Derived functor
Derived functor In mathematics, certain functors may be derived to obtain other functors closely related to ... a "long exact sequence". The concept of derived functors explains and clarifies many of these ... we are given a covariant left exact functor F : A → B between two ...
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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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Tor functor
Tor functor In mathematics, the Tor functors of homological algebra are the derived functors of the tensor product functor. They were first defined in generality to ... B. Then T is a right exact functor from Mod-R to the category of ... is commutative, it is a right exact functor from Mod-R to Mod-R) ...
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Ext functor
Ext functor In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first ... C. (This is a left exact functor (contravariant) so we want its right derived functors $R^nT$). To this ...
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Exact functor
Exact functor In homological algebra, an exact functor is one which preserves exact sequences. Formally ... categories, and let F:C→D be a functor. Let 0→A→B→C→0 be a short exact sequence ... is exact. If G is a contravariant functor from C to D, we can make ... A,X) defines a covariant left-exact functor from A to the category Ab ...
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Direct image functor
Direct image functor In mathematics, in the field of sheaf ... especially in algebraic geometry, the direct image functor generalizes the notion of a section of ... Sheaves}(Y)), then the direct image functor $f\_*\; :\; \backslash mathbf\; \{\; Sheaves\}(X)\backslash to\; \backslash mathbf\; ...\; mathcal\{G\}$, and this determines a functor. If $\backslash mathcal\{F\}$ is a ... of abelian groups on $X.$ Derived functors of direct image are called ...
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List of mathematics articles (D-F)
... examples) -- Derivative (generalizations) -- Derivative of a constant -- Derived category -- Derived functor -- Derived group -- Derived set (mathematics) -- Desargues' theorem -- Descartes' rule ...
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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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Coherent duality
... the continuing consideration of this area. Adjoint functor point of view While Serre duality uses ... as the existence of a right adjoint functor $f^!\backslash ;,$ now called twisted inverse ... a higher direct image with proper support functor $Rf\_!\backslash ;.$ Higher direct images are ... they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced ...
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Sheaf (mathematics)
... presheaf on X is then a contravariant functor from Top X to C. If F ... sheaves on X forms a category (a functor category). An isomorphism of sheaves on X ... general case. Let Top be the contravariant functor from the category of topological spaces Top ... X . Here Top(f) is a covariant functor from Top Y to Top X sending ... with Top(f) we obtain a contravariant functor from Top Y to C. A ...
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