For a category E with colimits, you can consider those objects x for which the representable E(x,-):E->Set preserves colimits. These have been given various names, including atomic, small-projective, and Cauchy. If E is a presheaf category then these are the retracts of representable functors. They were used by Bunge to characterize presheaf categories, and as far as I know this is where they first appeared. Among many other sources, you could also look at Kelly’s book on enriched category theory, Lawvere’s paper on generalized metric spaces, and Street’s work on absolute colimits. Best, Steve.
On 12 Jun 2024, at 6:52 pm, jschroed TV <jschroedtv@gmail.com> wrote:
Dear all,
I had a question regarding whether there is a known analogy of "join irreducibility in a complete join semi lattice" for colimit "irreducible elements" of Cocomplete categories. It is somewhat clear that at least for Presheaf categories one would want these to coincide with the representable functors. I have a few (naive) ideas as to what this could look like, but I wanted to know if this has been studied somewhere before.
Best, Quentin