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Stream: learning: questions

Topic: internalizing structures obtained by internalization

David Egolf (Aug 12 2024 at 18:56):

A monoid in the monoidal category $(\mathsf{Ab}, \otimes, \mathbb{Z})$ is a ring. It then in turn makes sense to talk about rings internal to appropriate categories, such as cartesian monoidal categories.

Can we do this in general? If we start with some structure $a$ that can be internalized in some category $C$, and thereby obtain the notion of some structure $a'$ - when can $a'$ be internalized in some category $C'$? And, if we can do this, how does $C'$ relate to $C$ and $a$? I'm currently primarily interested in the case where $a$ refers to the structure of a monoid, but I would welcome comments on other cases as well.

For example, "topological monoids" can be viewed as monoids in $\mathsf{Top}$. Can we then talk about topological monoids internal to certain categories? If so, what kind of categories support the notion of "internal topological monoid"?

As another example, quantales can be viewed as monoids in some appropriate category. What kind of categories support the notion of "internal quantale"?

After typing this out, I suspect this question has no easy answer in general. I would welcome any thoughts on the topic, though!

Morgan Rogers (he/him) (Aug 13 2024 at 09:38):

The way this works when it works is that the category $C$ is itself a category of structures in $\mathrm{Set}$, and internalizing the notion of "internal $a$ in $C$" amounts to identifying analogous structures to those in $C$ in another category $C'$. In particular, this will work as soon as $C'$ has enough structure to define internal $C$-things (because we can construct the category of those and then take an internal $a$ in that category). For the case of abelian groups, we need $C'$ to have finite products, and similarly for meet semi-lattices (see below). For the case of topological spaces, you need a lot more - the classical definition of topological space involves powersets/power objects, which are not especially common in categories which aren't toposes... but you can get around that by substituting that for a weaker or less demanding definition, depending on what kinds of property you want your category of "internal spaces" to have.
The main subtlety of this process is that the intermediate category of "internal $C$-things in $C'$" has properties that depend on those of $C'$. So if you want the "internal internal $a$-things" to behave as you're expecting, you need to check if the properties of $C'$ are compatible. For instance, if you take $C'$ to be $\mathrm{Set}^{\mathrm{op}}$, then the category of internal abelian groups is rather boring (it might be a fun exercise to figure out what I mean by that), although its properties are technically very nice for trivial reasons :') A more informative example might be that if you consider abelian groups in $\mathrm{FinSet}$, you lose the free abelian groups?

David Egolf (Aug 14 2024 at 17:03):

Thanks for your response! I am just starting to read it now.

David Egolf (Aug 14 2024 at 17:17):

Morgan Rogers (he/him) said:

In particular, this will work as soon as $C'$ has enough structure to define internal $C$-things (because we can construct the category of those and then take an internal $a$ in that category).

For example, let's say that $C = (\mathsf{Ab}, \otimes, \mathbb{Z})$ and that $a$ refers to a monoid. (And note that we can define a monoid internal to $C$). In this case, each object of $C$ is an abelian group. Then we are interested in a category $C'$ that supports the definition of internal $C$-things, which in this case is an internal abelian group.

So, let $C'$ be such that it supports a definition of internal abelian group. Presumably this means that $C'$ needs to be a monoidal category, at the least. (This monoidal structure could be provided by the existence of finite products in $C'$, for example, I think).

Then, I think you are claiming this: the category of internal abelian groups in $C'$ will itself support the definition of an internal monoid.

David Egolf (Aug 14 2024 at 17:19):

If the pattern you describe is true in general, that sounds like quite a powerful result! I am tempted to try and write a proof...

David Egolf (Aug 14 2024 at 17:24):

Your second paragraph is also quite interesting. If am understanding you correctly, sometimes even when the process of repeated internalization works, one can end up with something boring or limited at the end. So to get a nice result that behaves how you'd like requires some additional conditions on $C'$.

John Baez (Aug 14 2024 at 18:31):

David Egolf said:

So, let $C'$ be such that it supports a definition of internal abelian group. Presumably this means that $C'$ needs to be a monoidal category, at the least. (This monoidal structure could be provided by the existence of finite products in $C'$, for example, I think).

