Re: Notion of colimit irreducibility

[Enrico Vitale <enrico.vitale@uclouvain.be>]

Dear Quentin,
Under the name of absolutely presentable objects, these objects also appear in the book by Adamek, Rosicky and myself on algebraic theories. Bunge characterization is mentioned at the end of Chapter 6.
Best regards,
Enrico

> Le 13 juin 2024 à 00:43, Steve Lack <steve.lack@mq.edu.au> a écrit :
>
> 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
>



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