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
2-morphisms 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:
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
Richard Garner, Nick Gurski, The low-dimensional structures that tricategories form (arXiv:0711.1761)
Peter Guthmann, The tricategory of formal composites and its strictification (arXiv:1903.05777)
Last revised on September 6, 2022 at 08:29:21. See the history of this page for a list of all contributions to it.