There's a huge story about this stuff, but here are 4 items worth knowing:

1) If a category is monoidal it supports the definition of internal monoid... and monoidal categories are about the right level of generality for that, though you can get away with a multicategory.

2) If a category is braided monoidal it supports the definition of internal commutative monoid - we need to switch variables around to state the commutative law $ab = ba$... and this is about the right level of generality for that, though a braided multicategory should do. (If you don't know braided monoidal categories, the most famous kind are symmetric monoidal categories, and those are fine here though less general.)

3) Usually people say that to define internal groups we need a category with finite products, because we need to duplicate and delete variables to state the law $g g^{-1} = 1$. This is certainly a fine level of generality, though recently I saw a shocking way to define groups internal to a monoidal category, perhaps due to Tom Leinster, which I promptly forgot. (And no, I'm not talking about Hopf monoids, which are another end run around this problem.)

4) By 2) and 3) you can therefore also define abelian group objects in a category with finite products, and people are usually content with that level of generality.

Joe Moeller (Aug 14 2024 at 18:38):

I remember you talking about this strange internal group definition, and also don’t remember what it was. I’m super interested if anyone remembers.

John Baez (Aug 14 2024 at 19:01):

I texted Tom Leinster and he told me. In a category with finite products, a group object is a monoid object where the commutative square stating the associative law is a pullback. But this definition makes sense in any monoidal category: you have a monoid object, and you demand that the associativity square is a pullback! You don't even need all pullbacks to exist.

John Baez (Aug 14 2024 at 19:02):

I have no idea how practical this definition is.

John Baez (Aug 14 2024 at 19:14):

I added this idea to the nLab.

David Egolf (Aug 16 2024 at 01:29):

@John Baez Thanks for explaining those four items! That is interesting. I was particularly interested to learn that one can define commutative monoids internal to braided monoidal categories.

The concept of a monoid in a monoidal category seems so wonderful to me. So I guess I was hoping we could define a group internal to a "groupoidal" category - and that a category with finite products would just be one special kind of such a category.

David Egolf (Aug 16 2024 at 01:31):

Apparently a strict monoidal category is a monoid internal to $\mathsf{Cat}$. I am wondering now what a group internal to $\mathsf{Cat}$ is, and if such a category might possibly support the notion of a group internal to it.

David Egolf (Aug 16 2024 at 01:33):

In general, I would love to complete this list:

• monoid -> monoidal category
• commutative monoid -> braided monoidal category
• group -> ?? (possibly just categories with finite products, but maybe this can be generalized somehow?)

David Egolf (Aug 16 2024 at 01:35):

I don't know if the "shocking way" to define groups internal to a monoidal category should be included in this list.

John Onstead (Aug 16 2024 at 07:53):

Not sure if this will help but have you encountered the notion of a 2-group or the microcosm principle?
I'm curious about these things too so hopefully someone will elaborate more!

John Baez (Aug 16 2024 at 09:27):

David Egolf said:

Apparently a strict monoidal category is a monoid internal to $\mathsf{Cat}$. I am wondering now what a group internal to $\mathsf{Cat}$ is, and if such a category might possibly support the notion of a group internal to it.

We can define groups in any category with finite products. A group in Cat was traditionally called a [[categorical group]], but I renamed it a [[strict 2-group]] because there is also an extremely important 'weak' kind of [[2-group]] where the group laws hold up to natural isomorphism.

John Baez (Aug 16 2024 at 09:33):

2-groups were first intensively studied by Grothendieck's student Hoàng Xuân Sính, who wrote her thesis on them while Vietnam was being bombed by the US. I wrote this history and exposition of her work on 2-groups in honor of her 90th birthday:

• Hoàng Xuân Sính's thesis: categorifying group theory.

If you're mainly interested in strict 2-groups, i.e. groups in Cat, you might like the section on strict 2-groups here:

• John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups.

The first really exciting thing about strict 2-groups is that you can describe them as 'crossed modules' (see the paper for a definition), and use that to get lots of nice examples.

John Baez (Aug 16 2024 at 09:35):

And the answer to your second question is: yes. If $\mathsf{C}$ is a category with finite products, the category of groups in $\mathsf{C}$ also has finite products, so you can iterate the construction.

