This article is about coproducts in categories. For "coproduct" in the sense of comultiplication, see Coalgebra. For additional substances that result from the manufacturing of another product, see by-product.
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category.
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces...
a generalization of mathematical products Fibre product or pullback Coproduct or pushout Wick product of random variables Graph product Product (Brand...
the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This...
disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context...
zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects...
unital and associative. The unit is explicitly required to define the coproduct. Let V be a vector space over a field K. For any nonnegative integer k...
(and for representation theory in general), the coproduct must be linear. As a general rule, the coproduct in representation theory is reducible; the factors...
monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a...
coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is...
circles) gives a product map V ⊗ V → V {\displaystyle V\otimes V\to V} or a coproduct map V → V ⊗ V {\displaystyle V\to V\otimes V} , depending on how the boundary...
derived from oxygenation of isobutane, which affords t-butanol. This coproduct can be dehydrated to isobutene, converted to MTBE, an additive for gasoline...
case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as direct...
produced naturally on a large scale in volcanoes. It is manufactured as a coproduct in the production of phosphate fertilizers. The resulting hexafluorosilicic...
is dual to a coproduct defined on ⋀ ( V ) {\displaystyle {\textstyle \bigwedge }(V)} , giving the structure of a coalgebra. The coproduct is a linear function...
universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role...
compared to fish-fed salmon. Another possible alternative is a yeast-based coproduct of bioethanol production, proteinaceous fermentation biomass. Substituting...
amides into nitriles: P4O10 + RC(O)NH2 → P4O9(OH)2 + RCN The indicated coproduct P4O9(OH)2 is an idealized formula for undefined products resulting from...
component-wise. The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups. The coproduct of nonzero rings...
homomorphism. (A finite coproduct in the category of commutative algebras over a commutative ring is a tensor product of algebras. A coproduct in the category...
chloride gas is dissolved in water to produce hydrochloric acid as a useful coproduct of the reaction. Sodium bisulfate can be generated as a byproduct of the...
alcohol (typically methanol or ethanol) to produce biodiesel and an impure coproduct, glycerol. If the feedstock oil is used or has a high acid content, acid-catalyzed...
submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most...
Ozonolysis of oleic acid is an important route to azelaic acid. The coproduct is nonanoic acid: CH3(CH2)7CH=CH(CH2)7CO2H} + 4 O3 → HO2C(CH2)7CO2H} +...
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