In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, 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 Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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. The subject of weak ''n''-categories is an area of ongoing research.
History
There is currently 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 In mathematics, 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. Higher cate ...
. There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically (most remarkably
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
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
satisfying some universality properties).
In a terminology due to
John Baez
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
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
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
, 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 (in the case of characteristics zero) also via pretriangulated
A-infinity categories of
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
.
Quillen model categories are viewed as a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of an ; however not all can be presented via model categories.
See also
*
Bicategory
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomor ...
*
Tricategory In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.
Whereas a weak 2-category is said to be a ''bicategory'', a weak 3-category is said to be a ''tricategory'' (Gordon, Power & St ...
*
Tetracategory
*
Infinity category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
*
Opetope
In category theory, a branch of mathematics, an opetope, a portmanteau of "operation" and "polytope", is a shape that captures higher-dimensional substitutions. It was introduced by John C. Baez and James Dolan so that they could define a weak ...
*
Stabilization hypothesis
External links
''n''-Categories – Sketch of a Definitionby
John Baez
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
Lectures on n-Categories and Cohomologyby
John Baez
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
* Tom Leinster, Higher operads, higher categories
math.CT/0305049*
*
Jacob Lurie
Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow.
Life
When he was a student in the Science, Mathematics, and Computer Science ...
, Higher topos theory
math.CT/0608040 published version
pdf
Higher category theory
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