# nLab tricategory

Contents

### 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 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

## References

The original source:

refined in: