Bicategory
Encyclopedia
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 (strictly) associative, but only associative up to
an isomorphism. The notion was introduced in 1967 by Jean Bénabou.
Formally, a bicategory B consists of:
with some more structure:
The horizontal composition is required to be associative up to a natural isomorphism α between morphisms
and 
. Some more coherence axioms, similar to those needed for monoidal categories
, are moreover required to hold.
Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories
, and more generally to weak n-categories
for n-categories
.
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 bicategory is a concept in 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...
used to extend the notion of category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
an isomorphism. The notion was introduced in 1967 by Jean Bénabou.
Formally, a bicategory B consists of:
- objects a, b... called 0-cells;
- morphisms f, g, ... with fixed source and target objects called 1-cells;
- "morphisms between morphisms" ρ, σ... with fixed source and target morphisms (which should have themselves the same source and the same target), called 2-cells;
with some more structure:
- given two objects a and b there is a category B(a, b) whose objects are the 1-cells and morphisms are the 2-cells, the composition in this category is called vertical composition;
- given three objects a, b and c, there is a bifunctor
called horizontal composition.
The horizontal composition is required to be associative up to a natural isomorphism α between morphisms
and . Some more coherence axioms, similar to those needed for monoidal categoriesMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
, are moreover required to hold.
Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories
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 [Benabou 1967], a weak 3-category is said to be a tricategory .Tetracategories are the corresponding notion in dimension four...
, and more generally to weak n-categories
Weak n-category
In category theory, weak n-categories are a generalization of the notion of 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...
for n-categories
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:...
.