Topoi of automata I:
Four topoi of automata and regular languages

The same proof as theorem 2.12. ∎

Every slice category of a Grothendieck topos is again a Grothendieck topos. ∎

Since $\Sigma\text{-}\mathbf{FinSet}$ is a full subcategory of ${\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}}$, due to Giraud’s theorem, it is enough to show that finite $\Sigma$-sets form a generating set of ${\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}}$. This is easily implied by the definition of orbit-finiteness. ∎

The equivalence of topoi is due to lemma 3.5. Since we have proven that $({\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}},\mathcal{R})$ is an internal Boolean algebra and the equivalence $N\colon{\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}}\xrightarrow{{\simeq}}\mathbf{Sh}(\Sigma\text{-}\mathbf{FinSet},J)$ is given by

the corresponding internal Boolean algebra is given by $N(\mathcal{R})$

Then, proposition 2.24 implies

which completes the proof. ∎

The proof is almost the same as the proof of proposition 3.8. The only non-trivial part is proving that $\Sigma\text{-}\mathbf{FinMon}\hookrightarrow{\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}}$ is a generating full subcategory. It is enough to show that, for an arbitrary finite $\Sigma$-set $(Q,\delta)$ and an element $q_{0}\in Q$, there exists a finite $\Sigma$-monoid $(M,\{m_{a}\}_{a\in\Sigma})$ and a $\Sigma$-set homomorphism $f\colon(M,\overline{\delta})\to(Q,\delta)$ such that $q_{0}\in\mathrm{Im}(f)$. Let us consider $\mathrm{End}(Q)^{\mathrm{op}}$, the opposite monoid of the endofunction monoid $\mathrm{End}(Q)$. In other words, elements of the monoid $\mathrm{End}(Q)^{\mathrm{op}}$ are functions $Q\to Q$, and the multiplication $\phi\cdot\psi$ is defined to be $\psi\circ\phi$. The family of endomorphisms $\{\delta({-},a)\colon Q\to Q\}_{a\in\Sigma}$ makes it a finite $\Sigma$-monoid. The function

is a $\Sigma$-set morphism, since $(\phi\cdot\delta({-},a))(q_{0})=\delta(\phi(q_{0}),a)$. The element $q_{0}$ belongs to the image $\mathrm{Im}(f)$, since the identity function $\mathrm{id}_{Q}\in\mathrm{End}(Q)^{\mathrm{op}}$ is sent to $q_{0}$. ∎

The point $p\colon\mathbf{Set}\to\mathbf{Cont}(M)$ is decomposed into the composite of $\mathbf{Set}\to\mathbf{PSh}(M)\to\mathbf{Cont}(M)$, where $\mathbf{PSh}(M)$ denotes the topos of discrete actions. This decomposition allows us to calculate the internal Boolean algebra $B$ as a Boolean subalgebra of $\mathcal{P}(M)$. What we will prove is that $B$ coincides with the Boolean subalgebra $\mathrm{Clopen}(M)\subset\mathcal{P}(M)$.

The calculation of the direct image functor $\mathbf{PSh}(M)\to\mathbf{Cont}(M)$ implies that a subset $S\subset M$ belongs to $B$ if and only if every equivalence class of the equivalence relation

is open (See [[]Scholium 2.9.]rogers2023toposes for the details.). Here, $a^{-1}S$ denotes $\{m\in M\mid am\in S\}$.

First, we will prove that $B\subset\mathrm{Clopen}(M)$ (without the assumption of the compactness). Since for every $s\in S$, $s\sim_{S}b\iff s^{-1}S=b^{-1}S\implies e\in b^{-1}S\iff b\in S$, $S\in B$ implies that $S$ is open. $S$ is closed, because $M\setminus S$ also belongs to $B$.

We will prove $B\supset\mathrm{Clopen}(M)$ (using the assumption of compactness). Let $S\subset M$ be an arbitrary clopen subset, and $a\in M$ be an arbitrary element. It is enough to construct an open neighborhood $a\in U\subset M$ such that $\forall b\in U,\;a\sim_{S}b$. For each $m\in M$, we can take open neighborhoods $a\in U_{m}\subset M$ and $m\in V_{m}\subset M$ such that

since the multiplication map $M\times M\to M$ is continuous. The compactness of $M$ allows us to pick a finite subcover $M=V_{m_{1}}\cup\dots\cup V_{m_{n}}$. Take an arbitrary element $b$ of $U\coloneqq U_{m_{1}}\cap\dots\cap U_{m_{n}}$. What we need to prove is that $a\sim_{S}b$. Take an arbitrary element $m\in M$. For $1\leq i\leq n$ with $m\in V_{m_{i}}$, we have

which proves that $a\sim_{S}b$. ∎

Since $\Sigma\text{-}\mathbf{Set}$ is the topos of (continuous) actions of the discrete topological monoid ${{\Sigma}^{\ast}}$, the canonical inclusion $\iota\colon{{\Sigma}^{\ast}}\rightarrowtail\widehat{{{\Sigma}^{\ast}}}$ induces a geometric morphism $g\colon\Sigma\text{-}\mathbf{Set}\to\widehat{{{\Sigma}^{\ast}}}$, where $g^{*}$ is given by the restriction of the action along $\iota$:

By the construction, $g^{*}$ is faithful. The denseness of $\iota\colon{{\Sigma}^{\ast}}\to\widehat{{{\Sigma}^{\ast}}}$ implies that $g^{*}$ is full, i.e., the geometric morphism $g\colon\Sigma\text{-}\mathbf{Set}\to\mathbf{Cont}(\widehat{{{\Sigma}^{\ast}}})$ is connected. The compactness of $\widehat{{{\Sigma}^{\ast}}}$ implies that each orbit of continuous $\widehat{{{\Sigma}^{\ast}}}$ action is finite, i.e., the geometric morphism $g$ factors through the hyperconnected geometric morphism $h$ as follows.

Since $g$ and $h$ are connected, so is $e$. The remaining task is to prove $e^{*}$ is essentially surjective. Since $e^{*}$ is coreflective (and hence creates all colimits), it suffices to prove that the essential image of $e^{*}$ contains a generating set. By definition of profinite completion, every finite quotient monoid ${{\Sigma}^{\ast}}\twoheadrightarrow M$ is a continuous quotient of $\widehat{{{\Sigma}^{\ast}}}$, which implies that the canonical action $M\times\Sigma\to M$ belongs to the essential image of $e^{*}$. The same argument to proposition 3.11 completes the proof. ∎