mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a bicategory (or a weak 2-category) is a concept 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, ca ...

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, morphisms ...

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'' ...

'' 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. 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.

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

*:\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 natur ...

α between morphisms

h*(g*f)

and

(h*g)*f

. Some more coherence axioms, similar to those needed for monoidal categories, 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