nLab associative n-category

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

• An associative 0-category is a set

• An associative 1-category is an unbiased 1-category

• An associative 2-category is an unbiased strict 2-category

• An associative 3-category is an unbiased Gray 3-category

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.

Related concepts

• manifold-diagrammatic n-category

• semistrict n-category

• homotopy.io, Globular

• manifold diagram

• surface diagram

• globular set

• Gray-category

• Simpson's conjecture

References

• Christoph Dorn, Associative $n$-categories, talk at 103rd Peripatetic Seminar on Sheaves and Logic (pdf).

• Christoph Dorn, Associative $n$-categories, PhD thesis (arXiv:1812.10586).

• David Reutter, Jamie Vicary, High-level methods for homotopy construction in associative $n$-categories, LICS ‘19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune 62 (2019) 1–13 [arXiv:1902.03831, doi:10.1109/LICS52264.2021.9470575]

• Lukas Heidemann, David Reutter, Jamie Vicary, Zigzag normalisation for associative $n$-categories, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [arXiv:2205.08952, doi:10.1145/3531130.3533352]