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Stream: deprecated: monoids

Topic: what is this construction called?

Paolo Capriotti (Jan 15 2021 at 09:11):

If $M$ is a monoid and $X$ is a right $M$-set, the set of functions $X \to M$ has a monoid structure, defined by $(fg)(x) = f(x) g(x f(x))$. Does this have a name? Can it be expressed more abstractly?

There is also a group version of this. If $G$ is a group, and $X$ is now a $G$-set, we can take the subset of those $f\colon X \to G$ such that the map $x \mapsto x f(x)$ is invertible. This is then a subgroup of the above monoid.

Morgan Rogers (he/him) (Jan 15 2021 at 10:33):

When you say "the set of functions", are you referring to arbitrary set functions or $M$-set homomorphisms?

Paolo Capriotti (Jan 15 2021 at 10:34):

Arbitrary functions

Morgan Rogers (he/him) (Jan 15 2021 at 10:39):

So why does it matter that $X$ is a right $M$-set?
If you just consider sets $X$, this is a consequence of the fact that $\mathrm{Mon}$ is self-enriched and we have a free-forgetful adjunction between it and $\mathrm{Set}$. I don't have a nice name for the phenomenon, though.

Paolo Capriotti (Jan 15 2021 at 10:39):

I'm not sure what you mean. It matters that $X$ is an $M$-set because the action appears in the definition of the multiplication.

Morgan Rogers (he/him) (Jan 15 2021 at 10:41):

:face_palm: I can normally read, I swear

Morgan Rogers (he/him) (Jan 15 2021 at 10:46):

I'll come back when I find something useful to say :relieved:

Morgan Rogers (he/him) (Jan 15 2021 at 10:52):

It feels like it should be somehow adjoint to a semi-direct product operation, but I can't quite formulate the setting in which that would be the case...

Paolo Capriotti (Jan 15 2021 at 10:56):

It is some kind of semi-direct product, but non-canonically. At least in the group case, which is ultimately what I'm interested in, it should be isomorphic to some kind of wreath product of stabilisers.

Morgan Rogers (he/him) (Jan 15 2021 at 11:30):

I wonder if something related to this construction of the semi-direct product as an adjoint might give you a setting where the monoids/groups you describe come out naturally.

Joe Moeller (Jan 15 2021 at 18:38):

Maybe the right thing to say is that Mon is Set-powered: you can take powers of a monoid where the exponent is a set.