Spectral space

2013-12-02T10:22:49Z

159.149.107.58: /* Spectral maps */

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]].

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 <math>*:\mathbf{B}(b,c)\times\mathbf{B}(a,b)\to\mathbf{B}(a,c)</math> called ''horizontal composition''.
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.

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]].

== 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.

[[Category:Higher category theory]]

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Spectral space