Diagonal Map
Get Diagonal Map essential facts below. View Videos or join the Diagonal Map discussion. Add Diagonal Map to your PopFlock.com topic list for future reference or share this resource on social media.
Diagonal Map

In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the diagonal morphism



where is the canonical projection morphism to the -th component. The existence of this morphism is a consequence of the universal property that characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.

For concrete categories, the diagonal morphism can be simply described by its action on elements of the object . Namely, , the ordered pair formed from . The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism on the real line is given by the line that is the graph of the equation . The diagonal morphism into the infinite product may provide an injection into the space of sequences valued in ; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions that the image of the diagonal map will fail to satisfy.

See also


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



Music Scenes