# coproduct

## English

Commutative diagram showing the universal property of coproducts of two objects.

### Noun

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Wikipedia

coproduct (plural coproducts)

1. Any of multiple products that are produced at the same time, or by the same process
2. (mathematics) A structure constructed from several similar structures, equipped with appropriate inclusion maps, which generalises the disjoint union of sets, of topological spaces and the free product of groups
Considering the set of 15 axioms given in the "Predicate Calculus" article of the Encyclopedia of Mathematics, axioms 12 and 14 may be construed as stating that $\forall x \phi$ is the categorical product of all of the instantiations $\phi (x | t)$ of φ. Likewise, dually, axioms 13 and 15 may be construed as stating that $\exists x \phi$ is the categorical coproduct of all of the instantiations of φ. (Axiom 12 would give the projection morphisms, Axiom 13 the inclusion morphisms, and Axioms 14 and 15 the universal properties.)