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 is

  • Robert Gordon?, John Power, Ross Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558

This was refined in the thesis

which is probably the best current starting point to read about tricategories and from where to take pointers to the original work by Gordon-Power-Street.

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

  • Peter Guthmann, The tricategory of formal composites and its strictification, arXiv:1903.05777

Last revised on March 15, 2019 at 12:00:49. See the history of this page for a list of all contributions to it.