(no subject)

[timelets]

Usually the key aspect of an action of some X$X$ is that X$X$ itself carries an algebraic structure, such as being a group (or just a monoid) or being a ring or an associative algebra, which is also possessed by YY$Y^Y$ and preserved by the curried action actˆ$\widehat{act}$. Note that if Y$Y$ is any set then YY$Y^Y$ is a monoid,

https://ncatlab.org/nlab/show/action

also see Lawvere, 1986

"Historically the notion of monoid (or of group in particular) was abstracted from the actions, a pivotally important abstraction since as soon as a particular action is constructed or noticed, the demands of learning, development, and use mutate it into: 1) other actions on the same object, 2) actions on other related objects, and 3) actions of related monoids. "

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MxA-> A

UxA -> A