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 and 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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Universal property Central object of study in category theory

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Commutative diagram

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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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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 mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

In algebraic geometry, a prestackF over a category C equipped with some Grothendieck topology is a category together with a functor p: FC satisfying a certain lifting condition and such that locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.

In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : XY. The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.

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) , , .

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