# Pseudo-functor

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In mathematics, a pseudofunctorF is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that ${\displaystyle F(f\circ g)=F(f)\circ f(g)}$ and ${\displaystyle f(1)=1}$ do not hold as exact equalities but only up to coherent isomorphisms .

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The Grothendieck construction associates to a pseudofunctor a fibered category.

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In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra.

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In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".

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In mathematics the Karoubi envelope of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi.

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This is a glossary of properties and concepts in category theory in mathematics.

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The Grothendieck construction is a construction used in the mathematical field of category theory.

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In category theory, a branch of mathematics, a groupoid object in a category C admitting finite fiber products is a pair of objects together with five morphisms satisfying the following groupoid axioms

1. where the are the two projections,
2. (associativity)
3. (unit)
4. (inverse) , , .