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Stream: theory: category theory

Topic: Delooping of pseudomonoids

Matteo Capucci (he/him) (Oct 24 2020 at 21:02):

Hi everyone, is there a known notion for a delooping of a pseudomonoid in a monoidal 2-category?
It would be the 'formal' version of the delooping of a monoidal category, which is a pseudomonoid in $\mathbf{Cat}$
It would go more or less along these lines:
Let $(\mathbb X, I, \otimes)$ be a monoidal 2-cat, let $M$ be a pseudomonoid in $\mathbb X$. Then $\mathbb B M$ would be a 'weak 2-category' where

• there's just one dummy object $\bullet$
• 1-cells $\bullet \to \bullet$ are given by arrows $I \to M$ in $\mathbb X$, with identity given by $M$'s unit and composition by $M$'s monoidal product.
• 2-cells from $a: I \to M$ to $b:I\to M$ would be just $\mathbb X$'s 2-cells between them

In fact, one could describe this more succinctly as the delooping of $\mathbb X(I, M)$, whose monoidal structure is induced by $M$'s own monoidal structure.

Is this sensible? Is this known already?

sarahzrf (Oct 24 2020 at 21:06):

if this is sensible, it should be a categorification of some sort of delooping a monoid in a monoidal category

sarahzrf (Oct 24 2020 at 21:07):

but: as someone who definitely couldnt write down the laws for a pseudomonoid without looking them up: i mean, it sounds reasonable to me

sarahzrf (Oct 24 2020 at 21:08):

i'd imagine that homming in from $I$ will lose a fair amount of data though

sarahzrf (Oct 24 2020 at 21:08):

it might make sense to put the domain as a parameter somehow

sarahzrf (Oct 24 2020 at 21:09):

umm... if you have a monoid M in a monoidal category C... then C(A, M) is naturally a monoid for all A

sarahzrf (Oct 24 2020 at 21:10):

literally naturally—im gonna blindly guess that we get a monoid structure on C(-, M) under day convolution or something?

sarahzrf (Oct 24 2020 at 21:11):

oh wait no this only works for the cartesian monoidal structure, what am i saying lmfao
EDIT: oh wait no x2, certainly C(A, M) isn't gonna be a monoid in general, but C(-, M) will be a monoid under day convolution (yoneda embedding is monoidal if presheaves have day convolution, and monoidal functors turn monoids into monoids)

sarahzrf (Oct 24 2020 at 21:11):

sorry

sarahzrf (Oct 24 2020 at 21:21):

im curious what you want this for, though?

Reid Barton (Oct 24 2020 at 21:29):

I would assume that if you had a notion of "weak 2-category enriched in the monoidal 2-cat X", then BM would be an example of such a thing.

sarahzrf (Oct 24 2020 at 21:33):

ya—monoid = 1-object enriched category, pseudomonoid presumably = 1-object weakly enriched 2-category??

Reid Barton (Oct 24 2020 at 21:46):

I agree that $X(I, M)$ is a monoidal category so it has a delooping, and I also agree with @sarahzrf that it seems like it could lose most of the information about $M$, though it really depends on what $X$ is--the kinds of $X$ I'm used to thinking about are pretty close to Cat anyways, so then not that much would be lost.

Matteo Capucci (he/him) (Oct 25 2020 at 12:06):

Yeah, I'm really pissed off by having to hom with $I$. I thought also $X(-,M)$ could be a viable alternative, but I don't get anything similar to the usual delooping of a mon cat

Matteo Capucci (he/him) (Oct 25 2020 at 12:22):

An higher-level approach could be this: let $\mathbf{Mon}\mathbb{2Cat}$ be the 3-category of monoidal 2-categories, let $\mathbb{3Cat}$ the 3-category of 2-categories. Consider

1. The ??-functor $\operatorname{PSM} : \mathbf{Mon}\mathbb{2Cat} \to \mathbb{3Cat}$ which extracts pseudomonoids from a monoidal 2-category
2. The ??-functor $\Delta_{\mathbb{2Cat}} : \mathbf{Mon}\mathbb{2Cat} \to \mathbb{3Cat}$ which is constant of $\mathbb{2Cat}$

Then I claim delooping should be a ??-natural transformation $\operatorname{PSM} \to \Delta_{\mathbb{2Cat}}$, characterized by some universal property.

