Subject: categories: Inferring colimits

From: "David B. Benson" <[email-masked]>
Date: Tue, 15 Dec 1998 15:31:54 -0800

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.

---

From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Wed, 16 Dec 1998 10:55:59 +0000 (GMT)

> I have just defined as repletions?  If not, are there in any case some papers

I think the question as posed by David (relative to a particular category,
in which some but not all diagrams of a particular shape may have colimits)
is a very hard one. A lot is known about inferring the existence of
particular types of colimits, in arbitrary categories, from the existence
of other types: see the paper by Albert and Kelly "The closure of a class
of colimits" in JPAA 51 (1988), and subsequent references of which Max will
no doubt remind us. But when you work in a particular category, there are
so many ways of "mutilating" the category by omitting particular objects
which are required as the vertices of colimit cones, that I suspect there
is almost nothing you can say in general.

Incidentally, a similar comment applies to David's paper "Multilinearity
of sketches" in TAC 3 (1997): I tried to make this point in my review
(MR 98j:18006).

Peter Johnstone