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For weakly cohesive toposes, there exists a notion of contractability, and toposes with a subobject classifier $\Omega$ that is contractible are of special interest (see here).

It occured to me that one could view probability theory as a theory of (convex) interpolation of truth, which replaces boolean truth $\Omega = \{0,1\}$ by 'continuous truth' $\Omega = [0,1]$, so I wonder, is there an example of a topos with a subobject classifier given (in a suitable sense) by the real unit interval?

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  • $\begingroup$ To be clear, the question make sense, but having $\Omega = [0,1]$ is definitely not the right way of having a topos theoretic interpretation of probabilities: it would mean that any two propositions with the same probability of being true are equivalent! $\endgroup$
    – Simon Henry
    Commented Apr 14, 2023 at 12:05

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Here's an example which, I'm afraid, is not very interesting (and may not match your notion of “suitable sense”): let $X$ be the topological space $\mathopen]0,1\mathclose[$ (the open interval) with the (not at all separated) topology given by the $U_x := \mathopen]0,x\mathclose[$ for $x \in [0,1]$ (with $U_0 = \varnothing$, obviously). Since $U_x \cap U_y = U_{\min(x,y)}$ and $$ \bigcup_{i\in I} U_{x_i} = U_{\sup\{x_i : i\in I\}} $$ this is indeed a topology. Of course, $\mathcal{O}(X) := \{U_x : x\in[0,1]\}$ can be identified with $[0,1]$ as an ordered set (indeed, frame). The topos of sheaves on $(X, \mathcal{O}(X))$ then has this as subobject classifier.

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    $\begingroup$ Great! More precisely, $[0,1]$ is the frame $\Gamma(\Omega)$ of points of the subobject classifier. The $\Omega$ itself is the sheaf whose value on $U_x$ is $[0,x]$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 13, 2023 at 9:02
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    $\begingroup$ That seems to be an answer in the affirmative, even though I hoped for something different. I don't see an interesting meaning of this topos or a relation to probability theory, but perhaps that is just my ignorance. $\endgroup$
    – Nicolas Schmidt
    Commented Apr 13, 2023 at 9:06
  • $\begingroup$ @NicolasSchmidt You should indicate your requirements in the question, then. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 13, 2023 at 9:07
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    $\begingroup$ @NicolasSchmidt Yes, hence my admission that this example is “not very interesting”. The problem is mainly that $[0,1]$ has no sort of additive structure here: if you apply any increasing self-homeomorphism the topos doesn't see any difference, whereas in probability sums of probabilities have meaning. $\endgroup$
    – Gro-Tsen
    Commented Apr 13, 2023 at 9:30
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    $\begingroup$ Now maybe there is a way to make this topos more relevant to probability by considering some extra structure on it. E.g., if $F$ is an (external) finite set seen in the topos as a constant sheaf on $X$, then given an (external) probability measure $(p_i)_{i∈F}$ on $F$ we get an (internal) subset of the powerset of $F$ in the topos by taking a characteristic function $\chi\colon F \to [0,1]$ to $\sum_{i\in F} p_i \chi(i) \in [0,1]$: this seems to reflect probabilities at least somewhat, so maybe something can be done in that line of thought. $\endgroup$
    – Gro-Tsen
    Commented Apr 13, 2023 at 9:39

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