Functor

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word.

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word. F(\mathrm{id}_{X}) = \mathrm{id}_{F(X)}\,\! for every object X in C , F(g \circ f) = F(g) \circ F(f) for all morphisms f: X \to Y\,\! and g:Y\to Z in C . That is, functors must preserve identity morphisms and composition of morphisms. F(\mathrm{id}_X) = \mathrm{id}_{F(X)}\,\! for every object X in C , F(g \circ f) = F(f) \circ F(g) for all morphisms f: X\to Y and g: Y\to Z in C . Variance of functors (composite): The composite of two functors of the same variance: \mathrm{Covariant} \circ \mathrm{Covariant} \to \mathrm{Covariant} \mathrm{Contravariant} \circ \mathrm{Contravariant} \to \mathrm{Covariant} The composite of two functors of opposite variance: \mathrm{Covariant} \circ \mathrm{Contravariant} \to \mathrm{Contravariant} \mathrm{Contravariant} \circ \mathrm{Covariant} \to \mathrm{Contravariant} We see that (\{\text{Covariant},\text{Contravariant}\},\circ)\cong\mathbb Z/2\mathbb Z . Note that contravariant functors reverse the direction of composition. Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the opposite category C^\mathrm{op} . Some authors prefer to write all expressions covariantly. That is, instead of saying F: C\to D is a contravariant functor, they simply write F: C^{\mathrm{op}} \to D (or sometimes F: C \to D^{\mathrm{op}} ) and call it a functor. Contravariant functors are also occasionally called cofunctors. There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections \Gamma(TM) of a tangent bundle TM —as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections \Gamma\mathord\left(T^*M\right) of a cotangent bundle T^*M —as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as {x'}^{\,i} = \Lambda^i_j x^j for \mathbf{x}' = \boldsymbol{\Lambda}\mathbf{x} , or \omega'_i = \Lambda^j_i \omega_j for \boldsymbol{\omega}' = \boldsymbol{\omega}\boldsymbol{\Lambda}^\textsf{T}. In this formalism it is observed that the coordinate transformation symbol \Lambda^j_i (representing the matrix \boldsymbol{\Lambda}^\textsf{T} ) acts on the "covector coordinates" "in the same way" as on the basis vectors: \mathbf{e}_i = \Lambda^j_i\mathbf{e}_j —whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: \mathbf{e}^i = \Lambda^i_j \mathbf{e}^j ). This terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also covariance and contravariance of vectors. By definition, F^\mathrm{op} maps objects and morphisms in the identical way as does F . Since C^\mathrm{op} does not coincide with C as a category, and similarly for D , F^\mathrm{op} is distinguished from F . For example, when composing F: C_0\to C_1 with G: C_1^\mathrm{op}\to C_2 , one should use either G\circ F^\mathrm{op} or G^\mathrm{op}\circ F . Note that \left(F^\mathrm{op}\right)^\mathrm{op} = F . A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with n=2 . 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) , in this case \{\} \mapsto f(\{\}) = \{\} . For the other values, \begin{align} \{0\} \mapsto f(\{0\}) &= \{f(0)\} = \{\{\}\}, \\ \{1\} \mapsto f(\{1\}) &= \{f(1)\} = \{X\}, \\ \{0,1\} \mapsto f(\{0,1\}) &= \{f(0), f(1)\} = \{\{\}, X\}. \end{align} 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 x 0 is a point in X. A morphism from to is given by a continuous map with . To every topological space X with distinguished point x 0 , one can define the fundamental group based at x 0 , denoted . This is the group of homotopy classes of loops based at x 0 , with the group operation of concatenation. If is a morphism of pointed spaces, then every loop in X with base point x 0 can be composed with f to yield a loop in Y with base point y 0 . 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, Vect K , 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.

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