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Stream: theory: category theory

Topic: unitors to the left of me, unitors to the right

Matteo Capucci (he/him) (Feb 14 2022 at 17:00):

I have a question: is there a standard direction for 'invertible' constraints such as unitor?
nLab lists $\lambda : 1 \otimes x \to x$ and $\rho : x \otimes 1 \to x$ (apparently following MacLane), but then in this paper of Marmolejo for example, right and left unitors of a pseudomonad (another instance of pseudomonoid) are defined to point in different directions:
image.png

Matteo Capucci (he/him) (Feb 14 2022 at 17:02):

For pseudoalgebras, Marmolejo seems to prefer the 'left' convention: image.png

Matteo Capucci (he/him) (Feb 14 2022 at 17:04):

(look at $\psi$, going into the identity)

Matteo Capucci (he/him) (Feb 14 2022 at 17:05):

I notice that I actually tend to do like Marmolejo and dualize my unitors, it feels 'nicer'. But is there any formal reason that justifies this asymmetry/duality?

Joe Moeller (Feb 14 2022 at 17:06):

I think the Mac Lane one is pretty common. It seems natural to imagine reducing from more complicated expressions. I believe there is no conceptual difference so long as these maps are invertible.

Nathanael Arkor (Feb 14 2022 at 17:08):

The direction of the structural transformations comes from the axioms for a skew-monoidal category (more generally, lax-monoidal, but the skew-monoidal ones are the most well-behaved). It doesn't matter which of left-skew or right-skew you choose, but it makes sense to choose directions consistent with one of them.

John Baez (Feb 14 2022 at 17:09):

Matteo Capucci (he/him) said:

I have a question: is there a standard direction for 'invertible' constraints such as unitor?
nLab lists $\lambda : 1 \otimes x \to x$ and $\rho : x \otimes 1 \to x$ (apparently following MacLane)

That's the standard one - use that. It's the direction of 'simplification' and I believe people use it because it's more natural. For example if you wrote the laws of a monoid I think you'd write

$1 x = x$

not

$x = 1x$

John Baez (Feb 14 2022 at 17:10):

By the way, I believe it was my student Toby Bartels who invented the word 'unitor' to match 'associator'. Before, people said 'unit constraint'. (If Toby only reinvented it, I'd like to know.)

John Baez (Feb 14 2022 at 17:11):

Some later student invented 'leftor' and 'rightor' but I don't think those have caught on - people say 'left unitor' and 'right unitor'.

Matteo Capucci (he/him) (Feb 14 2022 at 17:12):

Nathanael Arkor said:

The direction of the structural transformations comes from the axioms for a skew-monoidal category (more generally, lax-monoidal, but the skew-monoidal ones are the most well-behaved). It doesn't matter which of left-skew or right-skew you choose, but it makes sense to choose directions consistent with one of them.

Can you elaborate on this? What are you referring to with 'structural transformations'?

John Baez (Feb 14 2022 at 17:14):

Associator, left and right unitor.

Matteo Capucci (he/him) (Feb 14 2022 at 17:15):

Mmh ok, that was a silly question indeed

John Baez (Feb 14 2022 at 17:15):

In a skew-monoidal category we make these morphisms non-invertible, and then choosing good conventions for directions becomes really crucial, because there are $2^3$ conceivable conventions, and I guess not all are equally good.

Matteo Capucci (he/him) (Feb 14 2022 at 17:16):

John Baez said:

In a skew-monoidal category we make these morphisms non-invertible, and then choosing good conventions for directions becomes really crucial, because there are $2^3$ conceivable conventions, and I guess not all are equally good.

Indeed

Matteo Capucci (he/him) (Feb 14 2022 at 17:16):

image.png

Matteo Capucci (he/him) (Feb 14 2022 at 17:16):

This is the first thing I checked and I was like 'nevermind'

John Baez (Feb 14 2022 at 17:16):

Ugh, names for all 8. :frowning:

John Baez (Feb 14 2022 at 17:17):

I'm more interested in why 2 are 'best' - the left and right skew monoidal categories. I've never looked into it. Maybe someone here can explain it in a human-friendly manner.

Matteo Capucci (he/him) (Feb 14 2022 at 17:17):

In general it seems that 'lax' structures have maps that pick, e.g. a lax monoidal functor has a map $J \to F(I)$, while oplax structures have maps that reduce, e.g. $F(I) \to J$

John Baez (Feb 14 2022 at 17:27):

The easiest way for me to remember what's a lax monoidal functor is that a lax monoidal functor does exactly the right thing to map monoid objects to monoid objects.

John Baez (Feb 14 2022 at 17:28):

(This is fun to work out two or three times, say once every 4 years.)

John Baez (Feb 14 2022 at 17:29):

Similarly, an oplax monoidal functor maps comonoid objects to comonoid objects.

John Baez (Feb 14 2022 at 17:30):

Also, a lax monoidal functor from $1$ to a bicategory picks out a monad in that bicategory.

John Baez (Feb 14 2022 at 17:30):

So "lax monoidal" goes nicely with "monoid" and "monad".

Matteo Capucci (he/him) (Feb 14 2022 at 17:50):

John Baez said:

The easiest way for me to remember what's a lax monoidal functor is that a lax monoidal functor does exactly the right thing to map monoid objects to monoid objects.

indeed

Matteo Capucci (he/him) (Feb 14 2022 at 17:51):

I also associate 'lax' with 'monoid/monad' in my head, and oplax with the dual

Nathanael Arkor (Feb 14 2022 at 17:53):

John Baez said:

I'm more interested in why 2 are 'best' - the left and right skew monoidal categories. I've never looked into it. Maybe someone here can explain it in a human-friendly manner.

The axioms for skew-monoidal categories give rise to nicer induced structures. For instance, in a left-skew-monoidal category, tensoring with the unit has the structure of a comonad, and the Kleisli morphisms (i.e. morphisms $I \otimes A \to B$) can be considered a notion of "weak morphism", which is useful for various applications. This is a theme, say, of Bourke and Lack's work on skew-monoidal categories.

Nathanael Arkor (Feb 14 2022 at 17:54):

Skew-monoidal structure also arises in many examples, whereas I haven't come across any real examples of lax-monoidal categories more generally (though perhaps someone can provide some).

Mike Shulman (Feb 14 2022 at 18:11):

Lax monoidal categories appear in https://arxiv.org/abs/0909.4715.

Joe Moeller (Feb 14 2022 at 19:22):

John Baez said:

Some later student invented 'leftor' and 'rightor' but I don't think those have caught on - people say 'left unitor' and 'right unitor'.

I think I was that later student, but it didn't even catch on with me :sweat_smile:

Mike Shulman (Feb 14 2022 at 19:26):

Nathanael Arkor said:

The axioms for skew-monoidal categories give rise to nicer induced structures. For instance, in a left-skew-monoidal category, tensoring with the unit has the structure of a comonad, and the Kleisli morphisms (i.e. morphisms $I \otimes A \to B$) can be considered a notion of "weak morphism", which is useful for various applications.

Personally, I would prefer to axiomatize the weak morphisms directly, if that's what we're interested in, and then recover the skew tensor product as a representing object with a certain universal property. Bourke and Lack did something along these lines in https://arxiv.org/abs/1708.06088, and I suggested a more general context for it in https://arxiv.org/abs/2106.15042. This also helps to explain the bizarre -looking orientations of the "associativity" and "unit" morphisms for the skew tensor, which are revealed as "actually" having some extra functors stuck in there breaking the obvious notion of associativity.