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Stream: learning: questions

Topic: monoidal coseparator?

Paolo Perrone (Apr 11 2024 at 09:25):

Hello all.
A coseparator or cogenerator in a category is an object $C$ which "has enough resolution to tell things apart".
In rigor, for all objects $X$ and $Y$, given two morphisms $f,g:X\to Y$, if for all morphisms $c:Y\to C$ we have that $c\circ f=c\circ g$, then $f=g$.

I'm now wondering if anyone has come across a sort of stronger, "monoidal" version of it.
That is, in a monoidal category, a monoid object $(C,m,e)$ such that for all objects $X,Y_1,Y_2$, given two morphisms $f,g:X\to Y_1\otimes Y_2$, if for all morphisms $h_1:Y_1\to C$ and $h_2:Y_2\to C$ we have that $m\circ (h_1\otimes h_2)\circ f=m\circ (h_1\otimes h_2)\circ g$, as morphisms $X\to C$, then $f=g$.
(And similarly for higher arity, using the n-ary multiplication.)

Has anyone seen anything like this? I believe that an example is the two-element set in $(\mathrm{Set},\times,1)$.

Paolo Perrone (Apr 11 2024 at 10:46):

A coseparator can be characterized by the fact that the corresponding presheaf is faithful (see the link above). I think the structure I'm considering can be characterized by the fact that the corresponding presheaf, which will be lax-monoidal, hence a multifunctor, it is a faithful multifunctor.

Paolo Perrone (Apr 11 2024 at 10:46):

Still, I wonder if anyone has studied these things before.

Matteo Capucci (he/him) (Apr 11 2024 at 13:01):

Paolo Perrone said:

A coseparator can be characterized by the fact that the corresponding presheaf is faithful (see the link above). I think the structure I'm considering can be characterized by the fact that the corresponding presheaf, which will be lax-monoidal, hence a multifunctor, it is a faithful multifunctor.

That's a neat characterization!

Matteo Capucci (he/him) (Apr 11 2024 at 13:01):

It's a multiplexing coseparator :star_struck:

Paolo Perrone (Apr 11 2024 at 13:11):

Yes, in a certain sense.
(Of course, morphisms into $Y_1\otimes Y_2$ could also be told apart by a traditional coseparator, just using a morphism $Y_1\otimes Y_2\to C$ which may not involve the multiplication. So this is about the particular way of telling them apart.)

Eric M Downes (Apr 11 2024 at 14:35):

It sounds like you are asking
"What monoidal categories have a coseparator that is a monoid object?"
Does that sound faithful to your question?

[Edit: if the category $\sf C=Mon$ Eckmann-Hilton arguments work, but fails for arbitrary monoidal categories, so the rest is also wrong.]
I think this is equivalent (Eckmann-Hilton) to "What coseparators are also commutative monoids?"...

Recalling that a semilattice is an idempotent commutative monoid, the subobject classifier of a topos with its heyting algebra sounds like an example of this kind of thing... except you probably already know that, especially as the coseparator article mentions topoi. If so, it might help to explicitly state which topos axioms you want to avoid?

Also, while I am unfamiliar with coseparators, it sounds like every arrow into a coseparator is monic, and that coseparators involve faithful functors to $\sf Set$... so I think we might further refine the question to "What objects are adjoint to cancellative commutative monoids in $\sf Set$?"

Cancellative monoids (like $\Bbb N$) are the kind that can be losslessly extended to a group by adjoining inverses to your monoid object $(C,m,e)\hookrightarrow(C',m',e')$ so that $m,e$ are sections of $m',e'$ and there is an involution $\zeta:C'\to C'$ with $m'\circ(\zeta\times id_{C'})\circ\Delta=e'$. So... maybe skip all of that and just buy a group-action object?! :) Or look for groups with submonoids; these will be infinite if the submonoid is not a subgroup.

(Or at least it may be clarifying to explain exactly how I'm being sloppy and a group action with a contravariant functor back out of ${\sf Set}^G$ wouldn't do at all.)

Eric M Downes (Apr 11 2024 at 15:59):

(and cue @Morgan Rogers (he/him) that a monoid $M$'s actions in $\sf Set$ are a presheaf category on the delooping of the monoid ${\bf B}M$ (e.g. the "internal category" of the monoid), so maybe you want a presheaf category over ${\bf B}M$ in which every arrow is monic, and then look at functors out of that?)

