Functor in a sentence

This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of X).

Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that : Naming conventions The use of "h A " for the covariant hom-functor and "h A " for the contravariant hom-functor is not completely standard.

Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups.

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous.

Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.

Given an object X, a functor G (taking for simplicity the first functor to be the identity) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism (so ).

It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects.

Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction.

That is, instead of saying is a contravariant functor, they simply write (or sometimes ) and call it a functor.

The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups.

The original category C is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory.

This follows, in part, from the fact the covariant Hom functor Hom(N, ) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid.

A continuous map of topological spaces X → Y determines a continuous functor O(Y) → O(X).

Additive functors If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab.

A functor F from C to D is a mapping that Jacobson (2009), p. 19, def. 1.2.

A functor G lifts limits of type J if it lifts limits for all diagrams of type J. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that G lifts limits if it lifts all limits.

A functor is an operation on spaces and functions between them.

Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint.

A morphism f:A → B of rings is a homological epimorphism if it is an epimorphism and it induces a full and faithfull functor on derived categories: D(f) : D(B) → D(A).