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Stream: theory: category theory

Topic: Skew monoidal functors

Emily (Mar 07 2024 at 21:25):

Does anyone know a reference (preferably carefully) defining skew monoidal functors?

Emily (Mar 07 2024 at 21:25):

I've found two really natural left-skew and right-skew monoidal category structures on the category $\mathsf{Sets}_*$ of pointed sets and pointed maps between them, and feel like the free pointed set functor

$$(-)^+\colon\mathsf{Sets}\to\mathsf{Sets}_*$$

is going to be skew monoidal with respect to those, like it is for $\wedge$ and $\times$ via

$$X^+\wedge Y^+\cong (X\times Y)^+$$

Emily (Mar 07 2024 at 21:26):

But I can't seem to find a definition of skew monoidal functor in the literature, and wanted to check here first before potentially reinventing the wheel

Nathanael Arkor (Mar 07 2024 at 22:24):

They're defined in the original paper on skew-monoidal categories, but the terminology is a little different: they're called "right-monoidal functors" therein, rather than "right-skew monoidal functors" (see Definition 2.2).

Nathanael Arkor (Mar 07 2024 at 22:26):

(More accurately, "skew-monoidal functor" is used, but as a collective term for both (lax) left-skew and right-skew monoidal functors.)

Emily (Mar 07 2024 at 22:55):

ahhh that explains why I couldn't find it :sweat_smile:

Emily (Mar 07 2024 at 22:55):

Thank you so much, Nathanael :)

Emily (Mar 07 2024 at 22:56):

By the way, have you ever seen a variation on the notion of monoidal functor whose isomorphisms go like

$$F(X)\otimes Y \to F(X\otimes Y)$$

or

$$X\otimes F(Y) \to F(X\otimes Y)$$

?

Emily (Mar 07 2024 at 22:56):

I ask because this is happening for the skew monoidal structures I'm trying to understand, and they feel exceedingly natural in a sense that if no such notion exists then this (and a couple other families of examples) would justify defining and studying it

Emily (Mar 07 2024 at 22:57):

Namely I have a left-skew monoidal product

$$\lhd\colon\mathsf{Sets}_*\times\mathsf{Sets}_*\to\mathsf{Sets}_*$$

on the category of pointed sets characterised by being the one that universally represents left-bilinear morphisms

$$f\colon X\times Y\to Z$$

of pointed sets, i.e. those satisfying $f(x_0,y)=z_0$ for all $y\in Y$.

Emily (Mar 07 2024 at 22:57):

so it's basically a version of the smash product that is only left-sided

Emily (Mar 07 2024 at 22:58):

And the thing is it satisfies the relation

$$X^+\lhd Y \cong (X\times Y)^+$$

just like the smash product satisfies

$$X^+\wedge Y^+ \cong (X\times Y)^+,$$

where $(-)^+$ is the free pointed set

Emily (Mar 07 2024 at 22:59):

So I was bit puzzled about this and whether something like this had already been studied before

Mike Shulman (Mar 07 2024 at 23:19):

Emily said:

By the way, have you ever seen a variation on the notion of monoidal functor whose isomorphisms go like

$$F(X)\otimes Y \to F(X\otimes Y)$$

or

$$X\otimes F(Y) \to F(X\otimes Y)$$

?

Of course that only makes sense if $F$ is an endofunctor. But it's a special case of a functor between different categories that preserves the [[action of a monoidal category]] on them, in the case of a monoidal category acting on itself by its tensor product.

Nathanael Arkor (Mar 07 2024 at 23:22):

Just to add to what Mike said, such functors are usually called [[strong functors]].

Nathanael Arkor (Mar 07 2024 at 23:25):

(In particular, right-strong functors, and left-strong functors respectively. You may see some people use "strong" and "costrong", though this convention is misleading.)

Emily (Mar 07 2024 at 23:41):

Mike Shulman said:

Of course that only makes sense if $F$ is an endofunctor.

Ahh, of course, what a silly mistake! :woman_facepalming:

Emily (Mar 07 2024 at 23:41):

I'll go back to trying to figure out what is going on with those products then :sweat_smile:

Emily (Mar 07 2024 at 23:42):

Thank you so much Mike and Nathanael! =)

Philip Saville (Mar 12 2024 at 08:32):

Emily said:

Namely I have a left-skew monoidal product

$$\lhd\colon\mathsf{Sets}_*\times\mathsf{Sets}_*\to\mathsf{Sets}_*$$

on the category of pointed sets characterised by being the one that universally represents left-bilinear morphisms

$$f\colon X\times Y\to Z$$

of pointed sets, i.e. those satisfying $f(x_0,y)=z_0$ for all $y\in Y$.

