tricategory in nLab

Contents

Contents

Idea

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).

Examples

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

• objects are RR-algebras;

• morphisms are algebra bimodules;

• 2-morphisms are bimodule homomorphisms.

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:

• Richard Garner, Nick Gurski: The low-dimensional structures formed by tricategories. Math. Proc. Camb. Phil. Soc. 146 (2009),pp. 551–589, (arXiv:0711.1761)

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

• Mike Shulman, Constructing symmetric monoidal bicategories (arXiv:1004.0993)

• strict 3-category
• bicategory
• tetracategory

References

The original source:

• Robert Gordon, A. John Power, Ross Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558 (ISBN:978-1-4704-0137-5)

refined in:

• Nick Gurski, An algebraic theory of tricategories, PhD thesis (2007) [pdf, pdf]

See also:

Textbook account:

• Niles Johnson, Donald Yau, Section 11.2 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

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

• Alex Hoffnung, Spans in 2-Categories: A monoidal tricategory (arXiv:1112.0560)

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)