I actually felt like the binary operation view is easier than a bunch of permutations! Simply because, you often want to think of it as combining two things, and the common examples are already written like binary operations (e.g. times, plus, function composition). Whereas I don’t have such intuitions to draw on for permutations, unless I translate that to “oh it’s a bijection” (which essentially gives the intuition for a group action).
There’s a generalization of Cayley’s theorem to category theory called Yoneda’s lemma, but I don’t understand it so don’t know how to apply it here.
I actually felt like the binary operation view is easier than a bunch of permutations! Simply because, you often want to think of it as combining two things, and the common examples are already written like binary operations (e.g. times, plus, function composition). Whereas I don’t have such intuitions to draw on for permutations, unless I translate that to “oh it’s a bijection” (which essentially gives the intuition for a group action).
There’s a generalization of Cayley’s theorem to category theory called Yoneda’s lemma, but I don’t understand it so don’t know how to apply it here.