|a kind of colimit
the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.
the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero)
|least upper bound
|base topological space
least upper bounds (joins)
|disjoint unions with their disjoint union topologies
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.