Change of monoid functors

One notable aspect of topoi of monoid actions is that they are always defined over a single ground monoid $M$. This can be a significant limitation, if we want to change from one monoid to another. Fortunately, we can get around this by defining a functor which maps one topos of monoid actions to another.

We start by defining $act : Mon \to Cat$ to be a topoi-valued contravariant functor. This maps each monoid object $M$ to its topoi of monoid actions $act(M)$. Furthermore, it maps each monoid homomorphism $f : L \to M$ to the change of monoid functor $act(f) : act(M) \to act(L)$.

Let $(S,\alpha)$ be an M-set with action $\alpha : M \times S \to S$. Then the change of monoid functor $act(f)$ maps this M-set $(S,\alpha)$ into an L-set $(S,\beta)$ with a L-action $\beta : L \times S \to S$ defined by $\beta(l,s) = \alpha(f(l),s)$. In other words, $act(f)$ produces an L-set with the same underlying set as $S$ but with an L-action defined by the M-action determined by the output of $f$ on an $l$ argument.

There are multiple different approaches to dealing with monoid actions. If we define an M-set by a monoid homomorphism $\alpha : M \to Sets$ then we can produce a new monoid homomorphism by the input action of $f : L \to M$ to get $\alpha \circ f : L \to Sets$. As $\alpha \circ f : L \to Sets$ is another monoid action, it is the output of the change of monoid functor $act(f)$ which changes the monoid action from the topoi of M-actions to the topoi of L-actions.

This fully defines the effect of the change of monoid functor $act(f)$ on objects, and in the case of monoids it produces the same underlying function of L-sets it had for M-sets. All that remains then is to demonstrate that equivariance is preserved by the effect of the change of monoid functor $act(f)$. This we can demonstrate by letting $g: S \to T$ be an equivariant map of M-sets $S$ and $T$. We will show that $act(f)(g) : S \to T$ is an equivariant map of $L$-sets.

Let $g'$ be equal to $act(f)(g)$. Then $g' : act(f)(S) \to act(f)(T)$ maps an L-set $S$ to an L-set $T$ by the exact same underlying function as $g$. We will show that $g'$ is equivariant. Let $l$ be an any element of the monoid $L$ and let $x$ be an element of the input set $S$. Then $lx$ is the $L$ action of $l$ on $x$, which by definition is the M-action of $f(l)$ on $x$, so since $g$ and $g'$ coincide, we can change $g'(lx)$ to $g(f(l)x)$. Then by equivariance of $g$ we can pull out $f(l)$. Then by changing the M-action of $f(l)$ back into the $l$ action we get $lg'(x)$. This demonstrates $g'(lx) = lg'(x)$ which means that $g'$ is equivariant. $$g'(l x)$$ $$= g( f(l) x)$$ $$= f(l) g(x)$$ $$= l g'(x)$$ If we sum all of this up, we get that $act : Mon \to Cat$ is a valid functor from the category of monoids to the category of categories which produces topoi from monoids and functors between topoi of monoid actions from monoid homomorphisms. This is similar to the change of topology functors of topoi of sheaves defined by continuous maps of topological spaces [1].

Topoi of monoid actions as concrete categories:
Let $\{1\}$ be the trivial monoid with a single element, which sends each element back to itself. Then the topos of actions of this monoid is equal to the topos of sets. This demonstrates that the topos $Sets$ is a topos of monoid actions. If we have another monoid $M$ then we can get a change of monoid functor associated to the embededing of the monoid homomorphism $f: {1} \to M$ which is the functor $act(M) \to Sets$.

This changes the monoid of a topoi of monoid actions from $M$ to the trivial monoid. In the process, this makes $act(M)$ into a concrete category. Therefore, each topoi of monoid actions is a concrete category, and the faithful concretization functor is a change of monoid functor.

Embedding sets into topoi of monoid actions:
If we have a monoid homomorphism $f : M \to \{1\}$ which maps each element of a given monoid to the single element of a trivial monoid then by the change of monoid functor this produces a functor from the topos $Sets$ to the topos of $M$-actions. This maps each set into the M-set in which each M-action is trivial and has no effect.

If we define an element of a topos by an element morphism $1 \to X$, then we see that the elements of a concrete topos don't need to coincide with their underlying set of elements. In the case in which they do coincide in an object, then that object is a set-like object.

In the case of topoi of monoid actions, the difference between the elements of an object of the topos and the elements of the underlying set, is the difference between fixed points of the action and the set of all elements. The set-like objects of a topos of monoid actions are therefore the M-sets in which every element is a fixed point. It is these set like objects that are the targets of the $Sets$ embedding.

Submonoid embeddings:
Let $A$ be a submonoid of a monoid $B$, then given a monomorphism embedding $f : A \to B$ we have a change of monoid functor which maps B-sets to A-sets defined by $act(f) : act(B) \to act(A)$. This change of monoid functor, simply takes the B-set $S$ to the A-set $S$ which has the A subset of actions of B acting on $S$. The concretization functor is simply a special case that reduces to a monoid to the trivial monoid, which is the subset consisting of no non-trivial actions. This submonoid reduction functor is very useful in actual examples of monoid actions, where in we often want to reduce the number of actions operating on a set.

References:
[1] https://stacks.math.columbia.edu/tag/008C