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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 8th 2009
   • PermaLink
   Author: Urs
   Format: Markdowncreated stubs for [[homotopy localization]] and [[A1 homotopy theory]]

   created stubs for

   homotopy localization

   and

   A1 homotopy theory

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2015
   • (edited Apr 27th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItex...

   …

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2015
   • (edited Apr 27th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry 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 $\mathbb{R}^n \times D \times \mathbb{R}^{0\vert q}$, where $D$ is an infinitesimally thickened points, $\mathbb{R}^{0|q}$ 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 $\mathbb{R}^{0\vert 1}$-homotopy localization of the $\infty$-topos over this site? I think that by direct analogy with the by now standard argument which shows that the $\mathbb{R}^1$-homotopy localization of the $\infty$-topos over the site of just the $\mathbb{R}^n$-s is the $\infty$-topos over the point, we get that the $\mathbb{R}^{0\vert 1}$-homotopy localization is the $\infty$-topos over the site of $\mathbb{R}^n \times D$s with the localization given on representables by removing the super-point part, $\mathbb{R}^n \times D \times \mathbb{R}^{0\vert q} \mapsto \mathbb{R}^n \times D$. Hence the corresponding reflector is the operation denoted $R$ [here](http://ncatlab.org/nlab/show/super%20formal%20smooth%20infinity-groupoid#Cohesion), I suppose.

   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 $\mathbb{R}^n \times D \times \mathbb{R}^{0\vert q}$, where $D$ is an infinitesimally thickened points, $\mathbb{R}^{0|q}$ 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 $\mathbb{R}^{0\vert 1}$-homotopy localization of the $\infty$-topos over this site?

   I think that by direct analogy with the by now standard argument which shows that the $\mathbb{R}^1$-homotopy localization of the $\infty$-topos over the site of just the $\mathbb{R}^n$-s is the $\infty$-topos over the point, we get that the $\mathbb{R}^{0\vert 1}$-homotopy localization is the $\infty$-topos over the site of $\mathbb{R}^n \times D$s with the localization given on representables by removing the super-point part, $\mathbb{R}^n \times D \times \mathbb{R}^{0\vert q} \mapsto \mathbb{R}^n \times D$.

   Hence the corresponding reflector is the operation denoted $R$ here, I suppose.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 28th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo 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 $\rightrightarrows$ to be a localization too. Also, does each modality have a negative modality? Well, I know that isn't true for $\emptyset$. But why do the negatives of those in corresponding positions appear, i.e., of $\flat$ and $\rightsquigarrow$? Is that observation about the negative of $\emptyset$ being the maybe monad give us a route out of the diagram to the linear/quantum world?

   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 $\rightrightarrows$ to be a localization too.

   Also, does each modality have a negative modality? Well, I know that isn’t true for $\emptyset$. But why do the negatives of those in corresponding positions appear, i.e., of $\flat$ and $\rightsquigarrow$?

   Is that observation about the negative of $\emptyset$ being the maybe monad give us a route out of the diagram to the linear/quantum world?