Algebraic structures over groups and monoids [closed]

Solution 1:

Vector spaces over fields are simply special cases of modules over rings, and modules over rings are in turn special examples of modules over monoid objects.

Definition: If $(C,\otimes,I)$ is a monoidal category, and $(X,\mu : X\otimes X\to X, e : I\to X)\in C$ is a monoid in $C$, a left module over $X$ is an object $N\in C$ together with a morphism $\rho : X\otimes N\to N$ such that $$ \rho\circ(\operatorname{id}_X\otimes\rho) = \rho\circ(\mu\otimes\operatorname{id}_N) $$ as morphisms $X\otimes X\otimes N\to N,$ and $\rho\circ(e\otimes\operatorname{id}_N) : I\otimes N\to N$ is equal to the unitor isomorphism.

Example: If $(C,\otimes, I) = (\mathsf{Ab},\otimes,\Bbb{Z})$ is the category of abelian groups with its usual tensor product, then a monoid $\mathcal{R}$ in $\mathsf{Ab}$ is equivalent to the data of a ring $R$, and the data of a left module over the monoid $\mathcal{R}$ is equivalent to the usual notion of a left module over the ring $R$. If $R = k$ is moreover a field, then the module is a $k$-vector space.

In this generality, monoids and groups may be thought of as special cases of monoid objects, in the monoidal category $(\mathsf{Set},\times,\{\ast\}).$ Unwinding the definitions above, a left module over a group or monoid $M$ is a set $S$ together with a function \begin{align*} -\cdot- : M\times S&\to S\\ (m,s)&\mapsto m\cdot s \end{align*} such that for all $m,n\in M$ and $s\in S,$ we have $m\cdot (n\cdot s) = (mn)\cdot s$ (this is the first condition in the definition), and if $e\in M$ is the identity, then $e\cdot s = s$ (this is the second condition).

If $M$ is a group, this is no more nor less than the notion of a group action on the set $S.$