modules over monoids：trouble in a specific example

Question

I‘m recently learning about monoidal categories and the monoid & module object in monoidal category. After reading the definitions, I hope to give a specific example about $kG$-mod category.($kG$ is the group algebra)

With the normal $\otimes_k$, I think the category of $kG$-modules can be a monoidal category. Then, what is the monoid object and module object in the monoidal category?

At first, I thought $kG$-mod can be a monoid, but there seems to be something wrong when I put $G$-action to the commutative diagram of definition of monoids. When I try to apply $G$-action by conjugation, it seems compatible.

I will be appreciate if you can help me and give me some specific information about the monoid in $kG$-mod monoidal category, and the module over the monoids. I have no ideal about the module object in this case. Thanks advance.

Answer 1

A monoid object in this category of $k[G]$-modules is by definition a $k$-module $A$ with a $k$-linear and $G$-equivariant associative product map $A\otimes_k A\to A$ and a unit $k[G]\to A$.

You can easily see that this just means $A$ has an algebra structure which is $G$-equivariant, meaning that $g(a\cdot b)=g(a)\cdot g(b)$ for $g\in G$ and $a,b\in A$.

And a module over this monoid object is just going to be an $A$-module $M$ with $G$-equivariant action (ie $g(a\cdot m)= g(a)\cdot g(m)$ for $g\in G$, $a\in A$ and $m\in M$).

Answer 2

More generally, if $(\mathcal{C},\otimes)$ is any monoidal category and $\mathcal{J}$ is a small category, the functor category $[\mathcal{J},\mathcal{C}]$ carries a monoidal structure as well. Everything is defined pointwise. $$(F \otimes G)(i) := F(i) \otimes G(i).$$ Exactly this is happening here with $\mathcal{J} = G$ (considered as a one-object category) and $(\mathcal{C},\otimes) = (\mathbf{Vect}_K,\otimes_K)$.

Since the operations are pointwise, it is not surprising at all that we have an isomorphism of categories $$\mathbf{Mon}([\mathcal{J},\mathcal{C}],\otimes) \cong [\mathcal{J},\mathbf{Mon}(\mathcal{C},\otimes)].$$ Also, if $\mathbf{Mod}(\mathcal{C},\otimes)$ denotes the category of pairs $(A,M)$ where $A$ is a monoid object and $M$ is an $A$-module in $\mathcal{C}$, then we have an isomorpism $$\mathbf{Mod}([\mathcal{J},\mathcal{C}],\otimes) \cong [\mathcal{J},\mathbf{Mod}(\mathcal{C},\otimes)].$$ For the example $\mathcal{J}=G$ we arrive at the description in the other answer.

Just to give another example: Let $\mathcal{J} = (\bullet \to \bullet \to \cdots)$. Then we have a monoidal category whose objects are sequences $X_0 \to X_1 \to \cdots$, and the tensor product of two such sequences is $X_0 \otimes Y_0 \to X_1 \otimes Y_1 \to \cdots$. A monoid in that monoidal category is a sequence $X_0 \to X_1 \to \cdots$ of monoids and monoid morphisms in $\mathcal{C}$.

As you see, different choices of the shape $\mathcal{J}$ yield different categories, but the pattern is always the same.