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A functor F: 1 → Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1 < sub > S </ sub > on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1 < sub > T </ sub > on T. Notice that 1 < sub > T </ sub > is the restriction of 1 < sub > S </ sub > to T. Consequently, subfunctors of F correspond to subsets of S.
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functor and F
A ( covariant ) functor F from a category C to a category D, written, consists of:
A functor F from C to D is a mapping that
We then define a contravariant functor F from C to D as a mapping that
The free functor F: Set → Grp sends every set X to the free group generated by X.
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.
Another way to describe G-sets is the functor category, where is the groupoid ( category ) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X = F and for every g in G ( i. e. for every morphism in ) induces a bijection F < sub > g </ sub >: X → X.
The categorical structure of the functor F assures us that F defines a G-action on the set X.
The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group
Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.
functor and 1
Identity functor in category C, written 1 < sub > C </ sub > or id < sub > C </ sub >, maps an object to itself and a morphism to itself.
If f: A < sub > 1 </ sub > → A < sub > 2 </ sub > and g: B < sub > 1 </ sub > → B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) → Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.
Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions which imitated the cohomology functor H < sup > 1 </ sup >.
In the case where C satisfies Grothendieck's axiom ( AB4 *), Jan-Erik Roos generalized the functor lim < sup > 1 </ sup > on Ab < sup > I </ sup > to series of functors lim < sup > n </ sup > such that
If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.
Note however that a right inverse of F ( i. e. a functor G such that FG is naturally isomorphic to 1 < sub > D </ sub >) need not be a right ( or left ) adjoint of F. Adjoints generalize two-sided inverses.
# For technical reasons, the category Ban < sub > 1 </ sub > of Banach spaces and linear contractions is often equipped not with the " obvious " forgetful functor but the functor U < sub > 1 </ sub >: Ban < sub > 1 </ sub > → Set which maps a Banach space to its ( closed ) unit ball.
In category theory, given any family P < sub > n </ sub > of invertible n-by-n matrices defining a similarity transformation for all rectangular matrices sending the m-by-n matrix A into P < sub > m </ sub >< sup >− 1 </ sup > AP < sub > n </ sub >, the family defines a functor that is an automorphism of the category of all matrices, having as objects the natural numbers and morphisms from n to m the m-by-n matrices composed via matrix multiplication.
A space M is a fine moduli space for the functor F if M represents F, i. e., the functor of points Hom (−, M ) is naturally isomorphic to F. This implies that M carries a universal family ; this family is the family on M corresponding to the identity map 1 < sub > M </ sub > ∈ Hom ( M, M ).
If we define Ω ( f ) = f < sup > − 1 </ sup > then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms.
The idea of derived functor here is that the sheaf of sections doesn't respect exact sequences as it is not right exact ; according to general principles of homological algebra there will be a sequence of functors H < sup > i </ sup > for i = 0, 1, ... that represent the ' compensations ' that must be made in order to restore some measure of exactness ( long exact sequences arising from short ones ).
But it turns out that ( if A is " nice " enough ) there is one canonical way of doing so, given by the right derived functors of F. For every i ≥ 1, there is a functor R < sup > i </ sup > F: A → B, and the above sequence continues like so: 0 → F ( A ) → F ( B ) → F ( C ) → R < sup > 1 </ sup > F ( A ) → R < sup > 1 </ sup > F ( B ) → R < sup > 1 </ sup > F ( C ) → R < sup > 2 </ sup > F ( A ) → R < sup > 2 </ sup > F ( B ) → ....
functor and →
For example, the Hom functor is of the type C < sup > op </ sup > × C → Set.
Constant functor: The functor C → D which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X.
Limit functor: For a fixed index category J, if every functor J → C has a limit ( for instance if C is complete ), then the limit functor C < sup > J </ sup >→ C assigns to each functor its limit.
Power sets: The power set functor P: Set → Set maps each set to its power set and each function to the map which sends to its image.
Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.
Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.
Forgetful functors: The functor U: Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.
Another example is the functor Rng → Ab which maps a ring to its underlying additive abelian group.
This is a functor which is contravariant in the first and covariant in the second argument, i. e. it is a functor Ab < sup > op </ sup > × Ab → Ab ( where Ab denotes the category of abelian groups with group homomorphisms ).
This defines a functor to Set which is contravariant in the first argument and covariant in the second, i. e. it is a functor C < sup > op </ sup > × C → Set.
The collection of all functors C → D form the objects of a category: the functor category.
It turns out that this functor cannot distinguish maps which are homotopic relative to the base point: if f and g: X → Y are continuous maps with f ( x < sub > 0 </ sub >)
functor and Set
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
Set is a category we understand well, and a functor of C into Set can be seen as a " representation " of C in terms of known structures.
If C is a locally small category ( i. e. the hom-sets are actual sets and not proper classes ), then each object A of C gives rise to a natural functor to Set called a hom-functor.
Let F be an arbitrary functor from C to Set.
Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that
In this way, Yoneda's Lemma provides a complete classification of all natural transformations from the functor Hom ( A ,-) to an arbitrary functor F: C → Set.
An important special case of Yoneda's lemma is when the functor F from C to Set is another hom-functor h < sup > B </ sup >.
Mapping each object A in C to its associated hom-functor h < sup > A </ sup > = Hom ( A ,–) and each morphism f: B → A to the corresponding natural transformation Hom ( f ,–) determines a contravariant functor h < sup >–</ sup > from C to Set < sup > C </ sup >, the functor category of all ( covariant ) functors from C to Set.
The meaning of Yoneda's lemma in this setting is that the functor h < sup >–</ sup > is fully faithful, and therefore gives an embedding of C < sup > op </ sup > in the category of functors to Set.
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.
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