In category theory, for any object a in any category C where the product a×a exists, there exists the diagonal morphism In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

δ_a: a → a×a,

satisfying

π_kδ_a = id_a for k=1,2, where

π_k

is the canonical projection morphism to the k-th component. The existence of this morphism is a consequence of the universal property which 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. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: There exists a morphism such that f = hg. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object a. Namely, δ_a(x) = (x,x), the ordered pair formed from x. The reason for the name is that the graph of such a diagonal morphism is diagonal (whenever it makes sense), for example the graph of the diagonal morphism R → R² on the real line is given by the line which is a graph of the equation y=x. The diagonal morphism into the infinite product X^∞ may provide an injection into the space of sequences valued in X; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions which the image of the diagonal map will fail to satisfy. In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... In mathematics, the real line is simply the set of real numbers. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... An injective function. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

In particular, the category of categories has products, and so one finds the diagonal functor Δ: C → C×C given by Δ(a) = (a,a), the ordered pair for any object a in C. This functor can be employed to give a succinct alternate description of the product of objects within the category C: a product a×b is a universal arrow from Δ to (a,b). The arrow comprises the projection maps. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

More generally, in any functor category C^J (here J should be thought of as a small index category), for each object a in C, there is a constant functor Δ_a which maps each object j in J to a Δ_a(j) = a and maps each morphism j → k in J to the identity morphism on a. The diagonal functor Δ: C → C^J assigns to each object of C the constant functor at that object (Δ(a) = Δ_a ∈ C^J), and to each morphism f: a → b in C the obvious natural transformation in C^J (given by η_j = f). In the case that J is a discrete category with two objects, the diagonal functor C → C×C is recovered. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Diagonal functors provide a way to define limits and colimits of functors. The limit of any functor F: J → C is a universal arrow from Δ to F and a colimit is a universal arrow F → Δ. If every functor from J to C has a limit (which will be the case if C is complete), then the operation of taking limits is itself a functor from C^J to C. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor from the discrete two object category is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ... In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...