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.
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 $R$ a commutative ring, there is a symmetric monoidal bicategory $Alg(R)$ whose
k-morphism are bimodule homomorphisms.
The monoidal product is given by tensor product over $R$.
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
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
Last revised on March 15, 2019 at 12:00:49. See the history of this page for a list of all contributions to it.