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Stream: learning: questions

Topic: Continuous group actions as functors

Bernd Losert (Feb 03 2024 at 14:26):

It is well known that every group action of a fixed group G on sets can be encoded as a functor {G} → Set, where {G} is the one-object-category version of G.

I am trying to work this out in the setting of continuous group actions. Now, I know this wont' work in Top since you require that the category be Cartesian closed, so let's use instead the category Conv of convergence spaces, which is Cartesian closed.

Also, it seems we need to work in an enriched setting: If G is a convergence group, then the one-object category {G} needs to be Conv-enriched. I don't think I need to enrich Conv though.

Is there anything else I need to do here? Also, any reference to this construction in the continuous setting would be greatly appreciated.

Morgan Rogers (he/him) (Feb 03 2024 at 16:40):

Bernd Losert said:

I am trying to work this out in the setting of continuous group actions. Now, I know this wont' work in Top since you require that the category be Cartesian closed, so let's use instead the category Conv of convergence spaces, which is Cartesian closed.

What is it that requires that the category to be cartesian closed?

Also, it seems we need to work in an enriched setting: If G is a convergence group, then the one-object category {G} needs to be Conv-enriched. I don't think I need to enrich Conv though.

Is there anything else I need to do here? Also, any reference to this construction in the continuous setting would be greatly appreciated.

A more concise version of this set-up is to consider group objects in Conv; you can directly apply the diagrammatic definition of group action that way (morphism $G \times X \to X$ satisfying some axioms for the interaction with the group structure morphisms).

Bernd Losert (Feb 03 2024 at 16:53):

What is it that requires that the category to be cartesian closed?

Because the functorial definition defines actions as morphisms of type G → (X → X) and you need uncurrying to turn this into G × X → X.

A more concise version of this set-up is to consider group objects...

Yes, but the goal here is to give a functorial representation.

Morgan Rogers (he/him) (Feb 03 2024 at 17:20):

Bernd Losert said:

What is it that requires that the category to be cartesian closed?

Because the functorial definition defines actions as morphisms of type G → (X → X) and you need uncurrying to turn this into G × X → X.

That makes these two things equivalent in a ccc, but you can still always use the latter if you merely have products (and this is what I normally consider the definition of a group action).

Morgan Rogers (he/him) (Feb 03 2024 at 17:22):

Ah I see you were already discussing something very similar here.

Morgan Rogers (he/him) (Feb 03 2024 at 17:26):

So I would consider @Kevin Arlin 's description of the Lawvere theory for monoid actions if you want a functorial representation. You can extract the category of actions of a specific monoid as a subcategory of that to avoid the problem you were encountering with the fixed monoid Lawvere theory where the monoid elements are discrete morphisms in the theory.

Bernd Losert (Feb 03 2024 at 17:31):

Yes, I could start using Lawvere theory, but I am still curious if the functorial representation {G} → Conv would work.

Morgan Rogers (he/him) (Feb 03 2024 at 17:32):

You can express it as an internal functor category, but cashing that out turns everything into products anyhow.

Bernd Losert (Feb 03 2024 at 17:33):

Do you happen to know of any paper or book that works through it?

Morgan Rogers (he/him) (Feb 03 2024 at 17:34):

I think Part C of Sketches of an Elephant should have you covered, although I would need to look into it to send you to the right section.

Morgan Rogers (he/him) (Feb 03 2024 at 17:35):

(it does it in the context of toposes but only finite limits are involved)

Morgan Rogers (he/him) (Feb 03 2024 at 17:39):

To make the Lawvere theory idea explicit, let $\mathcal{C}_{M}$ be the Lawvere theory of monoids, $\mathcal{C}_{M,X}$ the Lawvere theory of a generic monoid action. There is a functor $Z: \mathcal{C}_{M} \to \mathcal{C}_{M,X}$ picking out the monoid part (it's actually a full subcategory if I'm not mistaken), which induces a functor $Z^*: [\mathcal{C}_{M,X}, \mathcal{D}] \to [\mathcal{C}_{M}, \mathcal{D}]$ from a given action to its underlying monoid.
For a given monoid $M$ in $\mathcal{D}$ there is a functor $\ulcorner M \urcorner: 1 \to [\mathcal{C}_{M}, \mathcal{D}]$ identifying it, and actions of that monoid are the pullback of $Z^*$ along $\ulcorner M \urcorner$.

Morgan Rogers (he/him) (Feb 03 2024 at 17:47):

(it occurred to me after the fact that $[X,Y]$ in the above should be interpreted as product-preserving functors $X \to Y$).

Bernd Losert (Feb 03 2024 at 17:53):

I am going to need some time to parse that.

I think Part C of Sketches of an Elephant should have you covered, although I would need to look into it to send you to the right section

Doesn't Johnstone only deal with groups with a discrete topology?

Bernd Losert (Feb 03 2024 at 17:59):

Oops, not the group but the space itself.

