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Morgan Rogers (he/him) (Apr 19 2023 at 12:48):The adjective "equivariant" indicates a property or object which is unaffected by a particular group action. Is there a related term for invariance under the action of a monoid which is not necessarily a group, or is it appropriate to extend the meaning of "equivariant"?
Cole Comfort (Apr 19 2023 at 13:35):Morgan Rogers (he/him) said:
The adjective "equivariant" indicates a property or object which is unaffected by a particular group action. Is there a related term for invariance under the action of a monoid which is not necessarily a group, or is it appropriate to extend the meaning of "equivariant"?
Maybe I am misinterpreting your question, but I think I would just call it a module.
Morgan Rogers (he/him) (Apr 19 2023 at 13:37):I'm not asking for the name of an action of a monoid (that discussion has already happened here somewhere :rolling_on_the_floor_laughing: ) but rather the adjective which describes "being invariant under [a given action of a given monoid]".
Morgan Rogers (he/him) (Apr 19 2023 at 13:46):Thinking about it, I'm pretty sure that extending the meaning of "equivariant" seems not only fine but already a commonly done thing, so I should refine my question to: if I use the word "equivariant" in a context where the action is only implicitly specified, is there a risk of a reader erroneously assuming that I'm referring to a group action?
John Baez (Apr 19 2023 at 15:11):There's a risk, because I've never heard anyone say "equivariant" for a monoid action. So if were doing this I'd define equivariance for a monoid action and say I'll often be using the word in this generalized sense. (I can't think of a better word.)
John Baez (Apr 19 2023 at 15:13):Oh, I guess another word is "natural", since an equivariant map between monoid actions is a special case of a natural transformation between functors. But this word could easily have the wrong connotations!
Morgan Rogers (he/him) (Apr 19 2023 at 15:24):For context, I'm talking about an extension of a theory (in the logic sense), and calling it a "natural" extension, or the "natural theory of " (for a model of the original theory, not a monoid!) doesn't convey the relevant relationship. On the other hand, because automorphisms of models are so much more commonly considered than more general endomorphisms, equivariance might indeed be confusing.
Mike Shulman (Apr 19 2023 at 15:58):I don't think of "equivariant" as referring to something that is unaffected by a group action. I would call that "invariant". To me "equivariant" refers to something that varies in a specified way under a group action, e.g. a -set.
Nathanael Arkor (Apr 19 2023 at 15:59):Wood uses "equivariant" in the context of monoids/monads in Proarrows II.
John Baez (Apr 19 2023 at 18:22):Mike Shulman said:
I don't think of "equivariant" as referring to something that is unaffected by a group action. I would call that "invariant". To me "equivariant" refers to something that varies in a specified way under a group action, e.g. a -set.
I only use "equivariant" when referring to map between G-sets: f is equivariant if it's invariant under the action of G on such maps.
John Baez (Apr 19 2023 at 18:23):Or in other words, thinking of a G-set as a functor from BG to Set, an equivariant map between G-sets is precisely a natural transformation between such functors.
John Baez (Apr 19 2023 at 18:23):I wouldn't say that a G-set is equivariant.
John Baez (Apr 19 2023 at 18:25):I would often say that elements of certain G-sets are "covariant", e.g. a "covariant tensor". That's just a way of saying that we have a covariant functor from BG to Set. Similarly for "contravariant tensor".
Mike Shulman (Apr 19 2023 at 18:26):Yes, I agree about equivariant maps. But what about equivariant homotopy theory?
Mike Shulman (Apr 19 2023 at 18:27):Maybe you're right and "covariant" is the word for what I was thinking of, though.
John Baez (Apr 19 2023 at 18:28):Just checking, I see that Wikipedia agrees with me... here's how the article "Equivariance" starts:
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.
John Baez (Apr 19 2023 at 18:30):Mike Shulman said:
Yes, I agree about equivariant maps. But what about equivariant homotopy theory?
I don't know exactly why those folks call it that. It definitely sounds impressive, you gotta admit that.
Todd Trimble (Jul 31 2023 at 18:32):John Baez said:
Mike Shulman said:
I don't think of "equivariant" as referring to something that is unaffected by a group action. I would call that "invariant". To me "equivariant" refers to something that varies in a specified way under a group action, e.g. a -set.
I only use "equivariant" when referring to map between G-sets: f is equivariant if it's invariant under the action of G on such maps.
Meaning, then, invariant under the evident conjugation action.