nLab associative n-category

Redirected from "associative n-categories".
Note: associative n-category and associative n-category both redirect for "associative n-categories".
Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The notion of associative nn-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 nn-categories are strict, except for weak versions of higher exchange laws.

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

Composition operations in associative nn-categories naturally have stratified-geometric semantics in manifold diagrams, which makes them an instance of geometrical n n -categories. In particular, the weak coherence laws of an associative nn-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 nn-category with strict units. A (proof-wise) potentially more realistic version of this conjecture concerns the case of free associative nn-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 May 11, 2023 at 11:20:15. See the history of this page for a list of all contributions to it.