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Concept
Functor
Summary
In mathematics, specifically , a functor is a mapping between . 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 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.
Let C and D be . A functor F from C to D is a mapping that
associates each object in C to an object in D,
associates each morphism in C to a morphism in D such that the following two conditions hold:
for every object in C,
for all morphisms and in C.
That is, functors must preserve identity morphisms and composition of morphisms.
Covariance and contravariance (computer science)
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 in C with an object in D,
associates each morphism in C with a morphism in D such that the following two conditions hold:
for every object in C,
for all morphisms and 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 . Some authors prefer to write all expressions covariantly. That is, instead of saying is a contravariant functor, they simply write (or sometimes ) and call it a functor.
Contravariant functors are also occasionally called cofunctors.
There is a convention which refers to "vectors"—i.e.
Official source
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Functor
In mathematics, specifically , a functor is a mapping between . 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 is applied.
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a ) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the , whose objects are sets and whose arrows are functions. is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.
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In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
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