- Bicategory
In
mathematics , a bicategory is a concept incategory 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 byJean 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 :mathbf{B}(b,c) imesmathbf{B}(a,b) omathbf{B}(a,c) called "horizontal composition".The horizontal composition is required to be associative up to a natural isomorphism 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.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.References
*cite journal
last = Bénabou
first = Jean
title = Introduction to bicategories
journal = Reports of the Midwest Category Seminar
volume = 47
pages = 1-77
publisher = Springer
date = 1967
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