# How to define equivariance in the monoid category?

Let $$S$$ be a semigroup and $$X$$ a set. A left $$S$$-action structure on $$X$$ is a semigroup arrow $$S\to \mathsf{Set}(X,X)$$.

Let $$M$$ be an abelian group and $$R$$ a ring. A left $$R$$-module structure on $$M$$ is a ring arrow $$R\to\mathsf{Ab}(M,M)$$.

In both cases we are defining actions of semigroup/monoid objects in a category $$\mathsf C$$ within the category of internal monoids $$\mathsf{Mon}(\mathsf C)$$.

Usually, equivariance is defined in the category $$\mathsf {C}$$ itself using diagrams in $$\mathsf {C}$$ involving the monoid structure arrows of the acting object. Can we define equivariance in $$\mathsf{Mon}(\mathsf {C})$$?

For instance, can we define $$R$$-linear arrows of $$R$$-modules in the category of rings? Perhaps in the commutative case?

• How are you planning to make an arbitrary ring into an R-module? – Pedro Tamaroff Oct 25 '18 at 15:08
• @PedroTamaroff I'm not sure I understand your question. I think one can apply the forgetful functor $U$ from rings (monoid objects in abelian groups) to abelian groups and use the structure maps to give a ring arrow $R\to \mathsf{Ab}(UR,UR)$. – Arrow Oct 25 '18 at 15:12

If the ambient category is monoidal closed, the usual equivariance conditions may be expressed with internal homs. This expression does not live in $$\mathsf{Mon}(\mathsf C)$$, but I thought it's still worth recording.
Let $$M$$ be an internal monoid and consider actions $$M\to \underline{\mathsf{C}}(A,A)$$ and $$M\to \underline{\mathsf{C}}(B,B)$$. Observe any arrow $$f:A\to B$$ in $$\mathsf C$$ induces morphisms by pre-composition and post-composition. This arrow is equivariant precisely if the following diagram commutes.
$$\require{AMScd} \begin{CD} M @>>> \underline{\mathsf{C}}(A,A)\\ @VVV @VV{f_\ast}V\\ \underline{\mathsf{C}}(B,B) @>>{f^\ast}> \underline{\mathsf{C}}(A,B) \end{CD}$$
However, this still takes place in the category $$\mathsf C$$ and not $$\mathsf{Mon}(\mathsf C)$$ since the bottom right corner has no monoid structure.