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Stream: practice: terminology & notation

Topic: actions, modules

Dylan Braithwaite (Oct 06 2022 at 13:35):

How would you refer to the object being acted on, other than as a module? I get 'action' as a name for the morphism relating the monoid to the object, but referring to all of the data as an action seems off to me

Dylan Braithwaite (Oct 06 2022 at 13:36):

Though I agree it's annoying having to switch between 'action', 'module', and sometimes even 'representation' when searching for things like this

Morgan Rogers (he/him) (Oct 06 2022 at 13:38):

But a module is originally an abelian group equipped with the action of a ring (+axioms). The thing being called a "module object" is not a categorification of that, since it lacks any enforcement of the abelian group structure.

Morgan Rogers (he/him) (Oct 06 2022 at 13:40):

Dylan Braithwaite said:

I get 'action' as a name for the morphism relating the monoid to the object, but referring to all of the data as an action seems off to me

But this is a fair point, there is some ambiguity there which can be cumbersome to work around in writing...

Dylan Braithwaite (Oct 06 2022 at 13:41):

Is a module in the original sense not just a module object in $\mathbf{Ab}$?

Morgan Rogers (he/him) (Oct 06 2022 at 13:42):

Yes, and the abelian structure is pretty strongly tied into that. It seems strange to call a group acting on a topological space a "module" to me, for instance

Morgan Rogers (he/him) (Oct 06 2022 at 13:47):

A ring is "just" a monoid in $\mathbf{Ab}$, but it would be confusing to use the same name for both! (although that confusion doesn't bother everyone, based on @Joe Moeller 's anecdote :grimacing: )

Tobias Schmude (Oct 06 2022 at 13:48):

I'd argue that using the same name isn't only justified when generalizing from $\mathrm{Set}$, but also if we're motivated by some other sufficiently well known special case. As long as we don't have better names anyway.

Tobias Schmude (Oct 06 2022 at 13:48):

If we'd like to invent new names (which I don't really think is the right way), what could we do? When a monoid object $M$ acts on another object $C$, $M$ is the "actor", but what is $C$? The "stage" maybe? :upside_down:

Mike Shulman (Oct 06 2022 at 16:34):

For better or for worse, I think "module" now has a long history of being generalized away from the abelian context to mean any action of a monoid object. To those who haven't encountered this generalization before, it may seem weird, but in category theory at least I think it's pretty common.

Mike Shulman (Oct 06 2022 at 16:36):

I don't think "ring" has been nearly so widely generalized to refer to arbitrary monoids, probably partly because we already have the perfectly good term "monoid" to generalize, whereas there's no similarly good word for a — well, module object — in Set that could have been generalized alongside "monoid".

Mike Shulman (Oct 06 2022 at 16:38):

Also with "ring object" there could be more ambiguity, as you pointed out at least in a cartesian monoidal category a "ring object" could mean a [[ring object]]. But if you know that the thing you're talking about modules for is just a monoid object and not a ring object, then it's clear that your "module object" should just mean an action of that monoid object and not something with an extra addition operation.

Tobias Schmude (Oct 06 2022 at 16:48):

Are there nomenclature problems that we would encounter by just saying "action" and not having a separate name for the underlying object? We're letting all kinds of forgetful functors fall under the table in mathematical nomenclature anyway, why not this one? After all that's what we're doing in the $\mathrm{Ab}$-enriched case with the notion of a module as well.

Tobias Schmude (Oct 06 2022 at 16:48):

I'm not aware of a situation where the underlying object is of significance without the specified action. Maybe when we consider actions of different monoid objects on a single object? Do people do that?

Nathanael Arkor (Oct 06 2022 at 16:53):

Mike Shulman said:

I don't think "ring" has been nearly so widely generalized to refer to arbitrary monoids, probably partly because we already have the perfectly good term "monoid" to generalize, whereas there's no similarly good word for a — well, module object — in Set that could have been generalized alongside "monoid".

Don't both the terms "monoid" and "action" predate category theory, so that there should be no good reason to use "monoid" for a more general concept, and not also "action"? Monoids may be less commonly encountered than rings in "traditional" mathematical education, but certainly groups are commonplace, and one speaks of group actions (which are, after all, closer to monoids than rings) rather than group modules. So I do find it somewhat surprising to find uses of "module object" in this generality.

Nathanael Arkor (Oct 06 2022 at 16:54):

Tobias Schmude said:

Are there nomenclature problems that we would encounter by just saying "action" and not having a separate name for the underlying object? We're letting all kinds of forgetful functors fall under the table in mathematical nomenclature anyway, why not this one? After all that's what we're doing in the $\mathrm{Ab}$-enriched case with the notion of a module as well.

Doesn't "underlying object" serve as a sufficient separate name?

Mike Shulman (Oct 06 2022 at 16:55):

Yes, "action" is used in this generality as well. But it sounds weird to me to refer to the object being acted on as "an action".

