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Stream: learning: questions

Topic: proarrows

Gurkenglas (Oct 04 2021 at 10:04):

What do you call an arrow whose source and target lie in different categories? You can compose it with an arrow whose target is our source; you can compose it with an arrow whose source is our target.

Why do we view arrows as belonging to a category at all?

Can one have an internal category (a monad-of-spans-in-some-category) whose arrow object and object object lie in different categories?

Feel free to answer only a subset.

Zhen Lin Low (Oct 04 2021 at 10:15):

The fact that you titled the thread "proarrows" suggests you have an answer in mind...

Jules Hedges (Oct 04 2021 at 10:19):

I also call these things "heteromorphisms" by analogy to homomorphisms

Gurkenglas (Oct 04 2021 at 12:08):

yeah it seemed like an element of a set that is produced by a profunctor; i thought i had googled proarrow and failed to find anything, but it's right there... weird.

Joe Moeller (Oct 04 2021 at 12:25):

Ellerman also called them heteromorphisms:
https://ncatlab.org/nlab/show/heteromorphism

Matteo Capucci (he/him) (Oct 04 2021 at 13:02):

Yeah profunctors do exactly that. Iirc proarrows are something else though

Fawzi Hreiki (Oct 04 2021 at 13:19):

Aren’t proarrows just 2-categorical ‘formal’ abstractions of profunctors

John Baez (Oct 04 2021 at 17:51):

Yes.

Mike Shulman (Oct 04 2021 at 17:59):

Right, a "proarrow" is just an arrow in some bicategory where we are regarding the bicategory as analogous to $\rm Prof$. (For instance, it could be the target bicategory in a "proarrow equipment".) The analogy is "functor : arrow in a bicategory (like $\rm Cat$) :: profunctor : proarrow (in a bicategory like $\rm Prof$)".

John Baez (Oct 05 2021 at 18:36):

And then, of course, Mike combines these two bicategories into a single fibrant double category, with the proarrows as "loose" arrow and ordinary arrows as the "tight" arrows, and everyone lives happily ever after.

Matteo Capucci (he/him) (Oct 06 2021 at 10:57):

Are 'tight' and 'loose' catching on as terminology? I never remembered which is which

Mike Shulman (Oct 06 2021 at 11:38):

It doesn't help you to think of 'tight' as stricter and 'loose' as weaker?

Matteo Capucci (he/him) (Oct 06 2021 at 15:47):

Oh I see! But which ones are vertical and which one are horizontal?

Joe Moeller (Oct 06 2021 at 15:48):

I think the point is that vertical and horizontal are completely arbitrary, and in fact I think people use them in different ways. Whereas tight/loose give an impression of which has what property.

Matteo Capucci (he/him) (Oct 06 2021 at 15:49):

Yeah right, but the vertical and horizontal categories have a different role in a double category, iirc

Matteo Capucci (he/him) (Oct 06 2021 at 15:50):

I mean, squares are morphisms between morphisms of only one of the two (though this is hidden by the way you draw them)

Matteo Capucci (he/him) (Oct 06 2021 at 15:50):

To be fair, this is not clear even with the 'vertical' and 'horizontal' terminology, so it's not an argument against that

Joe Moeller (Oct 06 2021 at 15:53):

Right, so a double category is a weak internal category to Cat. So in this setup, we have an object category and a morphism category. The object category is a category, so the morphisms there are the tight morphisms. The objects of the morphism category are what are weak, so they're the loose morphisms.

John Baez (Oct 06 2021 at 15:54):

Matteo Capucci (he/him) said:

Oh I see! But which ones are vertical and which one are horizontal?

.

Nathanael Arkor (Oct 06 2021 at 15:56):

Matteo Capucci (he/him) said:

Oh I see! But which ones are vertical and which one are horizontal?

Most frequently, the horizontal arrows are the proarrows, i.e. the loose arrows, and the vertical arrows are the tight arrows. But different people follow different conventions.

Mike Shulman (Oct 06 2021 at 16:56):

Matteo Capucci (he/him) said:

Yeah right, but the vertical and horizontal categories have a different role in a double category, iirc

No, the tight and loose categories have a different role in a (pseudo) double category. Saying it this way is unambiguous, because you can specify the role of the tight morphisms and the role of the loose morphisms. Trying to specify the role of the "horizontal" morphisms requires first fixing a convention as to whether to draw the tight morphisms horizontally and the loose ones vertically or vice versa. That wouldn't be so much of a problem if there were a universally accepted convention, but there isn't. I tend to draw the loose morphisms horizontally, partly because it uses less space on the page when composing a large number of them, which I tend to do more frequently than composing a large number of tight morphisms, and also because then the "loose-globular" 2-cells are drawn in the same orientation as they would be in the bicategory of loose morphisms. But Bob Pare and his school use the opposite convention. However, we can all agree on which morphisms are tight and which are loose (at least, once we agree to use those words).

John Baez (Oct 06 2021 at 16:57):

This is why I found Matteo's question so demonic.

John Baez (Oct 06 2021 at 17:00):

You introduced "tight" and "loose" because nobody could agree which were "horizontal" and "vertical", and he said "great, that makes sense - but which ones are horizontal, and which are vertical?" :smiling_devil:

Mike Shulman (Oct 06 2021 at 17:24):

It's kind of like the conversation about $\mathbb{C}$ and $(-)^{\rm op}$ in the other thread, actually: there's a $\mathbb{Z}/2$ action and no canonical trivialization.

Matteo Capucci (he/him) (Oct 06 2021 at 18:32):

John Baez said:

Matteo Capucci (he/him) said:

Oh I see! But which ones are vertical and which one are horizontal?

.

I guess I deserve this :laughing:

Matteo Capucci (he/him) (Oct 06 2021 at 18:33):

I only now realize how silly my question sounds :laughter_tears:

John Baez (Oct 06 2021 at 19:48):

The "bwahaha" was because you sounded like you were demonically trying to force Mike into the very box he was trying to escape. (Of course I knew that's not what you were doing.)

Matteo Capucci (he/him) (Oct 07 2021 at 05:47):

That would've been arguably funnier

Jules Hedges (Oct 07 2021 at 09:21):

Heh, I learned about double categories from Mike's papers and for a couple of years had the impression that basically everybody had switched to the pseudo=horizontal, strict=vertical convention and abandoned the older opposite convention... and then I read David Jaz's book that uses the opposite convention, and my brain melted

Matteo Capucci (he/him) (Oct 16 2022 at 01:12):

John Baez said:

Matteo Capucci (he/him) said:

Oh I see! But which ones are vertical and which one are horizontal?

.

lmao just stumbled upon this thread again, how clueless I was