nLab associative n-category



Higher category theory

higher category theory

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Universal constructions

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1-categorical presentations



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** (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.


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 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.


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