John Baez (Aug 16 2024 at 09:39):

You can show the category of groups in the category of groups in $\mathsf{C}$ is equivalent to the category of abelian groups in $\mathsf{C}$. This is an application of the [[Eckmann-Hilton argument]].

John Baez (Aug 16 2024 at 09:40):

The category of groups in the category of abelian groups in $\mathsf{C}$ is equivalent to the category of abelian groups in $C$, so further iterations don't do anything.

John Baez (Aug 16 2024 at 09:41):

This business of internalization, and iterated internalization, is some of my favorite math!

David Egolf (Aug 18 2024 at 16:18):

Thanks for your comments! I look forward to reading and responding to them properly when I have some more energy.

David Egolf (Aug 30 2024 at 00:25):

[Apologies for the long delay. My health has been poor the last couple weeks.]

Thanks to both of you for your comments! The references you've provided look interesting. There is clearly a lot to learn about this.

I wasn't able to find the answer to this question in those references, although maybe I missed it: Can we define a group internal to any strict 2-group? (I'm also interested in the non-strict case, but I thought the strict case might provide a simpler starting point).

@John Baez noted above that, if $\mathsf{C}$ is a category with finite products, then the category of groups in $\mathsf{C}$ also has finite products. That means, I believe, that we can define a group internal to the category of groups in $\mathsf{C}$.

However, I believe that's a bit different from what I'm wanting to do: I'm not looking to define a group internal to some category of groups; I rather want to define a group internal to a strict 2-group (which is a group in $\mathsf{Cat}$).

David Egolf (Aug 30 2024 at 00:31):

A strict 2-group, which is a group in $\mathsf{Cat}$, I think should be the following together:

• a category $G$
• a "multiplication" functor $\mu: G \times G \to G$
• an "inverse" functor $i:G \to G$
• an "identity" functor $e:1 \to G$, where $1$ is the terminal category,

such that we have some commutative diagrams in $\mathsf{Cat}$ that express how associativity, unitality, and inverses work.

David Egolf (Aug 30 2024 at 00:35):

Taking a guess, a group in a strict 2-group $G$ might be something like all this data taken together:

• an object $G'$ of $G$
• a "multiplication" morphism $\mu':\mu(G',G') \to G'$ in $G$
• an "inverse" morphism $i':G' \to G'$ in $G$
• an "identity" morphism $e':1' \to G'$ in $G$ where $1'$ is some kind of identity with respect to $\mu$ (maybe $1'$ should be the object in the image of $e:1 \to G$?),

such that we have some commutative diagrams in $G$ that express how associativity, unitality, and inverses work.

David Egolf (Aug 30 2024 at 00:40):

I'm trying to imitate the definition of a monoid in a monoidal category, although it's very possible I made a mistake - I just typed up the above quickly without checking it carefully.

I am curious if this construction or one similar to it works out, enabling us to define a group internal to any strict 2-group.

David Egolf (Aug 30 2024 at 00:47):

If something like this works, I would be curious if the usual notion of group can be recovered as a group internal to some strict 2-group. (I am guessing that $\mathsf{Set}$ cannot be made into a strict 2-group!)

David Egolf (Aug 30 2024 at 01:02):

Section 4.3 of Higher-Dimensional Algebra III:n-Categories and the Algebra of Opetopes looks relevant, but doesn't explicitly talk about groups. It says this:

Here we show the following version of the microcosm principle: $O$-algebra objects can be defined in any $1$-coherent $O$-algebra. For example, monoid objects can be defined in any monoidal $1$-category, and commutative monoid objects can be defined in any stable $1$-category.

So, if I knew what an $O$-algebra object and a $1$-coherent $O$-algebra were, that paper might answer my question. :sweat_smile: However, I'll stop here for now.

John Baez (Aug 30 2024 at 01:09):

David Egolf said:

I wasn't able to find the answer to this question in those references, although maybe I missed it: can we define a group internal to any strict 2-group?

No, you can't. You can define a monoid internal to a 2-monoid (= monoidal category), which is a nice example of the [[microcosm principle]], but you can't define a group internal to a 2-group.

The microcosm principle works very nicely for structures that are algebras of operads, like monoids - that's what Jim and I were showing in that paper you quoted. But groups aren't algebras of an operad.

