In category theory, weak n-categories are a generalization of the notion of (strict) n-category where composition is not strictly associative but only associative up to coherent equivalence. There is currently much work to determine what the coherence laws should be for those. 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 (most remarkably the Michael Batanin's theory of weak higher categories) and those in which more topological models are used (e.g. a higher category as a simplicial set satisfying some universality properties).
In a terminology due to Baez and Dolan, a (n,k)-category is a weak n-category, such that all h-cells for h>k are invertible. Some of the formalism for (n,k)-categories 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 a 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 Kontsevich. Quillen model categories are viewed as a presentation of an (infinity,1)-category; however not all (infinity,1)-categories can be presented via model categories.