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Sorry for #2 (I had written there that we needed some other entry organization somewhere before seeing that we actually already did.)
Here is an actual technical issue: elswhere I had managed to mix myself upabout variances of the following.
Consider the site of objects of the form ℝn×D×ℝ0|q, where D is an infinitesimally thickened points, ℝ0|q is a superpoint, and maps are smooth maps extended to infinitesimal generators in the evident way. The coverage is induced from that of open covers on Cartesian spaces.
Then what is the ℝ0|1-homotopy localization of the ∞-topos over this site?
I think that by direct analogy with the by now standard argument which shows that the ℝ1-homotopy localization of the ∞-topos over the site of just the ℝn-s is the ∞-topos over the point, we get that the ℝ0|1-homotopy localization is the ∞-topos over the site of ℝn×Ds with the localization given on representables by removing the super-point part, ℝn×D×ℝ0|q↦ℝn×D.
Hence the corresponding reflector is the operation denoted R here, I suppose.
So should we expect a pattern in the full adjoint modalities diagram as to which are localizations? That’s why I was asking about R on the “Science of Logic” thread. Then we might expect ⇉ to be a localization too.
Also, does each modality have a negative modality? Well, I know that isn’t true for ∅. But why do the negatives of those in corresponding positions appear, i.e., of ♭ and ⇝?
Is that observation about the negative of ∅ being the maybe monad give us a route out of the diagram to the linear/quantum world?
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