Weak n-category
Encyclopedia
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 techically 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.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, weak n-categories are a generalization of the notion of (strict) n-category
N-category
In mathematics, n-categories are a high-order generalization of the notion of category. The category of n-categories n-Cat is defined by induction on n by:...
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
Higher category theory
Higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.- Strict higher categories :...
. 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 techically accessible formalisms of (infinity,1)-categories are now known. Now the most popular such a formalism centers on a notion of quasi-category
Quasi-category
In mathematics, a quasi-category is a higher categorical generalization of a notion of a category introduced by .André Joyal has much advanced the study of quasi-categories showing that most of the usual basic 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.
External links
- n-Categories – Sketch of a Definition by John Baez
- Lectures on n-Categories and Cohomology by John Baez
- Tom Leinster, Higher operads, higher categories, math.CT/0305049
- Carlos Simpson, Homotopy theory of higher categories, draft of a book arxiv/1001.4071 (alternative URL with hyperTeX-ed crosslinks: pdf)
- Jacob LurieJacob LurieJacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...
, Higher topos theory, math.CT/0608040, published version: pdf