Functor
Functor For functors in computer science, see the ... function object article. In category theory, a functor is a special type of mapping between ... Let C and D be categories. A functor F from C to D is a ... reverse composition". We then define a contravariant functor F from C to D as a ... that one can also define a contravariant functor as a covariant functor on the ...
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Functor category
Functor category In category theory, a branch of ... into a category; the morphisms in this functor category are natural transformations between functors. Functor categories are of interest for two main ... reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; a standard ...
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Hom functor
Hom functor In mathematics, specifically in category theory, Hom ... objects A in C we define a functor Hom(A,–) : C → Set to ... B in C we define a contravariant functor Hom(–,B) : C → Set as ... Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the ...
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Forgetful functor
Forgetful functor A forgetful functor is a type of functor in mathematics. The nomenclature is suggestive of such a functor's behaviour: given some object with structure ... is left as an empty list , the functor is simply to take the underlying ...
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Derived functor
Derived functor In mathematics, certain functors may be derived ... we are given a covariant left exact functor F : A → B between two abelian ... F. For every i≥1, there is a functor R i F: A → B, and ... we see that F is an exact functor if and only if R 1 F ... injective, then the sequence splits. Applying a functor to a split sequence results in ...
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Full and faithful functors
Full and faithful functors In category theory, a faithful functor (resp. a full functor) is a functor which is injective (resp. surjective) when ...
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Adjoint functors
... adjoint functors The idea of an adjoint functor was formulated by Daniel Kan in 1958 ... of abelian groups, where F was the functor –⊗A (i.e. take the tensor product with A), and G was the functor Hom(A,–). The use of the ... By itself, the generality of the adjoint functor concept isn't a recommendation to most ... to say that he promoted the adjoint functor concept in isolation: but recognition of ...
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Equivalence of categories
... An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common ... an algebraic setting, the composition of the functor and its "inverse" is not necessarily the ... categories where a strict form of inverse functor is required, but this is of much ... an equivalence of categories consists of a functor F : C -> D, a functor G : ...
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Function object
... object A function object, often called a functor, is a computer programming construct allowing an ... meaning may vary among programming languages. A functor used in this manner in computing bears little relation to the term functor as used in the mathematical field of ... theory. Description A typical use of a functor is in writing more intelligent callback functions ... more dynamic behavior of the function. A functor solves those problems since the function ...
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List of mathematics articles (D-F)
... Derivative of a constant -- Derived category -- Derived functor -- Derived group -- Derived set (mathematics) -- Desargues' theorem ... space -- Dirac measure -- Dirac operator -- Direct image functor -- Direct integral -- Direct limit -- Direct product -- Direct ... Enneagon -- Enneper-Weierstrass parameterization -- Enriched category -- Enriched functor -- Enriques-Kodaira classification -- Enriques surface -- Entire function ... sampling formula -- Exact coloring -- Exact differential -- Exact functor -- Exact sequence -- Exact trigonometric constants -- Examples of ... Expression -- Expression (mathematics) -- EXPSPACE -- EXPTIME -- Exsecant -- Ext functor -- Extended Euclidean algorithm -- Extended number -- Extended ...
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