Combining algebraic objects

We can define combinations of algebraic objects to construct new, “bigger” objects in the same category. We will use the concepts of categorical products and coproducts (AKA sums) in category theory to organize our presentation. While we will not go into exact definitions here, a categorical product can be thought of as the “most general” object with morphisms to its constituents, while a categorical coproduct can be thought of as the “most general” object with morphisms in the opposite direction, from the constituent objects to their coproduct. In certain categories, the product and coproduct of two objects coincide, in which case they are both called the biproduct (AKA direct sum). Even in these categories, however, the product and coproduct are distinct in the case of an infinite number of factors. Note that the common meaning of “direct sum” is not equivalent to the categorical direct sum (biproduct) in category theory, as we see below.

Product Coproduct
Sets Cartesian product \({A\times B}\) Disjoint union \({A\cup_{d}B}\)
Groups Direct product \({G\times H}\) Free product \({G*H}\)
Abelian groups Direct product \({G\times H}\) Direct sum \({G\oplus H}\)
Vector spaces Direct product \({V\times W}\) Direct sum \({V\oplus W}\)
Commutative rings with unity Direct product \({R\times S}\) Tensor product \({R\otimes S}\)

Notes: Coproducts in algebras and other categories can become quite complicated.

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