# nLab tricategory

### Context

#### Higher category theory

higher category theory

# 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 $R$ a commutative ring, there is a symmetric monoidal bicategory $Alg(R)$ whose

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 21 of

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

## References

The original source is

• Gordon, Power, 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

Revised on August 13, 2012 08:53:14 by Mike Shulman (71.136.235.154)