Functor Co-cone (extended)

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 all areas within mathematics to which 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.

Definition

Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that
* associates each object $X$ in ''C'' to an object $F(X)$ in ''D'',
* associates each morphism $f \colon X \to Y$ in ''C'' to a morphism $F(f) \colon F(X) \to F(Y)$ in ''D'' such that the following two conditions hold:
** $F(\mathrm_) = \mathrm_\,\!$ for every object $X$ in ''C'',
** $F(g \circ f) = F(g) \circ F(f)$ for all morphisms $f \colon X \to Y\,\!$ and $g \colon Y\to Z$ in ''C''.

That is, functors must preserve identity morphisms and composition of morphisms.

Covariance and contravariance

There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor ''F'' from ''C'' to ''D'' as a mapping that
*associates each object $X$ in ''C'' with an object $F(X)$ in ''D'',
*associates each morphism $f \colon X\to Y$ in ''C'' with a morphism $F(f) \colon F(Y) \to F(X)$ in ''D'' such that the following two conditions hold:
**$F(\mathrm_X) = \mathrm_\,\!$ for every object $X$ in ''C'',
**$F(g \circ f) = F(f) \circ F(g)$ for all morphisms $f \colon X\to Y$ and $g \colon Y\to Z$ in ''C''.

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 $C^\mathrm$. Some authors prefer to write all expressions covariantly. That is, instead of saying $F \colon C\to D$ is a contravariant functor, they simply write $F \colon C^ \to D$ (or sometimes $F \colon C \to D^$) 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 $^ = \Lambda^i_j x^j$ for $\mathbf' = \boldsymbol\mathbf$ or $\omega'_i = \Lambda^j_i \omega_j$ for $\boldsymbol' = \boldsymbol\boldsymbol^\textsf.$ In this formalism it is observed that the coordinate transformation symbol $\Lambda^j_i$ (representing the matrix $\boldsymbol^\textsf$) acts on the basis vectors "in the same way" as on the "covector coordinates": $\mathbf_i = \Lambda^j_i\mathbf_j$—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: $\mathbf^i = \Lambda^i_j \mathbf^j$). 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.

Opposite functor

Every functor $F \colon C\to D$ induces the opposite functor $F^\mathrm \colon C^\mathrm\to D^\mathrm$, where $C^\mathrm$ and $D^\mathrm$ are the opposite categories to $C$ and $D$. By definition, $F^\mathrm$ maps objects and morphisms in the identical way as does $F$. Since $C^\mathrm$ does not coincide with $C$ as a category, and similarly for $D$, $F^\mathrm$ is distinguished from $F$. For example, when composing $F \colon C_0\to C_1$ with $G \colon C_1^\mathrm\to C_2$, one should use either $G\circ F^\mathrm$ or $G^\mathrm\circ F$. Note that, following the property of opposite category, $\left(F^\mathrm\right)^\mathrm = F$.

Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type . It can be seen as a functor in ''two'' arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.

A multifunctor is a generalization of the functor concept to ''n'' variables. So, for example, a bifunctor is a multifunctor with .

Properties

Two important consequences of the functor axioms are:
* ''F'' transforms each commutative diagram in ''C'' into a commutative diagram in ''D'';
* if ''f'' is an isomorphism in ''C'', then ''F''(''f'') is an isomorphism in ''D''.

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 from ''A'' to ''C''. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.

A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.

Examples

; 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