Weak n-category

In category theory, a weak n-category is a generalization of the notion of strict n-category where composition and identities are not strictly associative and unital, but only associative and unital up to coherent equivalence.

This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to as bicategories, tricategories, and tetracategories.

Weak n-categories have become the main object of study in higher category theory.

In a terminology due to John Baez and James Dolan, a (n, k)-category is a weak n-category, such that all h-cells for h > k are invertible.

Now the most popular such formalism centers on a notion of quasi-category, other approaches include a properly understood theory of simplicially enriched categories and the approach via Segal categories; a class of examples of stable (infinity, 1)-categories can be modeled (in the case of characteristics zero) also via pretriangulated A-infinity categories of Maxim Kontsevich.