Examples of using Functors in English and their translations into Spanish
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So let's apply forgetful functors to the past.
Functors are everywhere and we can prove it.[…] Me.
Functions between orders become functors between categories.
Functors are everywhere and we can prove it.
However, in general, applicative functors are more general than this.
There is also a theory for Hochschild homology of certain functors.
The torsion functors Tor are the derived functors of the tensor product.
We will see the combineList function again later, when we consider Traversable functors. Exercises.
Nowadays, functors are used throughout modern mathematics to relate various categories.
If C is small and preadditive,then the category of all additive functors from C to A also forms an abelian category.
Applicative functors are an interesting abstraction which provide neat solutions to a number of problems.
Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product.
Applicative functors are the programming equivalent of lax monoidal functors with tensorial strength in category theory.
The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.
Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra,some of them examples of functors.
If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, and elsewhere as well.
MOF and UML are standards of the OMG Olog is a category theoretic approach to ontologies,emphasizing translations between ontologies using functors.
A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints.
Just as the study of groups is not complete without a study of homomorphisms,so the study of categories is not complete without the study of functors.
In abstract algebra,a completion is any of several related functors on rings and modules that result in complete topological rings and modules.
The example section below provides evidence of this; furthermore, universal constructions, which may be more familiar to some,give rise to numerous adjoint pairs of functors.
Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.
Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.
The individual models of the theory, i.e. the groups in our example,then correspond to functors from the encoding topos to the category of sets that respect the topos structure.
The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, andby using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
If η: F→ G and ε:G→ H are natural transformations between functors F, G, H: C→ D, then we can compose them to get a natural transformation εη: F→ H. This is done componentwise:(εη)X εXηX.
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
Natural transformations are,after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.
If η: F→ G is a natural transformation between functors F, G: C→ D and ε: J→ K is a natural transformation between functors J, K: D→ E, then the composition of functors allows a composition of natural transformations ε∗η: JF→ KG.
