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.
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.
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.
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]
Last revised on May 11, 2023 at 11:20:15. See the history of this page for a list of all contributions to it.