David Egolf (Aug 30 2024 at 01:44):

That is very interesting to know! When I have some more energy, I may try and better understand exactly what goes wrong with the construction I sketched above.

Your comment also gives me some motivation to learn about operads, at some point!

John Baez (Aug 30 2024 at 01:48):

To see the problems with your definition of "group in a strict 2-group", start by remembering that all morphisms in a 2-group are isomorphisms, so the consequences of your definition are a lot more drastic than you may realize!

John Baez (Aug 30 2024 at 01:58):

What you're calling the "multiplication" $\mu: G \times G \to G$, most people denote as $\otimes: G \times G \to G$, since $G$ is a monoidal category and that's how we write the multiplication in a monoidal category - as a tensor product.

So you're saying that $G' \in G$ is an object with $G' \otimes G' \cong G'$, but this implies that $G' \cong I$ where $I$ is the unit object for the tensor product $\otimes$. (I.e. $I$ is what you'd call $e$ of the one object in the terminal category.)

John Baez (Aug 30 2024 at 02:01):

So in trying to understand what a "group internal to a 2-group" amounts to, we can assume $G' = I$ without any real loss of generality. It'll be interesting to see what happens next!

John Baez (Aug 30 2024 at 02:01):

I haven't worked it out.

John Onstead (Aug 30 2024 at 06:28):

John Baez said:

The microcosm principle works very nicely for structures that are algebras of operads, like monoids - that's what Jim and I were showing in that paper you quoted. But groups aren't algebras of an operad.

I find this really interesting. However I don't understand operads very well, not enough to understand the paper at the very least. Is there any brief simplified explanation for why the microcosm principle shows up when doing algebra-like things with operads, but not always when we do algebra-like things with monads or Lawvere theories? What is so different in doing algebra with operads vs monads/lawvere theories?

John Baez (Aug 30 2024 at 14:41):

An operad describes an algebraic structure that has operations obeying equational laws that don't duplicate or delete variables. So monoids are described by some operad, since the associative law

$(x y) z = x (y z)$

has the same variables on each side, none duplicated and none missing from one side. But idempotent monoids are not described by an operad, since a variable is duplicated here:

$x x = x$

Monoids obeying the following extra law are also not described by an operad, since a variable is deleted:

$x y = x$

John Baez (Aug 30 2024 at 14:42):

Groups are right out, since here we are both duplicating a variable at left and deleting it at right:

$g g^{-1} = 1$

John Baez (Aug 30 2024 at 14:43):

You can duplicate and delete variables in a [[cartesian monoidal category]], but not in a general [[symmetric monoidal category]]. So you can define group objects in a cartesian monoidal category like $(\mathsf{Top}, \times)$, which are called topological groups, but not in a symmetric monoidal category like $(\mathsf{Vect}, \otimes)$. (There are workarounds like [[quantum groups]], but these "carry their own duplication and deletion operators with them".)

John Baez (Aug 30 2024 at 14:47):

All this makes structures defined by operads much more robust... and easier to categorify.

When an algebraic structure is described by some operad O, we say O is the operad for that algebraic structure, and we say structures of that type are algebras of O.

Jim and I defined for any operad O a kind of category that's like an algebra of O, but where all the laws hold only up to isomorphism - right now let me call it an O-category.

For example if O is the operad for monoids, then:

• an algebra of O is a monoid. (This is what "O is the operad for monoids" means.)
• an O-category is a monoidal category.

Then we defined a notion of an O-algebra object, which is like an algebra of the operad O, but living in an O-category. For example:

• an O-algebra object in an O-category is a monoid object in a monoidal category.

The microcosm principle says (among other things) that the perfect home for an O-algebra is an O-category.

John Baez (Aug 30 2024 at 14:48):

We went on and generalized this to n-categories, using an approach to n-categories based on operads.

John Baez (Aug 30 2024 at 14:51):

It's not an easy paper to read, but some of the examples are tons of fun!

John Baez (Aug 30 2024 at 14:54):

Why does the microcosm principle work more easily for operads than for Lawvere theories? I don't really know!

But I know that operads keep things really simple by working only with concepts that naturally arise in 'pure' n-category theory. Lawvere theories let us describe structures that obey laws with duplication and deletion of variables. But this means they only make sense in a 'cartesian' context, so any study of the microcosm principle for them would force us to ponder the world of 'cartesian n-categories'.

