# nLab associative n-category

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

The notion of associative $n$-categories (ANCs, Dorn 2018, Reutter & Vicary 2019) is a semi-strict algebraic model for higher category theory based on the geometric shape of globular sets with strictly associative composition: All higher coherence laws in associative $n$-categories are strict, except for weak versions of higher exchange laws.

Due to the tight control over coherence-issues, the theory of (free) associative $n$-categories lends itself to formal proof, for implementation of corresponding proof assistants see* homotopy.io* (and its precursor Globular).

Composition operations in associative $n$-categories naturally have stratified-geometric semantics in manifold diagrams, which makes them an instance of geometrical $n$-categories. In particular, the weak coherence laws of an associative $n$-category can be thought of in terms of a notion of homotopy between composites. This is similar to the case of a Gray-category, which is strictly associative and unital, but which has a weak exchange law. In this sense, ANCs can be seen as a generalization of Gray categories.

## Examples

A separate notion of ‘free’ associative n-categories has been developed. These are instances of geometric computads.

## Remarks

• It is conjectured that every weak n-category is weakly equivalent to an associative $n$-category with strict units. A (proof-wise) potentially more realistic version of this conjecture concerns the case of free associative $n$-categories, and is (as of 2023) a topic of active research.

• A related conjecture is Simpson's conjecture, which states that fully weak higher categories are (in an appropriate sense) equivalent to weakly unital higher categories.

• The underlying idea of this line of modelling higher structures can be traced back to work on Gray 3-categories and surface diagrams.

## References

Last revised on March 29, 2023 at 08:19:18. See the history of this page for a list of all contributions to it.