Morgan Rogers (he/him) (Apr 12 2024 at 07:37):

Some of those conclusions don't seem right @Eric M Downes . The quantifier in the definition of coseparator is a "for all" not "there exists"; the morphisms in won't necessarily be monic!

I also am not sure about your initial statement. Certainly an object satisfying Paolo's hypotheses is a coseparator in the usual sense that is also a monoid, but that's only sufficient if "enough" morphisms into the underlying object of the monoid factor as a tensor product of morphisms followed by the multiplication of the monoid.

Eric M Downes (Apr 12 2024 at 08:22):

Thanks.

You're right that I misapplied Eckmann-Hilton -- the arguments in my notes work for $\sf Mon$, the category of monoids, but not arbitrary monoidal categories. I guess we don't know if we can curry $m:C\otimes C\to C$ into morphisms in $\mathsf{C}$, they may just be internal to $C$, so its not clear if composition of $m$-actions distribute over $\otimes$; they may not even be comparable.

Please help me understand where I am mistaken, with the rest though....

What Paolo has described is a monoid object $(C,m,e)$ in a monoidal category $(\mathsf{C},\otimes,1)$. We're good there right?

Then he specifies that $C$ is a coseparator, which from the first paragraph of the definition on nLab [[cogenerator]]:
($C$ is a cogenerator / coseparator just when)
For any pair $g_1,g_2\in\mathsf{C}(X,Y)$ if they are indistinguishable by morphisms to $S$;
$\forall \theta:Y\to S;~\theta\circ g_1=\theta\circ g_2\implies g_1=g_2$

Is not everything after the $\forall$ the definition of a monomorphism $\theta$? So $\theta\mapsto c, ~S\mapsto C$ seems to imply that every $c:Y\to C$ is monic for maps $X\to Y$, and $X$ was arbitrarily chosen...

Paolo's definition is subtlety different I'll grant, he is specifying for all objects, given two morphisms... so I went to nLab, but the definition there seems pretty clear?

I don't mean to hijack this thread. We can move this elsewhere if nobody else finds this confusing.

Morgan Rogers (he/him) (Apr 12 2024 at 08:58):

What Paolo asked for is a strictly stronger property, where morphisms into a monoidal product $Y_1 \otimes Y_2$ are distinguished by morphisms of the form $m(h_1 \otimes h_2)$. Or to put it another way, $C$ being a coseparator requires that the collection of all morphisms $X \to C$ are jointly monic, whereas Paolo is asking for the stronger property that the collection of morphisms of the given form (as $h_1,h_2$ vary) are jointly monic.

Eric M Downes (Apr 12 2024 at 09:34):

Okay so they are monic, that's a relief.

Are you saying that $m\circ(h_1\otimes h_2)$ not only separates distinct functions to its domain $Y_1\otimes Y_2$, as a coseparator would, but it should also separate distinct functions to either $Y_1$ or $Y_2$ independently? I go back and forth on wether that is actually stronger or not.

I think I'm getting hung up on the idea that

1. coseparators act this way for every object
2. $Y_1\otimes Y_2$ is just another object, its internal structure doesn't matter that much?
3. So any combination of $h_1,h_2$ pre-composed with $m$ should also be monic.
4. so a coseparator that is also a monoid object, behaves exactly as you describe, not stronger or weaker.

But I'm clearly not understanding something, and its late here so I'll let people who might have useful answers to Paolo respond.

Morgan Rogers (he/him) (Apr 12 2024 at 09:52):

The point is that they're jointly monic, not individually so. There is no reason we can't have $m (h_1 \otimes h_2) = m (h_1 \otimes h_2')$. Or to be more specific, in $\mathrm{Set}$ the multiplication monoid structure on $\{0,1\}$ does the trick, since squaring is the identity (given a map $h:B \times C \to 2$ distinguishing $f,g:A\rightrightarrows B \times C$ we have $h = m(h \times h)$ being a distinguishing map of the required form). This monoid is not cancellative. Moreover, any monoid containing a submonoid with the property that Paolo mentioned will also have this property!

Eric M Downes (Apr 12 2024 at 14:05):

Thanks! That helps a lot.