Morgan Rogers (he/him) (Feb 03 2024 at 19:42):

I mean this is a special case of internal presheaves/copresheaves

Chris Grossack (they/them) (Feb 04 2024 at 17:29):

Here's an approach that seems closer to what you want (though it's possible you can get this from Morgan's comments, I haven't checked).

Let $\mathcal{C}$ be a cartesian closed category (as you were expecting) and $G$ a group object in $\mathcal{C}$. Then there's an associated one-object $\mathcal{C}$-enriched category $\mathrm{B}G$, where we take a single object $\star$ and define $\text{Hom}(\star,\star) = G$.

Chris Grossack (they/them) (Feb 04 2024 at 17:30):

Now we have two ways of defining an action of $G$ on an object $X \in \mathcal{C}$.

1. We could look at an arrow $\alpha : G \times X \to X$ in $\mathcal{C}$ satisfying the usual axioms
2. We could look at a $\mathcal{C}$-enriched functor $F : \mathsf{B}G \to \mathcal{C}$ sending $\star \mapsto X$.

In definition (2), we're crucially using the cartesian closed-ness in order to view $\mathcal{C}$ as being enriched over itself.

Chris Grossack (they/them) (Feb 04 2024 at 17:35):

The key way of translating back and forth between these two perspectives is to remember what a "$\mathcal{C}$-enriched functor" is. It has to send objects to objects (for us we send $\star \mapsto X$) and hom-objects to hom-objects. That is, there had better be a $\mathcal{C}$-arrow $\text{Hom}(\star, \star) \to \text{Hom}(F\star, F\star)$.

But $\text{Hom}(X,X) = \text{Hom}(F\star, F\star)$ is the $\mathcal{C}$-object given by the internal hom $[X,X]$. So taken together a $\mathcal{C}$-enriched functor $\mathsf{B}G \to \mathcal{C}$ is given by the data of an object $X \in \mathcal{C}$ and a $\mathcal{C}$-arrow $G \to [X,X]$.

Chris Grossack (they/them) (Feb 04 2024 at 17:36):

As you intuited, this is exactly the same story as the $\mathsf{Set}$-enriched case, since the data of such an enriched functor uncurries to exactly the data of an internal group action $G \times X \to X$ by CCC-ness of $\mathcal{C}$.

Now you can work in your favorite convenient category of topological spaces in order to play this game with continuous group actions. ^_^

Bernd Losert (Feb 04 2024 at 18:32):

Chris Grossack (they/them) said:

Here's an approach that seems closer to what you want (though it's possible you can get this from Morgan's comments, I haven't checked).

Thanks. This what I had in mind. If you have reference to this construction from a paper or a book, I would appreciate it.

Chris Grossack (they/them) (Feb 04 2024 at 19:05):

Hm... I don't know of a place this is written down, unfortunately. It's the kind of thing that's "obvious" to people familiar enough with enriched/internal categories and the usual relationship between groups and 1-object groupoids... One of the most frustrating parts of learning category theory is dealing with the number of interesting things that are stuck inside someone's head because they felt it was "too obvious to write down" (whether or not it's actually obvious to anyone else, lol)

Chris Grossack (they/them) (Feb 04 2024 at 19:06):

If you give me a minute, I can at least write up a blog post going into this construction in more detail ^_^. I'll link it once it's up.

Bernd Losert (Feb 04 2024 at 19:28):

Chris Grossack (they/them) said:

If you give me a minute, I can at least write up a blog post going into this construction in more detail ^_^. I'll link it once it's up.

Wow, thanks! By the way, I am writing a paper where I would like to use this kind of construction and would prefer having some literature to refer to, rather than working it out in the paper. Your blog post would be much appreciated.

Morgan Rogers (he/him) (Feb 05 2024 at 09:26):

Chris Grossack (they/them) said:

Here's an approach that seems closer to what you want (though it's possible you can get this from Morgan's comments, I haven't checked).

Yep, you can! For internal categories, see Section B2.3 of Sketches of an Elephant (for internal monoids, you take $C_0 = 1$, the terminal object; for groups, they're a special case of groupoids Example B2.3.12). The definition you want is Definition 2.3.11, probably in the special case where $mathbb{D} = \mathbb{S}$ is the base category with its canonical indexing... This section is not so easy to read because it is expressed in the language of indexed categories developed in the preceding section, but it might give some useful insights, and indexed categories are at least "functorial" in a sense that you might appreciate.

Chris Grossack (they/them) (Feb 06 2024 at 00:53):

(I haven't forgotten about this, but I have gotten distracted. I'll still let you know when the post is up!)

Chris Grossack (they/them) (Feb 18 2024 at 23:37):

Ok, sorry for the wait @Bernd Losert!

Once I started writing this post I wanted to turn it into something much more ambitious than the initial idea, haha. Hopefully it's helpful, or at the very least interesting!

https://grossack.site/2024/02/18/internal-enriched-groupoids