I'm having trouble with a definition.
We're working in the category of monoids. Take $A\in \mathfrak{Mon}$ and define a module over $A$ to be a set $M$ with an action: $A\times M \rightarrow M$ such that the pair $(a,m)\mapsto am$. The question is: how do I interprit $am$? A set doesn't usually come with an operation defined on it.
Sorry for the silly question, but I really can't make any sense of it. Thank you
The specific function $f:A\times M\to M$ is the action. That is, we define the expression $am$ to mean the element $f(a,m)\in M$.
Note also that the definition should include the requirement that $f$ is compatible with the monoid operation of $A$; i.e., you need $$f(a_1,f(a_2,m))=f(a_1a_2,m)\qquad f(1,m)=m$$ or, written with the more concise notation, $$a_1(a_2m)=(a_1a_2)m\qquad 1m=m$$ To give another approach, you can view an action as a function $\varphi:A\to \mathrm{End}(M)$ that is a monoid homomorphism, where $\mathrm{End}(M)$ is the monoid of all functions from the set $M$ to itself. The correspondence of this notion with the other notion of action is $$\varphi(a)\text{ is the function from $M$ to $M$ defined by }\quad \varphi(a)(m)=f(a,m)$$