Terminological conventions regarding group actions

Question

Suppose:

• $G$ is a group
• $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra

What conventions surround the phrase: "action of $G$ on $X$"?

In particular, does this mean:

• a homomorphism $G \rightarrow \mathrm{Aut}_{\mathrm{Mod}(T)}(X)$, or
• a homomorphism $G \rightarrow \mathrm{Aut}_{\mathbf{Set}}(X)$?

Answer 1

Don't take my answer as a reference, but from my experience, notably in French, "action of $G$ on $X$" usually refers to a set-action (i.e. an homomorphism $G \to \operatorname{Aut}_{\mathbf{Set}}(X)$) while "representation of $G$ on/in $X$" refers to an homomorphism $G \to \operatorname{Aut}_{\operatorname{Mod}(T)}(X)$.

For example, one talks about "linear representations" or "representations in Lie algebra". If you're writing something though, the better is still to precisely state your usage.