In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

to handle the cases where the composition of

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

s is not (strictly)

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

, but only associative ''

up to Two 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'' with respect to ''R'' a ...

'' an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

. The notion was introduced in 1967 by

Jean Bénabou Jean Bénabou (1932 – 11 February 2022) was a Moroccan-born French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré and Institut de mathémati ...

. Bicategories may be considered as a weakening of the definition of

2-categories In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categ ...

. A similar process for 3-categories leads to tricategories, and more generally to weak ''n''-categories for ''n''-categories.

Definition

Formally, a bicategory B consists of: *

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

''a'', ''b'', ... called 0-''cells''; *

s ''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

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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

*:\mathbf(b,c) \times \mathbf(a,b) \to \mathbf(a,c)

called ''horizontal composition''. The horizontal composition is required to be associative up to a

natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a nat ...

α between morphisms

h*(g*f)

and

(h*g)*f

. Some more

coherence axiom In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. ...

s, similar to those needed for

monoidal categories In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and ...

, are moreover required to hold: a monoidal category is the same as a bicategory with one 0-cell.

References

* J. Bénabou. "Introduction to bicategories, part I". In ''Reports of the Midwest Category Seminar'', Lecture Notes in Mathematics 47, pages 1-77. Springer, 1967.

External links

* Higher category theory {{categorytheory-stub