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Hi. I've been wondering whether there's any known results giving necessary and sufficient conditions on algebraic presentations of monads such that we can claim that the monad we are looking at is traced.
If someone knows about it that's @Joseph Collins
Bumping this thread because I'd also like to know the answer! I'm interested in the compact closed case as well (as in: conditions on the algebraic presentation of a monad that make its Kleisli category compact closed).
So you might be interested in my paper with Hassei:
https://arxiv.org/pdf/2208.06529.pdf
where we look at "traced monads" which are monads that lift traced monoidal structure to their Eilenberg-Moore categories.
Here are also some slides on the subject:
https://richardblute.files.wordpress.com/2021/11/lemay-novfest.pdf
and I'll be giving an updated talk about this at CMS2023 in a few weeks.
It remains an open question if "Traced monads" can be defined without mentioning the Eilenberg-Moore category. (Hassei has been working on this for 20+ years I believe) We give a solution in certain cases to this using Hopf monads.
Monads that lift compact closed structure to their Eilenberg-Moore categories are precisely Hopf monads:
https://arxiv.org/abs/1003.1920
https://arxiv.org/abs/math/0604180
http://www.tac.mta.ca/tac/volumes/33/37/33-37.pdf
Robin Piedeleu said:
Bumping this thread because I'd also like to know the answer! I'm interested in the compact closed case as well (as in: conditions on the algebraic presentation of a monad that make its Kleisli category compact closed).
I don't know the answer for Kleisli categories. But Hassei mentioned that he and others (I think he mentioned Paul-André Méliès?) looked at lifting trace to Kleisli categories and could only come up with nasty identities.
However I am very much interested in lifting trace/compact closed structure to Kleisli catgories as well.
(Donc @Robin Piedeleu si tu veux on peut en discuter plus!)
Thanks, @JS PL (he/him)!
I'm also interested in the question of what makes the Kleisli/EM category of a monad (over Set) traced/compact-closed. That is, when the underlying category is not necessarily compact nor traced. All Set-monads I know of whose Kleisli category is compact-closed are of the form for some semiring . It's somewhat mysterious to me. Why is that? Is that necessary? Sounds unlikely.
That's an interesting question! Here's a partial answer on when the Kleisli cat of a monad on is compact closed. First of all, in order for to be symmetric monoidal, needs to be a commutative monad. This at least makes into a commutative monoid (which in the semiring case is the underlying multiplicative monoid; if is in addition additive, then is even a semiring). Assuming compact closure, we then get natural isomorphisms:
If one could now argue that must be canonically self-dual, then this turns into your with . But I'm not sure how to complete the argument.
As an aside, a 2-dimensional version of this argument implies that the bicategory of distributors/profunctors is compact closed, since the bicategory of distributors is the Kleisli bicategory for the presheaf construction, which is pseudo-commutative, and (cf. Remark 5.9 of Di Liberti–Loregian's On the unicity of formal category theories).
Conjecture: A Kleisli category for a commutative monad on is compact closed if and only if is isomorphic to a monad of the form , where is a commutative quantale.
Here, the functoriality of such a monad on a function is the map given by , and the monad multiplication looks like . This recovers the usual powerset monad for .
Note also that the above conjecture really is an equivalence of properties rather than structures, since a monad on is commutative in at most one way, and being compact closed is also a mere property of a symmetric mnoidal category.
Thanks, that clarifies a lot!
I'm also inclined to believe your conjecture. I think to prove it will require some cardinality argument: for more general semirings, finite sets are always dualisable in the Kleisli category of . What seems to make the whole Kleisli category compact-closed is the ability to take arbitrary sums in your semiring (suprema for quantales).
Robin Piedeleu said:
That is, when the underlying category is not necessarily compact nor traced.
Ah that is a very interesting question! I'd be interested in the answer too.
Tobias Fritz said:
Conjecture: A Kleisli category for a commutative monad on is compact closed if and only if is isomorphic to a monad of the form , where is a commutative quantale.
Yes this seems quite plausible. It seems that neat argument you wrote above seems to point in that direction. And as @Robin Piedeleu says, having infinite sums is needed to define the cap.
Tobias Fritz said:
If one could now argue that must be canonically self-dual, then this turns into your with . But I'm not sure how to complete the argument.
This is very neat! However I also don't see how to argue . Of course, if you assume the Kleisli category is "self-dual compact closed" then your argument is finished.
For the traced story, the Kleisli category of the maybe monad is traced on the coproduct. So there seems to be more possible monads for this side of the story.
Tobias Fritz said:
But I'm not sure how to complete the argument.
If the duality in the Kleisli category is induced by an involution on the original category, then you get . (This is the case in the 2-dimensional example, for instance.)
Which suggests to me that either the duals in the compact closed structure should always be induced in this way or, in general, the correct expression is , and being able to remove the dual is a special situation.
For traced structure wrt coproduct on the Kleisli category, there is a whole bunch of work by @Sergey Goncharov, Stefan Milius, and others on all sorts of iterative monads, see for example Complete Elgot monads and coalgebraic resumptions.
Thanks, @Tom Hirschowitz for popularizing our work! We investigated in depth the case of coCartesian trace on the Kleisli category of a monad. According to Hyland and Hasegawa, such coCartesian trace is the same as a Conway fixpoint operator, i.e. for a monad , it has the profile , plus the laws of Conway iteration. These are generally known to be too weak, so a standard remedy is to add the so called uniformilty principle w.r.t. pure morphisms. We call the resulting monads Elgot monads. Regarding Cartesian tensor, let alone proper tensor, we did not analyse them in the context of monads.
Ah this is super interesting paper @Sergey Goncharov -- definitely on a topic that aligns with some of my current research projects near to my heart.
Sergey Goncharov said:
Regarding Cartesian tensor, let alone proper tensor, we did not analyse them in the context of monads.
Well one difficulty with providing a traced Cartesian structure on a Kleisli category is that unlike coproducts which come for free on a Kleisli category, products do not come for free. So there's even more extra structure needed here.
Nathanael Arkor said:
Tobias Fritz said:
But I'm not sure how to complete the argument.
If the duality in the Kleisli category is induced by an involution on the original category, then you get . (This is the case in the 2-dimensional example, for instance.)
I was under the impression that there was no involution on the category ?
JS PL (he/him) said:
Sergey Goncharov said:
Regarding Cartesian tensor, let alone proper tensor, we did not analyse them in the context of monads.
Well one difficulty with providing a traced Cartesian structure on a Kleisli category is that unlike coproducts which come for free on a Kleisli category, products do not come for free. So there's even more extra structure needed here.
Yes, exactly. From the semantics perspective, Cartesian trace is recursion , and technical aspects aside, Kleisli category is not the best place to interpret it, except for some corner cases, like the lifting monad on dcpo, for which Kleisli and EM coincide. Recursion is dual to iteration. But if we want to dualise iteration on monads, we also should dualize monads and get recursion on co-monads. At least, this is what semantic considerations suggest.
Another example is the power set monad on , whose Kleisli category is and this has a Cartesian trace, which is the same as the coCartesian trace because has biproducts. But this is an example where the Kleisli and the EM do not coincide.
Sergey Goncharov said:
But if we want to dualise iteration on monads, we also should dualize monads and get recursion on co-monads. At least, this is what semantic considerations suggest.
But yes, it is much more natural to consider Cartesian trace in coKleisli categories.