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| In [[mathematics]], a '''bicategory''' is a concept in [[category theory]] used to extend the notion of [[Category (mathematics)|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]].
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| Formally, a bicategory '''B''' consists of:
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| * objects ''a'', ''b''... called ''0-cells'';
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| * morphisms ''f'', ''g'', ... with fixed source and target objects called ''1-cells'';
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| * "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'';
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| with some more structure:
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| * 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'';
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| * given three objects ''a'', ''b'' and ''c'', there is a bifunctor <math>*:\mathbf{B}(b,c)\times\mathbf{B}(a,b)\to\mathbf{B}(a,c)</math> called ''horizontal composition''.
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| The horizontal composition is required to be associative up to a natural isomorphism α between morphisms <math>h*(g*f)</math> and <math>(h*g)*f</math>. Some more coherence axioms, similar to those needed for [[monoidal category|monoidal categories]], are moreover required to hold.
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| Bicategories may be considered as a weakening of the definition of [[2-categories]]. A similar process for 3-categories leads to [[tricategory|tricategories]], and more generally to [[weak n-category|weak ''n''-categories]] for [[n-category|''n''-categories]].
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| == References ==
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| * J. Bénabou. "Introduction to bicategories, part I". In ''Reports of the Midwest Category Seminar'', Lecture Notes in Mathematics 47, pages 1-77. Springer, 1967.
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| [[Category:Higher category theory]]
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