Mike Shulman (Oct 06 2022 at 16:56):

In my head (and, I suspect, those of many), the word "action" is analogous to "group structure", not to "group". You put a group structure on a set to get a group. You put an action on a set to get a... module.

Tobias Schmude (Oct 06 2022 at 16:58):

Nathanael Arkor said:

Tobias Schmude said:

Are there nomenclature problems that we would encounter by just saying "action" and not having a separate name for the underlying object? We're letting all kinds of forgetful functors fall under the table in mathematical nomenclature anyway, why not this one? After all that's what we're doing in the $\mathrm{Ab}$-enriched case with the notion of a module as well.

Doesn't "underlying object" serve as a sufficient separate name?

In a sense, yes. But it's rather bulky, and more like an informal description rather than a name.

Morgan Rogers (he/him) (Oct 06 2022 at 17:00):

I shouldn't have let this get derailed into a nomenclature discussion, I do actually need references for places that people have studied [whatever you'd like to call these] :sweat_smile:

Tobias Schmude (Oct 06 2022 at 17:07):

You're right, sorry. There's still something I'd like to say about this though, so I'll make a separate topic. Any idea where that might fit best? general: off-topic?

Nathanael Arkor (Oct 06 2022 at 17:09):

Perhaps we could use an entire stream relating to terminology :big_smile:

Tobias Schmude (Oct 06 2022 at 17:11):

Nathanael Arkor said:

Perhaps we could use an entire stream relating to terminology :big_smile:

Actually a very good idea :grinning_face_with_smiling_eyes:

Morgan Rogers (he/him) (Oct 06 2022 at 17:18):

Created from this discussion

Morgan Rogers (he/him) (Oct 06 2022 at 17:22):

Tobias Schmude said:

Nathanael Arkor said:

Perhaps we could use an entire stream relating to terminology :big_smile:

Actually a very good idea :grinning_face_with_smiling_eyes:

(Done!)

Morgan Rogers (he/him) (Oct 06 2022 at 17:26):

Since I can weigh in without derailing the discussion in the other stream now, I think act is a valid option for the entire structure (of which the "action" is the morphism part), and that underlying object is no more cumbersome than the "underlying set" of an ordinary group.

Nathanael Arkor (Oct 06 2022 at 17:31):

Given that people say that monoids have actions, it sounds odd to my ears to talk of "acts" rather than actions.

Morgan Rogers (he/him) (Oct 06 2022 at 17:34):

It seemed strange to me too some years ago, enough that I failed to find papers which contained results I'd reproven about acts in Set :silence: but it is one of the standard terms: https://www.degruyter.com/document/doi/10.1515/9783110812909/html?lang=en

Nathanael Arkor (Oct 06 2022 at 17:40):

Interesting, I hadn't encountered that term before.

Mike Shulman (Oct 06 2022 at 17:43):

Maybe that explains where people came up with "actegory".

Mike Shulman (Oct 06 2022 at 17:43):

I have to say I don't like either of them, though...

Tobias Schmude (Oct 06 2022 at 18:34):

Mike Shulman said:

In my head (and, I suspect, those of many), the word "action" is analogous to "group structure", not to "group". You put a group structure on a set to get a group. You put an action on a set to get a... module.

So to phrase it type theoretically:
If we have a dependent type, and a name <foo> for its $\Sigma$-type, a usual name for the dependent type itself is <foo>-structure (on a specified object).
Are there instances in mathematical terminology where it's the other way around: we have a name for the dependent type, and the name for the $\Sigma$-type is constructed from that in a way that we might generalize?

Beppe Metere (Oct 06 2022 at 20:23):

Well, I am not sure if this concerns terminology or notation. When you fix a group $G$ and you want to consider the category of sets acted by $G$, you can refer to it as the category $G$-$\mathbf{Set}$: objects are called $G$-sets and maps are equivariant maps. Of course you can extend this and consider the category of $\mathbf{Grp}$-$\mathbf{Set}$, where the acting group is not fixed, and maps are defined accordingly. I am not sure this terminology is common with monoid actions, but it does seem ok to me to call $M$-set a set acted by a monoid.

Beppe Metere (Oct 06 2022 at 20:25):

In the same fashion, if an object $X$ is acted by an internal monoid $M$, it can be named an $M$-object.

Beppe Metere (Oct 06 2022 at 20:32):

Concerning the use of the term actor for the acting guy, it is worth to observe that there is a fairly established convention in naming actor the object representing (internal) actions, as it is the case for instance with the group of automorphisms of a group (for groups acting on groups) and the Lie algebra of derivations (for Lie algebras acting on Lie algebras). I am not saying this is a clever choice of terminology, but only reporting that it exists.

Beppe Metere (Oct 06 2022 at 20:37):

Last, everybody knows there is a use of the term module - more precisely Beck module - to refer to abelian group objects in a slice of a category with pullbacks. This makes the abelianess embedded in the definition. But this does not causes clashes in terminology as far as the term Beck is mentioned.

