Topoi of monoid actions

Let $M$ be a monoid, then $M$ is associated with a category: the category of M-sets with equivariant maps between them. This produces a map of objects of categories: \[ act : Ob(Mon) \to Ob(Cat) \] The function act maps a monoid to its parent category of all categories. Its image in the category of categories consists entirely of topoi. Each monoid is associated with a different topoi, and topoi properties can be utliized to understand the characteristics of monoids.

Monoid actions can be characterized as set-valued functors $F : M \to Sets$. A category of M-sets is then a functor category, whose objects are set-valued functors of this sort. The morphisms of the functor category are natural transformations defined by component morphisms : \[ f: Ob(M) \to Arrows(Sets) \] But the domain of the component function of a natural transformation of a functor with monoid input is a single object, so the comonent function produces a single morphism in the output category. This component morphism is the equivariant map $f : X \to Y$ of monoid actions. Therefore, in the special case of natural transformations of functors with monoid input, we can clearly cut out the component function, and speak of the unique morphism of the unique object. Henceforth we shall refer to the morphisms of M-sets as equivariant maps.

Distributive lattice of subalgebras:
Every monoid action is associated with a preorder which has $x \sqsubseteq y$ when there exists some action $a \in M$ such that $ax = y$. This is the natural increasing action preorder of the monoid action, which describes the order that elements move in when acted on by the monoid. We can now clearly see that action-closed subsets of an M-set $X$ are fully determined by its action preorder, and so $Sub(X)$ consists of preorder ideals of the decreasing action preorder or dually filters of the increasing action preorder. These ideals/filters form an Alexandrov topology whose specialization preorder is the action preorder.

Action preorder $\to$ Alexandrov topology $\to$ Distributive lattice of subobjects

Every topology, including an Alexandrov topology forms a distributive lattice with respect to inclusion. In the case of a category, the poset of subobjects is the condensation of the mono input action preorder of an identity morphism. For the topoi of monoid actions, this mono input action preorder is determined by the action preorder of the monoid itself, which creates the distributive lattice of subobjects correspond to the distributive lattice ordered Alexandrov topology of action-closed sets. As every topoi has distributive lattices of subobjects, this explains why the special case of M-sets has distributive subobject lattices.

Images and inverse images
The topoi of sets is naturally associated with a covariant image functor and a contravariant inverse image functor. The former takes a function $f: X \to Y$ to a map $Im(f) : \wp(X) \to \wp(Y)$ and the later takes it to $Inv(f) : \wp(Y) \to \wp(X)$. The image functor is a join semilattice homomorphism, and the inverse image functor is a lattice homomorphism. In order to apply this to monoid actions, we should first review the operational definition of a monoid action: \[ a: M \times X \to X \] Given this function, we can form two different partial operations $L_x : X \to X$ and $R_x : M \to X$. The former is the transformation associated with an underlying monoid action, and the later is a map from the monoid to the action set. We want to apply the inverse image functor to $R_x$ to get: \[ R_x^{-1} : \wp(X) \to \wp(M) \] \[ R_x^{-1}(I) = \{m \in M : mx \in I \} \] We can define $M$ to be a monoid action on itself by left-multiplication. Then this function $R_x^{-1}$ maps action closed sets in $X$ to action closed sets in $M$. To see this, note that if $ax \in I$ then by action closure for any other $b \in M$ we have $bax \in I$, so $bax \in R_x^{-1}(I)$. It follows that $R_x^{-1}$ is action closed. As $R_x^{-1}$ maps closed sets to closed sets, it follows that for any monoid action $R_x$ is a continuous map of Alexandrov topologies.

Object of truth values:
The function $R_x^{-1} : \wp(X) \to \wp(M)$ is a map of subalgebras of $X$ and $M$, but in the special case that $X = M$ it is a transformation of the action closed sets of $M$. It would be nice, if actions of the form $R_x^{-1}$ formed on a monoid action on $Sub(M)$, and as we shall see they in fact do. This forms a monoid action $\omega : M \times Sub(M) \to Sub(M)$. First, we must check that the identity is preserved: \[ \omega_{e}(I) = \{ m : me \in I \} = \{ m \in I \} = I \] In general, given an identity function the contravariant inverse image functor preserves its role as an identity. Next we need to check composition closure. \[ \omega_a(\omega_b(I)) \] \[ = \omega_a(\{ m : mb \in I \}) \] \[ = \{ n : na \in \{ m : mb \in I \} \} \] \[ = \{ n : nab \in I \} \] \[ = \omega_{ab}(I) \] It follows that $(Sub(M), R_m^{-1})$ forms an M-set: the object of truth values $\Omega$. The truth element arrow is the function $1 \to \Omega$ which selects the set $M$ itself as an element of $Sub(M)$. We now have that $\Omega$ is an M-set whose monoid actions embed in the lattice endomorphism monoid $End(\Omega)$. We therefore have a functor from $M$ be the category of lattices and lattice homomorphisms.

Action on $\Omega$ is also monotone by the properties of inverse images, but it does not need to be increasing in general. There is one obvious case in which the monoid action on $\Omega$ must be increasing and that is when $x$ is a central element. In that case, we have that $I_x$ is $\{ a \in M : ax \in I \}$. Suppose $i \in I$ then we can plug $i$ in to get $ix \in I$ and by centrality we can flip this to get $xi \in I$ but since $I$ is action closed this is always the case, so $I \subseteq I_x$ which means that action by any central element $x$ is always increasing.

Therefore, suppose that $M$ is a commutative monoid, then the action on $\Omega$ is always increasing. It follows that the action $\Omega$ is a commutative J-trivial monoid, which is some further quotient of the condensation $\frac{M}{H}$ of the commutative monoid $M$ produced by the congruence determined by H classes. While this applies in the commutative case, in every case action on $\Omega$ is at least a lattice endomorphism.

Limits and colimits of monoid actions:
Most of the limits and colimits of monoid actions are some variant of products/coproducts which are inherited from the topoi of sets.

Empty coproduct (initial object): the empty coproduct is the monoid action on an empty set $(\emptyset, a)$.

Empty product (terminal object): the empty product is the monoid action on a singleton which always maps the single object back to itself $({x},a)$.

Product: suppose we have an underlying monoid $M$ then we have two actions on $S$ and $T$ defined by $\cdot : M \times S \to S$ and $\cdot : M \times T \to T$. Then we form a product on $S \times T$ by $\cdot : M \times (S \times T) \to (S \times T)$ which takes $m(s,t)$ to $(ms,mt)$. This is the monoid action that now acts on both M-sets simultaneouly. The action preorder of the product of monoid actions is the product of preorders.

Coproduct: the coproduct of two monoid actions $S + T$ is the disjoint union of monoid actions, which acts on $S$ and $T$ separately, depending upon which of them is given as an argument to an action function. The action preorder of the coproduct is the disjoint union of preorders.

Equalizer/coequalizer: in the category of sets given a pair of functions $f,g : S \to T$ we can clearly get a product function $f \times g : S \to T^2$. The equalizer is then the inverse image of the coreflexive component of the output relation and the coequalizer is the equivalence closure of the image relation. The equalizer is an input subobject and the coequalizer is an output quotient. The same is true in the category of monoid actions, and we can use this to define fiber products.

Fiber products: suppose we have morphisms $f : X \to Z$ and $g : Y \to Z$ then the fiber product $X \times_Z Y$ is a subobject of the product $X \times Y$ of monoid actions that equalizes the two morphisms $f$ and $g$. In the special case where $f : 1 \to Z$ is an element arrow then we have $1 \times_Z Y$ which acts essentially as an inverse image of $g$ by the selected element. In particular, we want to consider fiber products $1 \times_\Omega Y$ in order to construt subobjcet classifiers.

Fiber coproduct: the fiber coproduct $X +_Z Y$ is defined dually by the coequalizer quotient of the coproduct of monoid actions.

The limits/colimits of monoid actions can actually be defined this simply, using products/coproducts similar to the category of sets. This leads to the fact that the category of monoid actions is a bi-complete category.