John Onstead (Aug 30 2024 at 19:46):

Thanks for the breakdown! Now that I have some background I'll go back and read more of the paper!

David Egolf (Sep 02 2024 at 23:03):

I read recently that a Hopf monoid in any cartesian monoidal category is a group object in that category.

So, if one wished to consider group-like things in a more general setting, specifically in a braided monoidal category, one might take a look at the Hopf monoids in that setting.

John Baez (Sep 03 2024 at 00:18):

Right! And this is why people care about Hopf monoids. To be honest, mathematicians care most about Hopf monoids in $(\mathsf{Vect}, \otimes)$, which are called [[Hopf algebras]], and also Hopf monoids in categories rather similar to $(\mathsf{Vect}, \otimes)$.

For example, "cohomology" gives a symmetric monoidal functor from $(\mathsf{Top}, \times)$ to $(\mathsf{GrVect}, \otimes)$, the category of [[graded vector spaces]]. (Don't worry about the definition too much, just follow the flow here.) A group in $(\mathsf{Top}, \times)$ is called a topological group. For example the real line or the circle or the group of unitary $n \times n$ matrices is a topological group. But as you noted, this is same as a Hopf monoid in $(\mathsf{Top}, \times)$.

Since cohomology is a symmetric monoidal functor, it sends Hopf monoids to Hopf monoids. So the cohomology of a topological group is a Hopf monoid in $(\mathsf{GrVect}, \otimes)$. This is called a graded Hopf algebra.

So, any topological group gives a graded Hopf algebra!

Milnor, Kostant and Rosenberg did a detailed study of graded Hopf algebras, and their results shed a lot of light on topological groups. In particular, thanks to this, we know a lot about the cohomology of topological groups. This means that you can sometimes take a topological space and say "Is there any way to make this space into a topological group? No, because its cohomology can't be made into a graded Hopf algebra".

John Baez (Sep 03 2024 at 00:22):

(Stasheff put a lot of work into the question of when you can make a topological space into a topological group, and in the process he discovered the [[pentagon identity]], which Mac Lane later incorporated into the definition of monoidal category. So this question turned out to be very fruitful for mathematics. Stasheff wrote an influential book about it in 1970. But I digress.)

Mike Shulman (Sep 03 2024 at 01:22):

David Egolf said:

So, if one wished to consider group-like things in a more general setting, specifically in a braided monoidal category, one might take a look at the Hopf monoids in that setting.

And that's precisely what people do when they study "quantum groups", which John mentioned a bit ago.

Mike Shulman (Sep 03 2024 at 01:23):

Although there's a hidden dualization here, since co-homology is a contravariant functor, as are the functors that embed spaces in their generalizations used for quantum groups. But fortunately, the notion of Hopf monoid is self-dual!

John Baez (Sep 03 2024 at 01:26):

Oh, whoops, I hid that under the carpet by accident.

John Baez (Sep 03 2024 at 01:29):

By the way, @David Egolf, one reason that Hopf monoids are a bit different from group objects - even though they coincide in cartesian monoidal categories! - is that group objects can be defined by a Lawvere theory, while Hopf monoids cannot. The reason is that Lawvere theories describe algebraic gadgets with operations having a bunch of inputs and one output, while a Hopf monoid $H$ also involves a 'co-operation': that is, a thing $\Delta: H \to H \otimes H$ with one input and two outputs.

John Baez (Sep 03 2024 at 01:31):

There is a generalization of Lawvere theories that handles co-operations: they're called 'props'.

John Baez (Sep 03 2024 at 01:32):

So on your long list of "things to think about someday when there's nothing else to do", you can file Lawvere theories, operads and props right next to each other.

David Egolf (Sep 03 2024 at 17:39):

Very cool stuff! I look forward to (eventually) learning more about it!

Internalization is beautiful aesthetically - various interesting structures emerge by internalizing the same basic structure. But I hope that it can also be helpful in the study of very specific (and potentially less-studied) mathematical settings. (Such settings could arise when aiming to study a specific real-world system, for example). In such a setting, internalization could potentially help turbocharge the mathematical exploration process, by quickly giving some setting-specific structures to contemplate!