Matteo Capucci (he/him) (Oct 07 2022 at 17:51):

Related, but not equivalent, [[torsor]]

John Baez (Oct 09 2022 at 11:53):

Tobias Schmude said:

If we'd like to invent new names (which I don't really think is the right way), what could we do? When a monoid object $M$ acts on another object $C$, $M$ is the "actor", but what is $C$? The "stage" maybe? :upside_down:

I don't have any special word for this, but I just wanted to say: instead of saying

"$C$ is an action of $M$"

(which I agree sounds weird: for me the action is the map $\alpha : M \otimes C \to C$), I say

"$C$ is acted on by $M$",

which is just as quick.

John Baez (Oct 09 2022 at 11:55):

You might still want some shorter way to say

"What is the object that is acted on?"

like maybe

"What is the actee?"

but frankly I've managed to live to the age of 61 without needing any noun like "actee", and I'm hoping to hold out for the rest of my life.

Oscar Cunningham (Oct 09 2022 at 13:19):

By analogy to the term $G$-set for actions of a group, $C$ could be an $M$-object.

dusko (Oct 10 2022 at 09:08):

i think meerkats have highly standardized terminology. when the guy on the lookout screams: "Eagle", everyone runs down. when he screams: "Snake", everyone runs up. nonstandard terminology would be dangerous.

people probably started making sentences when they realized that they can never quite express what they have to say by words alone. so they approximate it by one sentence, and then stop, and then from the other direction in another sentence. and then mathematicians, when they need to manipulate fixed meanings started using disposable terminology $x$ or $\Theta$. the price of the precise local denotations is that the language is very primitive. (following the double articulation of natural language, sumerans had a doubly articulated numeral system. but we "progressed" to our flat numeral system, where a point is approximated at infinity, thank you very much.)

i think the terminology of category theory should consist of diagrams, not words.

Morgan Rogers (he/him) (Oct 10 2022 at 10:31):

@dusko how do you explain what the constituents in the diagrams mean?

dusko (Oct 10 2022 at 11:06):

Morgan Rogers (he/him) said:

dusko how do you explain what the constituents in the diagrams mean?

oh i missed that problem. who will ever convince students that pythagoras' theorem is true without telling them that the big side of the right triangle is called hypothenuse and that the litle ones are catheters.

Morgan Rogers (he/him) (Oct 10 2022 at 11:30):

You still have to tell them that the letters in the equation refer to the lengths of the sides of the triangle somehow. For Pythagoras you can draw a triangle and put in some suggestive arrows, but how do I tell you what the objects of the category I'm working with are supposed to be without words?

dusko (Oct 10 2022 at 11:46):

Morgan Rogers (he/him) said:

You still have to tell them that the letters in the equation refer to the lengths of the sides of the triangle somehow. For Pythagoras you can draw a triangle and put in some suggestive arrows, but how do I tell you what the objects of the category I'm working with are supposed to be without words?

and how could they ever get the concept of length from the picture alone, without me telling them the word "length".

sorry, i am kidding obviously. i am not saying that we should be proving things in silence. words help. even music would help. but imagining that the choice and standardization of words fix their meanings takes us back to early XIX century philology. in the meantime, how words evolve, and how they never hold the same meaning for too long in any narrative, these things have been widely studied. but for some reason, people like to form communities where belonging is tested through compliance. that is fine for many things, but it hasn't worked very well for category theory.

Morgan Rogers (he/him) (Oct 10 2022 at 11:57):

It's the converse, really: a well-chosen name can give you a lot of intuition for a concept in advance of its formal definition by situating it in relation to existing/previously established concepts. Some level of standardisation can reduce ambiguity, too. The real reason that terminology has been so tricky in CT is that theorists have generalized the same concepts in numerous directions, and each direction has inherited related but not necessarily compatible naming conventions.

Morgan Rogers (he/him) (Oct 10 2022 at 11:59):

That's why it's good to have a space like this to litigate terms (rather than dismissing the need to name things...) so that one can be aware of the different existing associations in making their choices.

John Baez (Oct 10 2022 at 12:14):

"Litigating" terms makes it sound like a legal affair.

I hereby accuse you of misusing the term "cartesian category"!

John Baez (Oct 10 2022 at 12:15):

So I don't like that particular way of thinking about it - but mathematicians can't resist discussing terminology, since we need to understand terminology to know what people are talking about.

Zhen Lin Low (Oct 10 2022 at 13:44):

Confucius said, "if names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success." This line of thought goes back a long long way.

Matteo Capucci (he/him) (Oct 10 2022 at 15:34):

dusko said:

i think meerkats have highly standardized terminology. when the guy on the lookout screams: "Eagle", everyone runs down. when he screams: "Snake", everyone runs up. nonstandard terminology would be dangerous.

i do indeed run away when someone shouts 'actegory' :laughing: