Re: Notion of colimit irreducibility

[Salvatore TRINGALI <salvo.tringali@gmail.com>]

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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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