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charset="utf-8" Content-Transfer-Encoding: quoted-printable Dear Quentin, An element $x$ in a (multiplicatively written) Dedekind-finite monoid $H$ i= s 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 contain= s the two-sided principal ideal generated by $a$). Considering that every c= ommutative monoid is Dedekind-finite: * A bounded semilattice with identity element $e$ is a commutative mono= id whose only unit is $e$, hence a join-irreducible element in a join-semil= attice with a bottom element is precisely an irreducible element as per the= above definition. * Regarding "colimit-irreducible elements" in a category $C$ with all f= inite colimits, the (equivalence) classes made up of the isomorphic objects= of $C$ form a commutative monoid $V(C)$ with a trivial group of units whe= n endowed with the binary operation $([A], [B]) \mapsto [A \coprod B]$, whe= re $[ \cdot ]$ 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]= . Best, Salvo Il giorno mer 12 giu 2024 alle ore 11:09 jschroed TV > ha scritto: Dear all, I had a question regarding whether there is a known analogy of "join irredu= cibility 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 functor= s. 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 You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group = | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --0000000000001fcada061aae3400 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
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 $x, y \i= n H$ 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 Dedek= ind-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-semilat= tice with a bottom element is precisely an irreducible element as per the a= bove definition.
  • Regarding "colimit-irreducible elements"= in a category $C$ with all finite colimits, the (equivalence) class= es 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 operatio= n $([A], [B]) \mapsto [A \coprod B]$, where $[ \cdot ]$ denotes the isomorph= ism class of an object. An object of $C$ is then colimit-irreducible (in the sense that it is isomorphic neither to t= he initial object nor to the coproduct of two non-initial objects) if and o= nly if its isomorphism class is irreducible in the monoid $V(C)$.
For further details, you may want to take a look at Sect. 1, Definitio= n 3.6, Remark 3.7(2), and Sect. 4.3 of [https://arxiv.org/abs/2102.01598].

Best,
Salvo

Il giorno mer 12 giu 2024 alle ore 11= :09 jschroed TV <jschroedtv@gmail.com> ha scritto:
Dear all,

I had a question regarding whether there is a known analogy of "j= oin irreducibility in a complete join semi lattice" for colimit "= irreducible elements" of Cocomplete categories. It is somewhat clear t= hat at least for Presheaf categories one would want these to coincide with the representable functors. I have a few (naive) id= eas 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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View&n= bsp;group files = ;  |   Leave group   |   <= /span>Learn more about Microsoft 365 Groups
 
 
 
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