nLab tricategory



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



A tricategory is a particular algebraic notion of weak 3-category. The idea is that a tricategory is a category weakly enriched over Bicat: the hom-objects of a tricategory are bicategories, and the associativity and unity laws of enriched categories hold only up to coherent equivalence.

Coherence theorems

One way to state the coherence theorem for tricategories is that every tricategory is equivalent to a Gray-category, which is a sort of semi-strict 3-category (everything is strict except for the interchange law).


For RR a commutative ring, there is a symmetric monoidal bicategory Alg(R)Alg(R) whose

The monoidal product is given by tensor product over RR.

By delooping this once, this gives an example of a tricategory with a single object.

The tricategory statement follows from Theorem 24 in either the journal version, or the arXiv:0711.1761v2 version of:

This, and that the monoidal bicategory is even symmetric monoidal is given by the main theorem in


The original source:

refined in:

See also:

Textbook account:

A discussion of monoidal tricategories, regarded by the discussion at k-tuply monoidal (n,r)-category as one-object tetracategories, is in section 3 of

See also

Last revised on February 16, 2023 at 10:33:50. See the history of this page for a list of all contributions to it.