I think this might be an instance of the fact that if you have a strong monad (+ some conditions which will likely hold over pointed sets) then its category of algebras is skew monoidal. If your monad is commutative you can define the universal thing classifying bilinear maps and that makes a tensor [by some results of Kock and Guitart]. But if your monad is only strong you can only define the universal thing classifying maps that are linear in one argument and that makes a skew monoidal tensor.

(This is based on work Marcelo Fiore and I presented at CT a while back: some slides that may help -- I think your universal property is the one on slide 18)

Emily (Mar 14 2024 at 11:27):

Philip Saville said:

Emily said:

Namely I have a left-skew monoidal product

$$\lhd\colon\mathsf{Sets}_*\times\mathsf{Sets}_*\to\mathsf{Sets}_*$$

on the category of pointed sets characterised by being the one that universally represents left-bilinear morphisms

$$f\colon X\times Y\to Z$$

of pointed sets, i.e. those satisfying $f(x_0,y)=z_0$ for all $y\in Y$.

I think this might be an instance of the fact that if you have a strong monad (+ some conditions which will likely hold over pointed sets) then its category of algebras is skew monoidal. If your monad is commutative you can define the universal thing classifying bilinear maps and that makes a tensor [by some results of Kock and Guitart]. But if your monad is only strong you can only define the universal thing classifying maps that are linear in one argument and that makes a skew monoidal tensor.

(This is based on work Marcelo Fiore and I presented at CT a while back: some slides that may help -- I think your universal property is the one on slide 18)

This sounds really fascinating! Have you and Marcelo written more about this since?

(Actually I was checking your homepage and found many of your works very interesting too; I plan to check them out carefully later)

Also, a further question: there are similar tensor products for monoids as the ones I described above for pointed sets, and in particular it seems to me that the tensor product for not necessarily commutative monoids described here should endow the category of monoids with that the nLab calls an unbiased oplax monoidal category structure.

Do you think this too could be an instance of a result analogous to the one you and Marcelo proved?

Philip Saville (Mar 19 2024 at 16:59):

This sounds really fascinating! Have you and Marcelo written more about this since?

We did get as far as a paper draft but then we realised things could be done in a nicer way, and we never got round to re-visiting it. I'd be happy to discuss over Zoom or similar if you're interested, though!

Also, a further question: there are similar tensor products for monoids as the ones I described above for pointed sets, and in particular it seems to me that the tensor product for not necessarily commutative monoids described here should endow the category of monoids with that the nLab calls an unbiased oplax monoidal category structure.

Without checking very much, I can believe the tensor being defined in this link is the one Marcelo and I talk about. I think there's a similar kind of tensor for semigroups, where the same kind of thing occurs.

I suppose every skew monoidal structure gives rise to an unbiased oplax monoidal structure. Of course it depends on exactly what you're trying to do, but I would suggest that the binary definition is likely more useful than the unbiased one in particular examples. At least, it's a lot less to check! Unbiased structure can be very nice to have, e.g. when defining certain structures in monoidal categories, but at least in the classical setting you can use coherence to see the unbiased and biased definitions are essentially the same. In the skew case I suspect this is not true. (There is some work by Tarmo Uustalu on coherence in the skew monoidal case, but the result is rather more subtle than Mac Lane's statement.)

Emily (Mar 28 2024 at 22:19):

Hi Philip, my apologies for taking a bit to reply!

Philip Saville said:

We did get as far as a paper draft but then we realised things could be done in a nicer way, and we never got round to re-visiting it. I'd be happy to discuss over Zoom or similar if you're interested, though!

That would be lovely! I plan to try to make my way through the material around this area (monoidal monads, bilinearity, etc.) in the future, and would love to understand your work with Marcelo then

(Well, to be honest, I've made a note to myself to actually go through _all_ your works carefully at some point, I've really found them truly interesting!)

Philip Saville said:

Without checking very much, I can believe the tensor being defined in this link is the one Marcelo and I talk about. I think there's a similar kind of tensor for semigroups, where the same kind of thing occurs.

When I initially wrote to you, I was under the impression that this product likely would not admit a natural skew monoidal structure, and only an oplax monoidal one (whose axioms are a pain to prove, hence why I was hoping for a more clean approach like the one you developed with Marcelo for skew monoidal structures).

But thinking about it, I could very well see this happening; I'll try to construct one and see how that goes!

Thank you so much again, Philip :)