, 1-cells between objects, 2-cells between 1-cells, and 3-cells between 2-cells. The composition at each level exists but is not strictly associative or unital: these laws hold up to specified invertible higher cells, and those higher cells themselves satisfy coherence conditions.

Concretely, for any pair of objects A and B there is a category T(A,B) whose objects are 1-cells f: A→B and whose morphisms are 2-cells α: f⇒g. For A→B→C there is a horizontal composition functor T(B,C) × T(A,B) → T(A,C). The associativity of this horizontal composition is given by an invertible 2-cell a_{f,g,h}: (h∘g)∘f ⇒ h∘(g∘f), and there are left and right unit 2-cells λ_f and ρ_f. These 2-cells are linked by invertible 3-cells (modifications) that enforce coherence among different ways of composing several 1-cells.

The defining coherence axioms include higher-dimensional analogues of the pentagon and triangle identities, ensuring compatibility among all composites of associators and unitors up to specified 3-cells. In particular, associativity and unit constraints need not commute strictly, only up to coherent 3-cells.

Tricategories generalize bicategories and strict 3-categories: a bicategory is a tricategory with all 3-cells identities, and a strict 3-category has all associativity and unit laws strictly equal. A standard example is the tricategory of bicategories, homomorphisms, pseudo-natural transformations, and modifications.