Tricategory
Encyclopedia
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
[Benabou 1967], a weak 3-category is said to be a tricategory (Gordon, Power & Street 1995; Baez & Dolan 1996; Leinster 1998).
Tetracategories
are the corresponding notion in dimension four. Dimensions beyond three are seen as increasingly significant to the relationship between knot theory
and physics. John Baez, R. Gordon, A. J. Power and Ross Street have done much of the significant work with categories beyond bicategories thus far.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a tricategory is a kind of structure of category theory
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...
studied in higher-dimensional category theory.
Whereas a weak 2-category
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
is said to be a bicategory
Bicategory
In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B...
[Benabou 1967], a weak 3-category is said to be a tricategory (Gordon, Power & Street 1995; Baez & Dolan 1996; Leinster 1998).
Tetracategories
Tetracategory
In category theory, a tetracategory is a weakened definition of a 4-category.- External links :* by Todd Trimble....
are the corresponding notion in dimension four. Dimensions beyond three are seen as increasingly significant to the relationship between knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
and physics. John Baez, R. Gordon, A. J. Power and Ross Street have done much of the significant work with categories beyond bicategories thus far.