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The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum; namely, we mod out by the subgroup consisting of pairs (f(z), −g(z)). Thus we have "glued" along the images of Z under f and g.
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The typical diagram of the definition of a universal morphism. In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them.
The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the pushout.
Universal property of the product Whether a product exists may depend on C {\displaystyle C} or on X 1 {\displaystyle X_{1}} and X 2 . {\displaystyle X_{2}.} If it does exist, it is unique up to canonical isomorphism , because of the universal property, so one may speak of the product.
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Characteristic property of disjoint unions. This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff f i = f o φ i is continuous for all i in I. In addition to being continuous, the canonical injections φ i : X i → X are ...