This claim is gibberish since I barely know my 2-cat theory, let alone 3-cats

John Baez (Oct 25 2020 at 17:00):

@Matteo Capucci - your idea is not surprising to me, but I don't know if anyone has written it down - I've never seen it done.

When you say "weak 2-category" I think you might want the concept of "pseudo-category":

• Nelson Martins Ferreira, Pseudo-categories.

John Baez (Oct 25 2020 at 17:02):

He writes:

Defining a pseudo-category we begin with a 2-category, take the definition of an internal category there, and replace the equalities in the associativity and identity axioms by the existence of suitable isomorphisms which then have to satisfy some coherence conditions.

John Baez (Oct 25 2020 at 17:05):

One level down we have this situation. Categories internal to a category C can be defined when C has pullbacks, or at least when the pullbacks you happen to need exist. Monoids internal to C can be defined when C is monoidal. These situation overlap when your category C has a terminal object and binary products. Then it's monoidal, but also the necessary pullbacks exist to talk about internal categories where the object of objects is the terminal object - since then the pullbacks we need are just binary products!

John Baez (Oct 25 2020 at 17:07):

So, if there's anything you don't like about this situation - and admittedly it is a bit annoying how our general monoidal structure is forced to be cartesian - you should fix it before ascending to the categorified situation you're interested in!

John Baez (Oct 25 2020 at 17:09):

If you want to think about "categories internal to a category that doesn't have pullbacks", you may want to think about quantum categories, but this seems like a rather strenuous digression from what you're actually thinking about.

John Baez (Oct 25 2020 at 18:15):

sarahzrf said:

as someone who definitely couldnt write down the laws for a pseudomonoid without looking them up:

If you know the laws for a monoidal category you (secretly, unbeknownst to yourself) know the laws for a pseudomonoid, since they're the same... just written out in a monoidal 2-category other than Cat.

You've got your tensor product, your unit object, your associator, your pentagon identity, your left and right unitors, your triangle identity for those... and that's all!

sarahzrf (Oct 25 2020 at 23:33):

John Baez said:

sarahzrf said:

as someone who definitely couldnt write down the laws for a pseudomonoid without looking them up:

If you know the laws for a monoidal category you (secretly, unbeknownst to yourself) know the laws for a pseudomonoid, since they're the same... just written out in a monoidal 2-category other than Cat.

You've got your tensor product, your unit object, your associator, your pentagon identity, your left and right unitors, your triangle identity for those... and that's all!

i appreciate the encouragement, but i doubt i could write down the laws for a monoidal category :upside_down:

sarahzrf (Oct 25 2020 at 23:33):

well, okay, i could probably work them out if you really pushed me, but i would strongly prefer not to

sarahzrf (Oct 25 2020 at 23:34):

in any case, i'd rly have to strain to figure out how to internalize them into a monoidal bicategory

sarahzrf (Oct 25 2020 at 23:34):

especially w/o assuming that it's a strict 2-category ;_;

John Baez (Oct 26 2020 at 03:48):

Remembering the defnition of a monoidal bicategory is one level harder than remembering the definition of a monoidal category. I'm kinda stunned you don't know what a monoidal category is, because you seem to now quite a bit of category theory. Anyway, the pentagon and triangle axioms are the only non-obvious parts, but they're so memorable once you see them and ponder them that they're impossible to forget.

John Baez (Oct 26 2020 at 03:49):

(Actually the hardest part is that the triangle axiom is one of three obvious coherence laws for the left and right unitors, and it's not at all obvious that the "middle" one of these laws implies the other two. Mac Lane didn't notice that, so he included all three.)

sarahzrf (Oct 26 2020 at 07:50):

i know perfectly well what a monoidal category is :weary:

sarahzrf (Oct 26 2020 at 07:51):

i just couldnt write down the coherence bullshit off the top of my head

sarahzrf (Oct 26 2020 at 07:52):

i cant say i relate at all to finding the pentagon and triangle memorable lmao

sarahzrf (Oct 26 2020 at 07:52):

maybe i just havent pondered them properly

sarahzrf (Oct 26 2020 at 07:53):

maybe the problem is that i dont know my homotopy theory and hence lack a proper taste for coherence

Matteo Capucci (he/him) (Oct 26 2020 at 09:33):

John Baez said:

Matteo Capucci - your idea is not surprising to me, but I don't know if anyone has written it down - I've never seen it done.

