|generalisation||a kind of limit|
|group||the product is given by the cartesian product with multiplication defined componentwise.|
|Grp (abelian)||direct sum|
|vector space||direct sum|
|poset||greatest lower bound
|base topological space|
greatest lower bounds (meets)
|Top||the space whose underlying set is the cartesian product and which carries the product topology|
tensor products are not categorial products.
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.
Products for groups are discussed on this page.