Notion of colimit irreducibility

Notion of colimit irreducibility

[jschroed TV]

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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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Re: Notion of colimit irreducibility

[Salvatore TRINGALI]

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Dear Quentin,

An element $x$ in a (multiplicatively written) Dedekind-finite monoid $H$ is a non-unit $a \in H$ such that $a \ne xy$ for all non-units $x, y \in H$ such that $x$ and $y$ are proper divisors of $a$ in $H$ (meaning that each of the two-sided principal ideals generated by $x$ and $y$ properly contains the two-sided principal ideal generated by $a$). Considering that every commutative monoid is Dedekind-finite:

  *   A bounded semilattice with identity element $e$ is a commutative monoid whose only unit is $e$, hence a join-irreducible element in a join-semilattice with a bottom element is precisely an irreducible element as per the above definition.
  *   Regarding "colimit-irreducible elements" in a category $C$ with all finite colimits, the (equivalence) classes made up of the isomorphic objects of  $C$ form a commutative monoid $V(C)$ with a trivial group of units when endowed with the binary operation $([A], [B]) \mapsto [A \coprod B]$, where $[ \cdot ]$ denotes the isomorphism class of an object. An object of $C$ is then colimit-irreducible (in the sense that it is isomorphic neither to the initial object nor to the coproduct of two non-initial objects) if and only if its isomorphism class is irreducible in the monoid $V(C)$.

For further details, you may want to take a look at Sect. 1, Definition 3.6, Remark 3.7(2), and Sect. 4.3 of [https://arxiv.org/abs/2102.01598<https://url.au.m.mimecastprotect.com/s/cCPpCxngGkf7Ajy3I8Y1vp?domain=arxiv.org>].

Best,
Salvo

Il giorno mer 12 giu 2024 alle ore 11:09 jschroed TV <jschroedtv@gmail.com<mailto:jschroedtv@gmail.com>> ha scritto:
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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Re: Notion of colimit irreducibility

[Steve Lack]

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 

Re: Notion of colimit irreducibility

[Enrico Vitale]

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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