When you say "weak 2-category" I think you might want the concept of "pseudo-category":

• Nelson Martins Ferreira, Pseudo-categories.

Thanks a lot for the reference. This shows how lost I am in the world of 2-categories: everytime I think I grasped more or less what's going on, I get jeopardize by a definition I didn't foresee.
I've just learned that 'double categories' are cats internal to Cat, and I get the point of that. Also strict 2-cats I understand they are 'cats enriched in Cat', and I see why.
But now I'm failing to understand why would a 'weak 2-category' (by which I mean a category with both 1- and 2-cells such that associativity and unitality are satisfied up to invertible 2-cells [which probably ought to satisfy some coherence law as well]) be internal to Cat.

Matteo Capucci (he/him) (Oct 26 2020 at 09:43):

John Baez said:

One level down we have this situation. Categories internal to a category C can be defined when C has pullbacks, or at least when the pullbacks you happen to need exist. Monoids internal to C can be defined when C is monoidal. These situation overlap when your category C has a terminal object and binary products. Then it's monoidal, but also the necessary pullbacks exist to talk about internal categories where the object of objects is the terminal object - since then the pullbacks we need are just binary products!

Is this a consequence of the fact that you suggest $\mathbb B M$ should be an internal (2-)category in $\mathbb X$?

Matteo Capucci (he/him) (Oct 26 2020 at 09:56):

Btw it looks like this is exactly what I'm after when I think 'weak 2-cat': https://ncatlab.org/nlab/show/bicategory

Matteo Capucci (he/him) (Oct 26 2020 at 10:23):

On the side, I just realized that there's a correspondence between monoidal lax/pseudoactions of $M$ and monoidal lax/pseudofunctors $\mathbb B M \to \mathbb X$.

Matteo Capucci (he/him) (Oct 26 2020 at 10:34):

This generalizes to monoidal 2-cats a classic characterization of monads, which can be seen either as lax monoidal actions $1 \to [\mathbf C, \mathbf C]$ or as lax monoidal functors $\mathbb B1 \to \mathbf{Cat}$

John Baez (Oct 26 2020 at 16:00):

sarahzrf said:

i cant say i relate at all to finding the pentagon and triangle memorable lmao

Okay, I guess when I got interested in higher categories I became fascinated with coherence laws, because those are the big mystery of the subject. The pentagon is a law of nature: you can just start trying to write down all reasonable coherence laws for associators

$a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z)$

in a big enormous diagram, and you'll discover it's built from pentagons (and more boring commutative squares).

John Baez (Oct 26 2020 at 16:02):

And the reason is that there are 5 ways to bracket a product of 4 things.

It's just like how the 2 ways to bracket a product of 3 things gives birth to the associative law!

John Baez (Oct 26 2020 at 16:24):

Matteo Capucci said:

John Baez said:

When you say "weak 2-category" I think you might want the concept of "pseudo-category":

• Nelson Martins Ferreira, Pseudo-categories.

Thanks a lot for the reference. This shows how lost I am in the world of 2-categories: everytime I think I grasped more or less what's going on, I get jeopardized by a definition I didn't foresee.

I just learned about this paper from a referee of a paper of mine. But I didn't feel jeopardized: the idea has been floating around for a while; this is just where someone wrote it down.

I've just learned that 'double categories' are cats internal to Cat, and I get the point of that.

Okay, if you've just learned this, like last week, then the idea of a "weak category" internal to Cat might come as a shock.

The point is that we can define a category internal to C whenever C is a category with pullbacks. Cat is a category with pullbacks, so we can define a category internal to Cat - and that's a double category.

But Cat is much more than a category: it's a 2-category, thanks to natural transformations. And it's often really evil to treat Cat as a mere category: that amounts to requiring that various diagrams of functors commute "on the nose", instead of "up to natural isomorphism". In real life that's rarely how things work.

So, we should want a concept of "weak category internal to a 2-category with pullbacks". And that's what Ferreria calls a pseudocategory.

In particular we can look at a pseudocategory internal to Cat, and this is what some people call a pseudo double category.

But in fact pseudo double categories are so much more useful than mere double categories - they include such important examples not included by the earlier concept - that Mike Shulman has started using "double category" to mean "pseudo double category", and so do I now . Of course we warn the reader that we're doing this. It just gets tiring to write "pseudo double category" 100 times in one paper.

sarahzrf (Oct 26 2020 at 16:38):

John Baez said:

sarahzrf said:

i cant say i relate at all to finding the pentagon and triangle memorable lmao

Okay, I guess when I got interested in higher categories I became fascinated with coherence laws, because those are the big mystery of the subject.

honestly same, but i havent gotten around to actually learning any of the stuff yet :sob:

sarahzrf (Oct 26 2020 at 16:39):

its less that i think the coherence laws for monoidal categories are boring and unimportant, and more that i have a sense that they're the tip of an iceberg that i really want to appreciate fully at some point and which i cannot remember until then

Matteo Capucci (he/him) (Oct 26 2020 at 16:40):

Alright. So delooping a monoidal category gives you a pseudo double category, and not a bicategory? My confusion stems from the fact I don't see 'vertical' and 'horizontal' arrows arising in a natural way here. I guess you recover a bicategory when horizontal morphisms are trivial, so that squares collapse to the usual picture of a filling arrow between two parallel morphisms.

sarahzrf (Oct 26 2020 at 16:42):

i periodically look up some definition like pseudonatural transformations for some purpose or other and then think to myself "geez look at all of these giant coherence diagrams that are going in one of my ears and out the other, these sure do seem mysterious and important, it really seems like understanding where they come from would explain a ton, god i should learn some homotopy theory, sighhhh maybe later im in the middle of something"

sarahzrf (Oct 26 2020 at 16:42):

and i have yet to dig into that stuff

Shea Levy (Oct 26 2020 at 16:43):

sarahzrf said:

its less that i think the coherence laws for monoidal categories are boring and unimportant, and more that i have a sense that they're the tip of an iceberg that i really want to appreciate fully at some point and which i cannot remember until then

I'm not confident this is actually correct, but I've been stumbling across a lot of coherence diagrams lately, and every time I do it seems a special case of associativity of some unbiased operator

John Baez (Oct 26 2020 at 16:45):

Matteo Capucci said:

Alright. So delooping a monoidal category gives you a pseudo double category, and not a bicategory?

No, it gives a bicategory, at least in the approach I've always seen. This is because a monoidal category is exactly the same thing as a one-object bicategory.

Matteo Capucci (he/him) (Oct 26 2020 at 16:47):

John Baez said:

Matteo Capucci said:

Alright. So delooping a monoidal category gives you a pseudo double category, and not a bicategory?

No, it gives a bicategory, at least in the approach I've always seen. This is because a monoidal category is exactly the same thing as a one-object bicategory.

Great to hear. So that's what in my head was a 'weak 2-category', so my terminology was way off.

Reid Barton (Oct 26 2020 at 16:52):

Matteo Capucci said:

so my terminology was way off.

Not really, since bicategories are weak 2-categories. Or as the nLab says:

A bicategory is a particular algebraic notion of weak 2-category (in fact, the earliest to be formulated, and still the one in most common use).

Reid Barton (Oct 26 2020 at 16:56):

Your original message is completely correct in its current form except that $\mathbb{B} M$ isn't really the delooping of $M$ itself, but rather the delooping of the monoidal category you get by "forgetting the $\mathbb{X}$-structure", namely, by applying $\mathbb{X}(I, -)$.

Reid Barton (Oct 26 2020 at 16:58):

I thought the question was: what kind of thing is the delooping of $M$ itself.

Matteo Capucci (he/him) (Oct 26 2020 at 17:03):

Reid Barton said:

Matteo Capucci said:

so my terminology was way off.

Not really, since bicategories are weak 2-categories. Or as the nLab says:

A bicategory is a particular algebraic notion of weak 2-category (in fact, the earliest to be formulated, and still the one in most common use).

Oh well. Good to hear.

Matteo Capucci (he/him) (Oct 26 2020 at 17:12):

Reid Barton said:

Your original message is completely correct in its current form except that $\mathbb{B} M$ isn't really the delooping of $M$ itself, but rather the delooping of the monoidal category you get by "forgetting the $\mathbb{X}$-structure", namely, by applying $\mathbb{X}(I, -)$.

Indeed. This is what I observed here:
Matteo Capucci said:

In fact, one could describe this more succinctly as the delooping of $X(I, M)$, whose monoidal structure is induced by $M$'s own monoidal structure.

Regarding this:
Reid Barton said:

I thought the question was: what kind of thing is the delooping of $M$ itself.

I guess there could be different ways to look at it. One is to try to get this to be actually true (I believe I've been to fast to claim it's actually true, I will now check the details)
Matteo Capucci said:

On the side, I just realized that there's a correspondence between monoidal lax/pseudoactions of $M$ and monoidal lax/pseudofunctors $\mathbb B M \to \mathbb X$.

Maybe it's better phrased in term of '"lifting" $M$ from $\mathbb X$ to $\mathbb X\mathbf{Cat}$.

Matteo Capucci (he/him) (Oct 26 2020 at 17:21):

Matteo Capucci said:

Maybe it's better phrased in term of '"lifting" $M$ from $\mathbb X$ to $\mathbb X\mathbf{Cat}$.

Now I realize this is a good hint to myself. I was referring to the fact the definition I gave in my first message was of an $\mathbb X$-enriched category, but it's not! This observation though may be the key to solve it all.
In fact let $\mathbb B M$ be the $\mathbb X\mathbf{Cat}$ on a single object $\bullet$ and whose object of morphisms $\bullet \to \bullet$ is $M$ itself.
This doesn't forget anything about $M$! We are not homming with the hom-$I$-nous (I couldn't not make this pun, excuse me) $I$ anymore. And now $\mathbb B M \to \mathbb X$ can actually be an $\mathbb X$-enriched ??-functor, which is even nicer.

Matteo Capucci (he/him) (Oct 26 2020 at 17:23):

Plus I think it's nice to notice $[\mathbb B M, \mathbb X]$ is a (co)presheaf category, namely the category (topos, actually, but I'm not sure about the enriched case?) of $M$ actions in $\mathbb X$, which is where it all started.

Matteo Capucci (he/him) (Oct 26 2020 at 17:30):

I'm wondering now if $\mathbb B M$ ought to be some kind of bicategory anyway? I can't think of a natural notion of 2-cells now. I guess it's a feature of categories to have them, and it's not a general phenomenon.

John Baez (Oct 26 2020 at 17:37):

Matteo Capucci said:

John Baez said:

Matteo Capucci said:

Alright. So delooping a monoidal category gives you a pseudo double category, and not a bicategory?

No, it gives a bicategory, at least in the approach I've always seen. This is because a monoidal category is exactly the same thing as a one-object bicategory.

Great to hear. So that's what in my head was a 'weak 2-category', so my terminology was way off.

Bicategories are the same as weak 2-categories, but somehow we drifted off into discussing a much less well-known concept, 'weak categories', also known 'pseudocategories'. I'm not sure why we started talking about that - I think you asked about them.

Reid Barton (Oct 26 2020 at 17:37):

I'm not sure exactly what of "2-"/"bi-"/"weak" to put where, but fundamentally, the delooping of $M$ should be a category (weakly) enriched in $\mathbb{X}$. I haven't seen a definition of this concept but I would guess that you can reverse engineer what it should be by taking all the structure that the delooping has and then allowing multiple objects.

Matteo Capucci (he/him) (Oct 26 2020 at 17:38):

We do need 2-cells in $\mathbb B M$, since they're needed to state the coherence laws of $M$. I'm not sure how to describe their totality though.

John Baez (Oct 26 2020 at 17:39):

sarahzrf said:

its less that i think the coherence laws for monoidal categories are boring and unimportant, and more that i have a sense that they're the tip of an iceberg that i really want to appreciate fully at some point and which i cannot remember until then

Okay, that's an understandable attitude. I just remembered that you've been specializing in categories where all diagrams commute.

Matteo Capucci (he/him) (Oct 26 2020 at 17:39):

John Baez said:

Matteo Capucci said:

John Baez said:

Matteo Capucci said:

Alright. So delooping a monoidal category gives you a pseudo double category, and not a bicategory?

No, it gives a bicategory, at least in the approach I've always seen. This is because a monoidal category is exactly the same thing as a one-object bicategory.

Great to hear. So that's what in my head was a 'weak 2-category', so my terminology was way off.

Bicategories are the same as weak 2-categories, but somehow we drifted off into discussing a much less well-known concept, 'weak categories', also known 'pseudocategories'. I'm not sure why we started talking about that - I think you asked about them.

I think I said 'weak 2-categories' meaning 'bicategories', while you know your 2-cats and understood, well, 'pseudocategories'.

John Baez (Oct 26 2020 at 17:43):

Looking back I have no idea why I started talking about pseudocategories. I just have weak categories, aka pseudocategories, on my mind these days.

Matteo Capucci (he/him) (Oct 26 2020 at 17:44):

:laughing: no worries! I still appreciate the willingness to help

Matteo Capucci (he/him) (Oct 26 2020 at 17:45):

Plus I learned things, so it's good anyway

John Baez (Oct 26 2020 at 17:45):

Now I think you were asking something like this: if you have a pseudomonoid internal to a monoidal 2-category, can we deloop it and get a bicategory internal to that monoidal 2-category?

Matteo Capucci (he/him) (Oct 26 2020 at 17:45):

Yes. But replace the second 'internal' with 'enriched'.

John Baez (Oct 26 2020 at 17:46):

Okay. That's actually a lot easier.

John Baez (Oct 26 2020 at 17:46):

So then the answer is just "sure".

John Baez (Oct 26 2020 at 17:48):

So Reid was right:

I would assume that if you had a notion of "weak 2-category enriched in the monoidal 2-cat X", then BM would be an example of such a thing.

... where "weak 2-category" = "bicategory".

John Baez (Oct 26 2020 at 17:50):

I don't know if I've seen the definition of "bicategory enriched in a monoidal 2-category C", but I could write it down: you just copy the definition of bicategory, writing and "hom(x,y) $\in$ C" wherever you write "set hom(x,y)", and $\otimes$ lots of places where you normally write $\times$, and so on.

John Baez (Oct 26 2020 at 17:52):

Okay, here it is:

https://ncatlab.org/nlab/show/enriched+bicategory

John Baez (Oct 26 2020 at 17:52):

The definition was first written down, apparently independently, by Carmody and Lack in 1995.

Reid Barton (Oct 26 2020 at 17:53):

And another way to say it which will end up with the same result is to take the definition of a pseudomonoid in a monoidal 2-category and write $C(a, b)$ in place of $M$, with the multiplication map $M \otimes M \to M$ becoming $C(a, b) \otimes C(b, c) \to C(a, c)$ (or perhaps $C(b, c) \otimes C(a, b) \to C(a, c)$), and so on.

Reid Barton (Oct 26 2020 at 17:53):

Well, maybe don't use the letter $C$ if your monoidal 2-category is also called $C$.

sarahzrf (Oct 26 2020 at 17:59):

John Baez said:

sarahzrf said:

its less that i think the coherence laws for monoidal categories are boring and unimportant, and more that i have a sense that they're the tip of an iceberg that i really want to appreciate fully at some point and which i cannot remember until then

Okay, that's an understandable attitude. I just remembered that you've been specializing in categories where all diagrams commute.

maybe less so recently, but i mean that wasnt for no reason

Matteo Capucci (he/him) (Oct 26 2020 at 18:02):

John Baez said:

Okay, here it is:

https://ncatlab.org/nlab/show/enriched+bicategory

Yes, this seem to be the right one. I was confused by 2-cells for the last 30 minutes but now I'm pretty sure I was right when I wrote this:
Matteo Capucci said:

I'm wondering now if $\mathbb B M$ ought to be some kind of bicategory anyway? I can't think of a natural notion of 2-cells now. I guess it's a feature of categories to have them, and it's not a general phenomenon.

There's no 2-cells: the coherence morphisms live in a different world, namely $\mathbb X$.