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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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Representable and functors
Category: Representable functors
Category: Representable functors
functors and We
We note that an inverse system in a category C admits an alternative description in terms of functors.
We note that a direct system in a category 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.
Hence, a natural transformation can be considered to be a " morphism of functors ".
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.
In a pre-abelian category, exact functors can be described in particularly simple terms.
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.
In abstract algebra, one uses homology to define derived functors, for example the Tor functors.
** 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, there are several forgetful functors from the category of commutative rings.
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.
For example, the Grothendieck spectral sequence of a composition of two functors
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.
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