element

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. Definition Suppose C is any category and A, T are two objects of C. A T-valued point of A is simply a morphism . 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. Properties of morphisms Many properties of morphisms can be restated in terms of points. For example, a map is said to be a monomorphism if For all maps , , if then . Suppose and in C. Then g and h are A-valued points of B, and therefore monomorphism is equivalent to the more familiar statement f is a monomorphism if it is an injective function on points of B. Some care is necessary. f is an epimorphism if the dual condition holds: For all maps g, h (of some suitable type), implies . In set theory, the term "epimorphism" is synonymous with "surjection", i.e. Every point of C is the image, under f, of some point of B. This is clearly not the translation of the first statement into the language of points, and in fact these statements are not equivalent in general. However, in some contexts, such as abelian categories, "monomorphism" and "epimorphism" are backed by sufficiently strong conditions that in fact they do allow such a reinterpretation on points. Similarly, categorical constructions such as the product have pointed analogues. Recall that if A, B are two objects of C, their product A × B is an object such that There exist maps , and for any T and maps , there exists a unique map such that and . In this definition, f and g are T-valued points of A and B, respectively, while h is a T-valued point of A × B.