Element (category theory)

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points. In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points. Suppose C is any category and A, T are two objects of C. A T-valued point of A is simply an arrow p : T → A {displaystyle pcolon T o A} . The set of all T-valued points of A varies functorially with T, giving rise to the 'functor of points' of A; according to the Yoneda lemma, this completely determines A as an object of C. Many properties of morphisms can be restated in terms of points. For example, a map f {displaystyle f} is said to be a monomorphism if Suppose f : B → C {displaystyle fcolon B o C} and g , h : A → B {displaystyle g,hcolon A o B} in C. Then g and h are A-valued points of B, and therefore monomorphism is equivalent to the more familiar statement Some care is necessary. f is an epimorphism if the dual condition holds: