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Stream: learning: questions

Topic: Enrichment in a closed category

Nathaniel Virgo (Jul 17 2024 at 03:41):

We can define enrichment in a monoidal category $\mathcal{V}$, without making any additional assumptions on $\mathcal{V}$. However, in practice, when people want to enrich in a monoidal category $\mathcal{V}$, they often first prove that $\mathcal{V}$ is monoidal closed.

I guess this is because enrichment in a monoidal closed category is in some way nicer than enrichment in a mere monoidal category. But can this be stated more precisely? Is there a theorem saying what enrichment in a (symmetric?) monoidal closed category gives you that enrichment in a monoidal category doesn't, which justifies the use of monoidal closed categories in practice?

Nathaniel Virgo (Jul 17 2024 at 03:57):

Ah nvm, I guess it's because being monoidal closed means $\mathcal{V}$ can be seen as enriched in itself, which lets you talk about enriched presheaves and the [[enriched Yoneda lemma]]. I guess I'm still interested in hearing if there's another reason for it.

Alexander Campbell (Jul 17 2024 at 06:46):

@Nathaniel Virgo Amusingly, any closed category $\mathcal{V}$ is enriched over itself, one doesn't even need it to be monoidal for that. In fact, one still gets an enriched Yoneda lemma in this setting (see Street's paper Skew-closed categories).

Nathaniel Virgo (Jul 17 2024 at 06:49):

Interesting, ok, so you can define enrichment in terms of the closed structure instead of the monoidal structure. I guess if it is monoidal closed the definitions coincide?

Nathaniel Virgo (Jul 17 2024 at 06:52):

On the other side, I'm pretty sure there are results for categories enriched in monoidal (not closed) categories that could be called "the Yoneda lemma". Without closedness you can't define a profunctor as a $\mathcal{V}$-functor into $\mathcal{V}$, but you can still define it as something that gets acted on by two $\mathcal{V}$-categories, and I think there's a version of the Yoneda lemma that you can prove from that.

Alexander Campbell (Jul 17 2024 at 06:56):

That's right. Hinich recently had a paper that proceeds in such a manner.

Alexander Campbell (Jul 17 2024 at 06:57):

(Maybe 2016 isn't as "recent" as it feels to me.)

Nathanael Arkor (Jul 17 2024 at 08:21):

Perhaps the pertinent point is that while you can define enriched presheaves without assumptions on the base of enrichment, and consequently define an ordinary category of enriched presheaves, to actually form an enriched category of enriched presheaves, one requires at least some closure, to form the objects of enriched natural transformations that form the hom-objects of the enriched presheaf category.

John Baez (Jul 17 2024 at 12:41):

I also thought you needed $\mathcal{V}$ to be closed, with some other good properties too, to define a $\mathcal{V}$-object of natural transformations between two $\mathcal{V}$-functors between two $\mathcal{V}$-categories.

Morgan Rogers (he/him) (Jul 18 2024 at 15:35):

Alexander Campbell said:

Nathaniel Virgo Amusingly, any closed category $\mathcal{V}$ is enriched over itself, one doesn't even need it to be monoidal for that. In fact, one still gets an enriched Yoneda lemma in this setting (see Street's paper Skew-closed categories).

This reference arrived with very convenient timing for me, thanks @Alexander Campbell

Morgan Rogers (he/him) (Jul 19 2024 at 11:08):

TIL that monoidal categories were derived from considering monoidal closed categories. Eilenberg and Kelly present closed categories first!

Nathanael Arkor (Jul 19 2024 at 11:16):

Monoidal categories are not due to Eilenberg and Kelly. They are due to Bénabou and Mac Lane c. 1963. However, I believe the Eilenberg and Kelly paper is the first place the terminology "monoidal category" appears.

Morgan Rogers (he/him) (Jul 19 2024 at 11:19):

Thanks for the correction @Nathanael Arkor . Indeed, scrolling down the references in the nLab page for [[monoidal category]], one finds the references to Benabou's "Categories with multiplication", but the first line of the references presents E&K as "the first monograph". I find that somewhat misleading!

John Baez (Jul 19 2024 at 11:21):

James Dolan pointed me to Eilenberg and Kelly's paper a long time back. Their approach makes a certain sense because category theory is all about the homs. So it's interesting how unpopular it is to introduce closed categories first and treat monoidal categories as closed categories with extra properties, rather than the reverse.

John Baez (Jul 19 2024 at 11:23):

Btw Benabou's definition of monoidal category ("categorie avec multiplication") is flawed: none of the most famous monoidal categories obey the coherence law in his definition. I explain why on pages 5-6 here.

John Baez (Jul 19 2024 at 11:24):

I hope the nLab mentions this.

John Baez (Jul 19 2024 at 11:25):

His paper is still important.

John Baez (Jul 19 2024 at 11:27):

Also, it's amusing that in Mac Lane's paper, the first correctly defining monoidal categories, he didn't call them that. He called them"bicategories"!

Morgan Rogers (he/him) (Jul 19 2024 at 11:42):

John Baez said:

Btw Benabou's definition of monoidal category ("categorie avec multiplication") is flawed: none of the most famous monoidal categories obey the coherence law in his definition. I explain why on pages 5-6 here.

Aha, I couldn't see right away what was wrong with Benabou's condition, so I'm glad you provided a link.

John Baez (Jul 19 2024 at 11:49):

I learned this from @Mike Shulman, btw.

Morgan Rogers (he/him) (Jul 19 2024 at 11:50):

John Baez said:

Also, it's amusing that in Mac Lane's paper, the first correctly defining monoidal categories, he didn't call them that. He called them"bicategories"!

In the introduction to that paper he acknowledges some work of Stasheff and Epstein preceding his, so maybe Mac Lane doesn't deserve the full credit for this that he is typically given? Does anyone know the background for that? (I can get a pdf of Part II of Stasheff's work, but not Part I, which Mac Lane cites)

John Baez (Jul 19 2024 at 11:56):

I think Mac Lane deserves plenty of credit!

It's well-known that Stasheff came up with the pentagon condition earlier than Mac Lane, in his work on when an H-space is homotopy equivalent to a group. Mac Lane surely got the pentagon law from him: I believe they were in communication. But Stasheff wasn't talking about categories, and he didn't mention the unit coherence laws.

I don't know what Epstein did. I would like to know!

The main result of Mac Lane's paper is not the definition of monoidal category: it's his coherence theorem justifying that definition. This theorem is hard to state correctly, and fairly hard to prove.

John Baez (Jul 19 2024 at 12:01):

Mac Lane is not quite correct in his paper when he says that his advance over Benabou is finding a "finite set" of coherence laws for monoidal categories. Maybe Mac Lane is being deliberately polite, or maybe he just didn't notice, but not only does Benabou's definition give an infinite set of coherence laws, it also gives infinitely many coherence laws that don't hold in, say $(\mathsf{Set}, \times)$.

Morgan Rogers (he/him) (Jul 19 2024 at 12:04):

It's entirely possible that he didn't notice. Does the kind of equality of objects that causes the problem actually happen in $\mathrm{Set}$?
I'm really glad to have easy access to people who know this history. I'm working on something that requires me to talk with some authority about monoidal categories, so having a solid historical account of where the definition came from feels important.

John Baez (Jul 19 2024 at 12:13):

Morgan Rogers (he/him) said:

It's entirely possible that he didn't notice. Does the kind of equality of objects that causes the problem actually happen in $\mathrm{Set}$?

Whoops, quite possibly not. This kind of equality of objects happens in a skeletal category equivalent to $\mathsf{Set}$. If it really matters to you that the "bad coherence law" on page 5 of my paper actually fails there, you should check. But this check will actually be rather annoying!

Here's why: $\mathsf{FinSet}$ is equivalent to a category that's both strict and skeletal, and there the "bad coherence law" actually does hold because all the objects you need to be equal are equal, and all the associators are identity. But $\mathsf{Set}$ cannot be made both strict and skeletal. So then we should get cases of this "bad coherence law" that fail. But finding them may be a pain in the butt, since they will probably involve infinite sets.

John Baez (Jul 19 2024 at 12:15):

(As usual I edited my answer after you read it and :thumbs_up:ed it.)

John Baez (Jul 19 2024 at 12:16):

It's probably safer to just say that Benabou's axiomatization includes axioms that don't follow from Mac Lane's (and have 'no right to be true').

John Baez (Jul 19 2024 at 12:27):

I've improved the nLab's discussion of the history near the end of [[monoidal category]].

John Baez (Jul 19 2024 at 12:45):

Why do you need expertise in the history of monoidal categories, @Morgan Rogers (he/him)? As you can see, I needed it when I wrote a history of Hoàng Xuân Sính's work on 2-groups. I wanted to know what someone in Vietnam, learning math from Grothendieck, might know about monoidal categories. She wrote her thesis from around 1967 to 1972, in the middle of a war. She knew Benabou's work, but luckily also knew the correct definition due to Mac Lane, and she used the pentagon identity to get a 3-cocycle in group cohomology from any 2-group.

Morgan Rogers (he/him) (Jul 19 2024 at 12:48):

I'm putting together a survey of settings in which we can talk about monoids and their actions. It may develop into a monograph, but for the time being everything I'm doing is known (and old!) so I want to keep track of some of the historical context.

John Baez (Jul 19 2024 at 12:49):

Nice! It can be really fun to dig into the history of math. You just have to always keep in mind that the usual stories we hear are oversimplified - getting to the truth is a bit like detective work.

Morgan Rogers (he/him) (Jul 19 2024 at 12:52):

Indeed, it can take as much time to dig through the historical web of references as to read them!

John Baez (Jul 19 2024 at 13:05):

Luckily, you usually wind up learning a bunch of interesting stuff.

Kevin Carlson (Jul 19 2024 at 21:10):

John Baez said:
But finding them may be a pain in the butt, since they will probably involve infinite sets.

I think that in a cartesian monoidal skeletal category equivalent to Set, where we have an object $N$ with $N^2=N$ and product projections $\pi_1,\pi_2:N\to N,$ the associator $N=(N\times N)\times N\to N\times (N\times N)=N$ must be given on the first component by $\pi_1^2.$ Your "bad coherence condition" in this case becomes $a_{N,N,N}^2=a_{N,N,N}$ and in particular says that $\pi_1^4=\pi_1^2.$ We construct $\pi_1$ by composing a bijection $\mathbb N\to \mathbb N^2$ with the standard projection; using the traditional zig-zag isomorphism you can end up with $\pi_1$ sending $0,1,3,6,10,\ldots\mapsto 0; 2,4,7,11,\ldots\mapsto 1; 5,8,12,\ldots \mapsto 2, \ldots .$ Then $\pi_1$ is strictly decreasing so eventually sends any particular number to $0,$ but of course not immediately; for instance $\pi_1^2(5)=1$ but $\pi_1^4(5)=0.$ So this coherence doesn't always hold. Perhaps it never does but I'm not sure how easy that might be to show.

John Baez (Jul 20 2024 at 08:46):

Impressive! That argument still counts as a "pain in the butt" in my book - since I don't know what the "traditional zig-zag isomorphism" is, and I'm still trying to understand how the triangular numbers 0, 1, 3, 6, 10 showed up - but it's quite admirable, and I'll probably even enjoy it once I understand it.

Todd Trimble (Jul 20 2024 at 10:54):

I would guess it's refering to a coding of pairs of natural numbers by a single natural number, something like

$(x, y) \mapsto \binom{x+y+1}{2} + y$

(something like that -- this is off the top of my head). One of those bijections $\mathbb{N}^2 \to \mathbb{N}$ where you count along SE to NW diagonals.

John Baez (Jul 20 2024 at 11:14):

Got it! Nice!

Kevin Carlson (Jul 21 2024 at 05:08):

Yeah, I can’t immediately read Todd’s formula but I meant to identify a possible inverse by what I think everybody does when they’re asked to write a bijection down $\mathbb N\to \mathbb N^2$, which is zigzag through diagonals like Todd describes.

Todd Trimble (Jul 21 2024 at 14:06):

The only difference is that whereas you were zigzagging, I'm constantly zagging. Both of us go along diagonals parallel to $x + y = 0$, but in my formula, each diagonal starts from the $x$-axis. Not as pictorially nice as the "mowing the lawn" zigzagging description, but nicer in terms of writing down an explicit formula. :-)

Todd Trimble (Jul 21 2024 at 14:15):

Funnily, it's harder to generalize this bijection $\mathbb{N}^2 \to \mathbb{N}$ to general infinite sets. The exercise is to start from a well-ordered set $X$ and use the well-ordering to produce a bijection $X^2 \to X$. What turns out to work is a bijection which is built up from an ever-increasing union of "squares" (cartesian squares of initial segments) rather than an ever-increasing union of triangles, as in the zigzagging above. (I guess this is off-topic enough, so I won't resist adding that the fact that every infinite set can be put in bijection with its square is equivalent to the axiom of choice. (-: )

Morgan Rogers (he/him) (Jul 23 2024 at 13:22):

Todd Trimble said:

I would guess it's refering to a coding of pairs of natural numbers by a single natural number, something like

$(x, y) \mapsto \binom{x+y+1}{2} + y$

(something like that -- this is off the top of my head). One of those bijections $\mathbb{N}^2 \to \mathbb{N}$ where you count along SE to NW diagonals.

Just to add a little without going further off-topic: I found a nice derivation of the $k$-variable version of this formula on MathOverflow today.