Inferring colimits

["David B. Benson" <dbenson@eecs.wsu.edu>]

Dear category theorists,

I have a question to which answers will be most appreciated.

To set the stage for the question, consider a category A for which
diagrams D':G'-->A and D'':G''-->A have colimits, colim D' and colim D'',
respectively.  Suppose the sum colim(D')+colim(D'') exists in A.
Then the obvious diagram [D',D'']:G'+G''-->A has a colimit, the sum
mentioned just above, and irrespective of whether any other sums
may exist in A.

So from the existence of some colimits, the existence of others may
be inferred.

Definition:
Relative to a base category A,
for each collection K of (small) diagrams on A with colimits,
the collection of all inferable (small) diagrams with colimits
is said to be a <<repletion>> of K.

Example:
For every nonempty category C,
and every (small) category G with terminal object,
every diagram D:G-->C is in the repletion of the empty collection,
hence in the repletion of every collection of diagrams on C with colimits.

Now for the question.  Has there been any systematic study of what
I have just defined as repletions?  If not, are there in any case some papers
I should consider?

Thank you very much!

Season's Greetings,
David

Post Script (in the traditional sense):

Writers of textbooks in category theory may wish to consider
including the following as an exercise --

For all small categories C and all functors F:C-->Sets, the left Kan
extension of F along Id:C-->C is (isomorphic to) F.