Subobject classifer:
We previously covered the object of truth values and its element arrow, and an aspect of this was functions of the sort $R_x^{-1}$ defined by inverse images of the action funtion. We can apply this again to get the characteristic arrow of a subobject $k : X \to Y$. This produces a map $\chi : Y \to \Omega$. \[ \chi(y) = \{ a \in M : ay \in X \} = R_y^{-1}(X) \] We can clearly see that $\chi(y) = M$ is logically equivalent to the condition that $y \in X$ since this the identity function preserves the membership of $y$ in $X$ which clearly means $y$ is itself a member of $X$. Therefore, $X$ is a pullback of $1 \times_\Omega Y$ which means that $\chi$ is a valid characteristic arrow of a subobject classifier.

In order to see that this is an equivariant map we need to get $\chi(by)$ which is equal to $\{ a \in M : a(by) \in X \}$. We then get $\{ a \in M : ab \in \{a' \in M : a'y \in X \}\}$ which is equal to $\{ a \in M : (ab)y \in X \}$ which is $b\chi(y)$. It follows that $\chi$ is an equivariant map.

The only thing that remains to be proven is the $\Omega$-axiom that $\chi$ is the unique equivariant map that satisfies this property. In order to do that, note that $aI = M$ implies that $a \in I$ with respect to actions on $Sub(M)$ sets of the object of truth values $\Omega$. The expression $aI$ is equal to $\{ m \in M : ma \in I \}$ but in order for that to equal $M$ we must have that the identity acting on $a$ is in $I$ which means $a \in I$.

Suppose that $y \in X$ then as described previously, $\chi(y) = M$. Suppose that $y \not\in X$ and let $a \in M$ be some element such that $ay \in X$ then $\chi(ay) = M$ and by equivariance we have $chi(ay) = M = a\chi(y)$. In this case, $a\chi(y) = M$ is logically equivalent to the condition that $a \in \chi(y)$ becaue $a$ is acting on an element of $\Omega$ as previously described. This means that $a \in chi(y)$ is logically equivalent to the condition that $ay \in X$, which is the definition of the subobject classifier previously provided. Therefore, this definition of a subobject classifier is unique so M-sets satisfy the $\Omega$ axiom.

Exponentation:
The idea of exponentation is that we can take any hom class of morphisms between two objects, and we can then turn these hom classes into objects of a category in their own right that have an evaluation arrow associated with them. In the case of monoid actions, we will establish an exponential object of this sort from morphisms from $X \to Y$ by the class of all morphisms of the sort $M \times X \to Y$.

The exponentation object $Y^X$ therefore is a set consisting of all morphisms $M \times X \to Y$, or $Hom(M \times X, Y)$. An M-action is established on this hom class, defined by compositional input actions. We can take any $(m,y)$ and then we can multiply it by $a \in M$ to get $(am,y)$, this produces a scalar multiplication $s$ of the first component of the ordered pair. The M-action on $Y^X$ is defined by composition $t(f) = f \circ s$. In other words, it takes a morphism and produces another morphism by input action, that first acts on the first component of the input ordered pair.

We can define an evaluation map $ev : Y^X \times X \to Y$ based upon this by selecting an identity element $e$ of the monoid $M$ and then setting $ev(f,x) = f(e,x)$. This produces an evaluation arrow associated to the exponential object.

In order to validate that this is an evaluation arrow, we lastly need to get an exponential adjoint $\hat{g} : C \to X^Y$ for a map $g : C \times X \to Y$. This can be defined by $\hat{g}(c) = \lambda (m,x) g(mc,x)$. This produces a function in the associated exponential object. Then, notice that the evalutaion function sets $m$ to the identity if we do that we get $\lambda (m,x) g(m,x)$ which is just $g$ so the function $g$ filters through the exponential adjoint and its product with the identity, making this a valid exponential object. Therefore, categories of monoid actions have exponential objects.

Topoi theory:
Let $M$ be a monoid, then the category $act(M)$ is bicomplete, has exponentation, and subobject classifiers. That means that the category of actions $act(M)$ of a monoid $M$ is a topoi. The topoi theoretic properties of monoids can be used to better understand them. For example, a monoid action is a group action if and only if its topoi of actions is classical. The topoi theoretic properties of monoids is a topic for further discussion.