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Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom ( X, Y ) of morphisms from X to Y.

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## Some Related Sentences

Representable and functors

functors and We

__We__note that an inverse system

**in**a

**category**

**C**admits an alternative description

**in**terms

**of**

__functors__

**.**

__We__may therefore form its right derived

__functors__; their values are abelian groups and they are denoted by H < sup > n </ sup >( G

**,**M ), "

**the**n-th cohomology group

**of**G with coefficients

**in**M ".

__We__write M < sup > G </ sup > for

**the**subgroup

**of**M consisting

**of**all elements

**of**M that are held fixed by G

**.**This is a left-exact functor

**,**and its right derived

__functors__are

**the**group cohomology

__functors__

**,**typically written as H < sup > i </ sup >( G

**,**M ).

functors and can

From about 1955 he started

**to**work on sheaf theory and homological algebra**,**producing**the**influential " Tôhoku paper "**(**Sur quelques points d ' algèbre homologique**,**published**in**1957**)**where he introduced Abelian categories and applied their theory**to**show that sheaf cohomology__can__be defined as certain derived__functors__**in**this context**.**
Note that arrows between categories are called

__functors__**,**subject**to**specific defining commutativity conditions ; moreover**,**categorical diagrams and sequences__can__be defined as__functors__**(**viz**.**
Basic constructions

**,**such as**the**fundamental group or fundamental groupoid**of**a topological space**,**__can__be expressed as fundamental__functors__**to****the****category****of**groupoids**in**this way**,**and**the**concept is pervasive**in**algebra and its applications**.**
* Adjoint

__functors__**:**A functor__can__be left**(**or right**)**adjoint**to**another functor that maps**in****the**opposite direction**.**
Such a

**pair****of**adjoint__functors__typically arises**from**a construction defined by a universal property ; this__can__be seen as a more abstract and powerful view on universal properties**.**
One

__can__compose__functors__**,**i**.**e**.**if F is a functor**from**A**to**B and G is a functor**from**B**to****C**then**one**__can__form**the**composite functor G ∘ F**from**A**to****C****.**Composition**of**__functors__is associative where defined**.**
This shows that

__functors____can__be considered as**morphisms****in**categories**of**categories**,**for**example****in****the****category****of**small categories**.**
More generally

**,**if**C**is an arbitrary abelian**category**that has enough injectives**,**then so does**C**< sup > I </ sup >, and**the**right derived__functors__**of****the**inverse limit functor__can__thus be defined**.**
One

__can__use**Hom**__functors__**to**relate limits and colimits**in**a**category****C****to**limits**in**Set**,****the****category****of**sets**.**
The notion

**of**a natural transformation is categorical**,**and states**(**informally**)**that a particular map between__functors____can__be done consistently over an entire**category****.**
If and are natural transformations between

__functors__**,**then we__can__compose them**to**get a natural transformation**.**
If is a natural transformation between

__functors__**,**and is another functor**,**then we__can__form**the**natural transformation by defining
If

**C**is**any****category**and I is a small**category****,**we__can__form**the**functor**category****C**< sup > I </ sup > having as**objects**all__functors__**from**I**to****C**and as**morphisms****the**natural transformations between those__functors__**.**
As a result

**,**general theorems about left / right adjoint__functors__**,**such as**the**equivalence**of**their various definitions or**the**fact that they respectively preserve colimits / limits**(**which are also found**in****every**area**of**mathematics ),__can__encode**the**details**of**many useful and otherwise non-trivial results**.**
If

**C**is a complete**category****,**then**the**__functors__with left adjoints__can__be characterized by**the**adjoint functor theorem**of**Peter J**.**Freyd**:**G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied**:**for**every**object**Y****of**D there exists a family**of****morphisms**
* If is a topological space

**(**viewed as a**category**as above**)**and is some small**category****,**we__can__form**the****category****of**all contravariant__functors__**from****to****,**using natural transformations as**morphisms****.**

functors and generalize

Additive

__functors__between preadditive categories__generalize__**the**concept**of**ring homomorphism**,**and ideals**in**additive categories**can**be defined as sets**of****morphisms**closed under addition and under composition with arbitrary**morphisms****.**
Monads are used

