In mathematics, a bicategory (or a weak 2category) 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.
Bicategories may be considered as a weakening of the definition of 2categories. A similar process for 3categories leads to tricategories, and more generally to weak ncategories for ncategories.
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Category Theory 9.2: bicategories

Category Theory 1.7: Horizontal Composition and 2Categories

Evan Patterson: "A Short Introduction to Categorical Logic"
Transcription
Definition
Formally, a bicategory B consists of:
 objects a, b, ... called 0cells;
 morphisms f, g, ... with fixed source and target objects called 1cells;
 "morphisms between morphisms" ρ, σ, ... with fixed source and target morphisms (which should have themselves the same source and the same target), called 2cells;
with some more structure:
 given two objects a and b there is a category B(a, b) whose objects are the 1cells and morphisms are the 2cells. 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 categories, are moreover required to hold: a monoidal category is the same as a bicategory with one 0cell.
Example: Boolean monoidal category
Consider a simple monoidal category, such as the monoidal preorder Bool^{[1]} based on the monoid M = ({T, F}, ∧, T). As a category this is presented with two objects {T, F} and single morphism g: F → T.
We can reinterpret this monoid as a bicategory with a single object x (one 0cell); this construction is analogous to construction of a small category from a monoid. The objects {T, F} become morphisms, and the morphism g becomes a natural transformation (forming a functor category for the single homcategory B(x, x)).
References
 ^ Fong, Brendan; Spivak, David I. (20181012). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].
 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
 Bicategory at the nLab