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The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a countable basis of $X$ and $v:\mathbb{N} \rightarrow \mathcal{O}$ a total surjection. Then we say that $f:X \rightarrow Y$ is computable if given any enumeration of all basic open sets containing $x$, one can compute (in the sense of Turing) an enumeration of all basic open sets contaning $f(x)$. Equivalently we can state that for any basic open set $O$ of $Y$, we have that $f^{-1}(O)$ is a recursively enumerable open in the sence that $O=\cup_n f^{-1}(O)=\cup_n O_n$ where $O_n$ is a recursively enumerable sequence of basic open sets of $X$

This brings some sort of a generalization of Turing reduction, when $X$ and $Y$ are both the Cantor space.

Now, I am wondering if it is possible to have a categorical point of view on all this. My research brought me to Topoi, the effective topos and the recursive topos. I have some difficulties to understand what is the effective topos (as I have to understand what is the recursive topos).

Before I spend a lot of time to study them, I would like to know if there are somehow related to what I've stated above (i.e., does one of them contains in some way the computable functions between topological spaces ? is there only continuous functions between topological spaces in these Topos ? can we even talk about topological spaces in these topos ?) or are hey something completely different ?

I apologize if my question seems too general. Any answer will be appreciated :)
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The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a coutable countable basis of $X$ and $v:\mathbb{N} \rightarrow \mathcal{O}$ a total surjection. Then we say that $f:X \rightarrow Y$ is computable if given any enumeration of all basic open sets containing $x$, one can compute (in the sense of turingTuring) an enumeration of all basic open sets contaning $f(x)$. Equivalently we can state that for any basic open set $O$ of $Y$, we have that $f^{-1}(O)$ is a recursively enumerable open in the sence that $O=\cup_n O_n$ where $O_n$ is a recursively enumerable sequence of basic open sets of $X$

This brings some sort of a generalization of turing rectibilityTuring reduction, when $X$ and $Y$ are both the Cantor space.

Now, I am wondering if it is possible to have a categorical point of view on all this. My research brought me to Topoi, the effective topos and the recursive topos. I have some difficulties to understand what is the effective topos (as I have to understand what is the recursive topos).

Before I spend a lot of time to study them, I would like to know if there are somehow related to what I've stated above (i.e., does one of them contains in some way the computable functions between topological spaces ? is there only continuous functions between topological spaces in these Topos ? can we even talk about topological spaces in these topos ?) or are hey something completely different ?

I apologize if my question seems too general. Any answer will be apreciated appreciated :)
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Effective topos and computability in topological spaces

The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a coutable basis of $X$ and $v:\mathbb{N} \rightarrow \mathcal{O}$ a total surjection. Then we say that $f:X \rightarrow Y$ is computable if given any enumeration of all basic open sets containing $x$, one can compute (in the sense of turing) an enumeration of all basic open sets contaning $f(x)$. Equivalently we can state that for any basic open set $O$ of $Y$, we have that $f^{-1}(O)$ is a recursively enumerable open in the sence that $O=\cup_n O_n$ where $O_n$ is a recursively enumerable sequence of basic open sets of $X$

This brings some sort of a generalization of turing rectibility, when $X$ and $Y$ are both the Cantor space.

Now, I am wondering if it is possible to have a categorical point of view on all this. My research brought me to Topoi, the effective topos and the recursive topos. I have some difficulties to understand what is the effective topos (as I have to understand what is the recursive topos).

Before I spend a lot of time to study them, I would like to know if there are somehow related to what I've stated above (i.e., does one of them contains in some way the computable functions between topological spaces ? is there only continuous functions between topological spaces in these Topos ? can we even talk about topological spaces in these topos ?) or are hey something completely different ?

I apologize if my question seems too general. Any answer will be apreciated :)