Weak n-category
In category theory in mathematics, 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 or coherent isomorphism. A weak 0-category is just a set, and a weak 1-category is a ordinarily category. 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.
History
There is much work to determine what the coherence laws for weak n-categories should be. Weak n-categories have become the main object of study in higher category theory. There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically and those in which more topological models are used.In a terminology due to John Baez and James Dolan, a is a weak n-category, such that all h-cells for h > k are invertible. Some of the formalism for are much simpler than those for general n-categories. In particular, several technically accessible formalisms of (infinity, 1)-categories are now known. 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 can be modeled also via pretriangulated A-infinity categories of Maxim Kontsevich. Quillen model categories are viewed as a presentation of an ; however not all can be presented via model categories.