Examples
In the category of sets, the product (in the category theoretic sense) is the cartesian product. Given a family of sets Xi the product is defined as
with the canonical projections
Given any set Y with a family of functions
the universal arrow f is defined as
Other examples:
- In the category of topological spaces, the product is the space whose underlying set is the cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous.
- In the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication.
- In the category of groups, the product is the direct product of groups given by the cartesian product with multiplication defined componentwise.
- In the category of relations (Rel), the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets (Set) is a subcategory of Rel.)
- In the category of algebraic varieties, the categorical product is given by the Segre embedding.
- In the category of semi-abelian monoids, the categorical product is given by the history monoid.
- A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).
Read more about this topic: Product (category Theory)
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