**in****the**theory**of**pairs**of**adjoint__functors__**,**and they__generalize__closure operators on partially ordered sets**to**arbitrary categories**.**

functors and previous

In terms

**of**inverse image__functors__**the**condition**of**being a splitting means that**the**composition**of**inverse image__functors__corresponding**to**composable**morphisms**f**,**g**in**E equals**the**inverse image functor corresponding**to**f o g**.**In other words**,****the**compatibility isomorphisms c < sub > f**,**g </ sub >**of****the**__previous__section are all identities for a split**category****.**

functors and example

This is formally

**the**tensor-hom adjunction**,**and is an archetypal__example__**of**a**pair****of**adjoint__functors__**.**
The class

**of**Abelian categories is closed under several categorical constructions**,**for__example__**,****the****category****of**chain complexes**of**an Abelian**category****,**or**the****category****of**__functors__**from**a small**category****to**an Abelian**category**are Abelian as well**.**
** The earlier

__example__**of**G-sets**can**be seen as a special case**of**functor categories**:****every**group**can**be considered as a one-object**category****,**and G-sets are nothing but__functors__**from**this**category****to**Set
Hence

**,**given**the**information that**the**identity__functors__form an equivalence**of**categories**,****in**this__example__**one**still**can**choose between two natural isomorphisms for each direction**.**
For

__example__**,****in****the**above__example__**of**commutative rings**,****in**addition**to**those__functors__that delete some**of****the**operations**,**there are__functors__that forget some**of****the**axioms**.**
This

**can**sometimes be done by ad hoc means**:**for__example__**,****the**left derived__functors__**of**Tor**can**be defined using a flat resolution rather than a projective**one****,**but it takes some work**to**show that this is independent**of****the**resolution**.**
As an

__example__**,**for each topological space there is**the****category****of**vector bundles on**the**space**,**and for**every**continuous map**from**a topological space**X****to**another topological space**Y**is associated**the**pullback functor taking bundles on**Y****to**bundles on**X****.**Fibred categories formalise**the**system consisting**of**these categories and inverse image__functors__**.**

functors and any

It allows

**the**embedding**of**__any__**category**into a**category****of**__functors__**(**contravariant set-valued__functors__**)**defined on that**category****.**
* For

__any__**category****C**and object A**of****C****the****Hom**functor**Hom****(**A ,–)**:****C**→ Set preserves all limits**in****C****.**In particular**,****Hom**__functors__are continuous**.**
Moreover

**,**if both**C**and D are additive categories**(**i**.**e**.**preadditive categories with all finite biproducts ), then__any__**pair****of**adjoint__functors__between them are automatically additive**.**
If

**the**chain complex depends on**the**object**X****in**a covariant manner**(**meaning that__any__morphism**X**→**Y**induces a morphism**from****the**chain complex**of****X****to****the**chain complex**of****Y**), then**the**H < sub > n </ sub > are covariant__functors__**from****the****category**that**X**belongs**to**into**the****category****of**abelian groups**(**or modules ).
Most constructions that

**can**be carried out**in**D**can**also be carried out**in**D < sup >**C**</ sup > by performing them " componentwise ", separately for each object**in****C****.**For instance**,**if__any__two**objects****X**and**Y****in**D have a product**X**×**Y****,**then__any__two__functors__F and G**in**D < sup >**C**</ sup > have a product F × G**,**defined by**(**F × G )( c**)**
Moreover

**,**since__any__function between discrete or indiscrete spaces is continuous**,**both**of**these__functors__give full embeddings**of**Set into Top**.**
This is enough

**to**show that right derived__functors__**of**__any__left exact functor exist and are unique up**to**canonical isomorphism**.**
* The truncation

__functors__**,**or**in**fact for__any__n**,**which are obtained by translating**the**argument**of****the**original two__functors__**.**
Its relationship

**to****the**truncation__functors__is that they are defined so that for__any__complex A**,**for < math > i < 0 </ math > and is unchanged for ; likewise for ;**in**particular**,**is not independent**of**them**,**but**in**fact**.**
These properties carry over without change

**to**__any__t-structure**,****in**that if D is a t-category**,**then there exist truncation__functors__into its core**,****from**which we obtain a cohomology functor taking values**in****the**core**,**and**the**above properties are satisfied for both**.**
It

**can**be shown**(**see Grothendieck**(**1971**)**section 8**)**that**,**inversely**,**__any__collection**of**__functors__f < sup >*</ sup >: F < sub > S </ sub > → F < sub > T </ sub > together with isomorphisms c < sub > f**,**g </ sub > satisfying**the**compatibilities above**,**defines a cleaved**category****.**0.404 seconds.