;
Diagram: For categories
C and
J, a diagram of type
J in
C is a covariant functor D \colon J\to C. ;
(Category theoretical) presheaf:For categories
C and
J, a
J-presheaf on
C is a contravariant functor D \colon C\to J.In the special case when
J is
Set, the category of sets and functions,
D is called a
presheaf on
C. ; Presheaves (over a topological space): If
X is a
topological space, then the
open sets in
X form a
partially ordered set Open(
X) under inclusion. Like every partially ordered set, Open(
X) forms a small category by adding a single arrow if and only if U \subseteq V. Contravariant functors on Open(
X) are called
presheaves on
X. For instance, by assigning to every open set
U the
associative algebra of real-valued continuous functions on
U, one obtains a presheaf of algebras on
X. ; Constant functor: The functor which maps every object of
C to a fixed object
X in
D and every morphism in
C to the identity morphism on
X. Such a functor is called a
constant or
selection functor. ; : A functor that maps a category to that same category; e.g.,
polynomial functor. ; : In category
C, written 1
C or id
C, maps an object to itself and a morphism to itself. The identity functor is an endofunctor. ; Diagonal functor: The
diagonal functor is defined as the functor from
D to the functor category
DC which sends each object in
D to the constant functor at that object. ; Limit functor: For a fixed
index category J, if every functor has a
limit (for instance if
C is complete), then the limit functor assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the
right-adjoint to the
diagonal functor and invoking the
Freyd adjoint functor theorem. This requires a suitable version of the
axiom of choice. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant). ; Power sets functor: The power set functor maps each set to its
power set and each function f \colon X \to Y to the map which sends U \in \mathcal{P}(X) to its image f(U) \in \mathcal{P}(Y). One can also consider the
contravariant power set functor which sends f \colon X \to Y to the map which sends V \subseteq Y to its
inverse image f^{-1}(V) \subseteq X. For example, if X = \{0,1\} then F(X) = \mathcal{P}(X) = \{\{\}, \{0\}, \{1\}, X\}. Suppose f(0) = \{\} and f(1) = X. Then F(f) is the function which sends any subset U of X to its image f(U), which in this case means \{\} \mapsto f(\{\}) = \{\}, where \mapsto denotes the mapping under F(f), so this could also be written as (F(f))(\{\})= \{\}. For the other values, \{0\} \mapsto f(\{0\}) = \{f(0)\} = \{\{\}\},\ \{1\} \mapsto f(\{1\}) = \{f(1)\} = \{X\},\ \{0,1\} \mapsto f(\{0,1\}) = \{f(0), f(1)\} = \{\{\}, X\}. Note that f(\{0, 1\}) consequently generates the
trivial topology on X. Also note that although the function f in this example mapped to the power set of X, that need not be the case in general. ; : The map which assigns to every
vector space its
dual space and to every
linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed
field to itself. ; Fundamental group: Consider the category of
pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where
X is a topological space and
x0 is a point in
X. A morphism from to is given by a
continuous map with . To every topological space
X with distinguished point
x0, one can define the
fundamental group based at
x0, denoted . This is the
group of
homotopy classes of loops based at
x0, with the group operation of concatenation. If is a morphism of
pointed spaces, then every loop in
X with base point
x0 can be composed with
f to yield a loop in
Y with base point
y0. This operation is compatible with the homotopy
equivalence relation and the composition of loops, and we get a
group homomorphism from to . We thus obtain a functor from the category of pointed topological spaces to the
category of groups. In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the
fundamental groupoid instead of the fundamental group, and this construction is functorial. ; Algebra of continuous functions: A contravariant functor from the category of
topological spaces (with continuous maps as morphisms) to the category of real
associative algebras is given by assigning to every topological space
X the algebra C(
X) of all real-valued continuous functions on that space. Every continuous map induces an
algebra homomorphism by the rule for every
φ in C(
Y). ; Tangent and cotangent bundles: The map which sends every
differentiable manifold to its
tangent bundle and every
smooth map to its
derivative is a covariant functor from the category of differentiable manifolds to the category of
vector bundles. Doing this constructions pointwise gives the
tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise,
cotangent space is a contravariant functor, essentially the composition of the tangent space with the
dual space above. ; 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. Likewise, a functor from
G to the
category of vector spaces,
VectK, is a
linear representation of
G. In general, a functor 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. ; Lie algebras: Assigning to every real (complex)
Lie group its real (complex)
Lie algebra defines a functor. ; Tensor products: If
C denotes the category of vector spaces over a fixed field, with
linear maps as morphisms, then the
tensor product V \otimes W defines a functor which is covariant in both arguments. ; Forgetful functors: The functor which maps a
group to its underlying set and a
group homomorphism to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed
forgetful functors. Another example is the functor which maps a
ring to its underlying additive
abelian group. Morphisms in
Rng (
ring homomorphisms) become morphisms in
Ab (abelian group homomorphisms). ; Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor sends every set
X to the
free group generated by
X. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See
free object. ; Homomorphism groups: To every pair
A,
B of
abelian groups one can assign the abelian group Hom(
A,
B) consisting of all
group homomorphisms from
A to
B. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor (where
Ab denotes the
category of abelian groups with group homomorphisms). If and are morphisms in
Ab, then the group homomorphism : is given by . See
Hom functor. ; 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 of morphisms from
X to
Y. This defines a functor to
Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor . If and are morphisms in
C, then the map is given by . Functors like these are called
representable functors. An important goal in many settings is to determine whether a given functor is representable. == Relation to other categorical concepts ==