Sum 


generalisation  a kind of colimit  
set example  disjoint union {a,b,c}+{x,y}= 

group  free product 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. 

Grp (abelian)  direct sum the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero) 

vector space  direct sum  
poset  least upper bound join 

base topological space  wedge  
POS 

least upper bounds (joins) 
Rng  
Top  disjoint unions with their disjoint union topologies  
Grf  
category 
Sum
When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.
Product
Products for groups are discussed on this page.