Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 7th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPutting together $$ SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint) $$ and [[complex analytic ∞-groupoid]], presumably there's nothing to stop $$ Super \mathbb{C} Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd \rtimes SuperPoint) $$ to generalise [[complex supermanifold]].

   Putting together

   $$SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)$$

   and complex analytic ∞-groupoid, presumably there’s nothing to stop

   $$Super \mathbb{C} Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd \rtimes SuperPoint)$$

   to generalise complex supermanifold.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that should work without problem.

   Yes, that should work without problem.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWould we still say that shape is $loc_{\mathbb{R}}$ and that $R$ is $loc_{\mathbb{R}^{0|1}}$ in this complex setting?

   Would we still say that shape is $loc_{\mathbb{R}}$ and that $R$ is $loc_{\mathbb{R}^{0|1}}$ in this complex setting?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo in the complex setting shape would not be localization at $\mathbb{R}$. It might be that it is now localization at $\mathbb{C}$, but I haven't really thought this through. But $Rh \simeq loc_{\mathbb{R}^{0|1}}$ would still hold, and I would still write it this way, yes.

   So in the complex setting shape would not be localization at $\mathbb{R}$. It might be that it is now localization at $\mathbb{C}$, but I haven’t really thought this through.

   But $Rh \simeq loc_{\mathbb{R}^{0|1}}$ would still hold, and I would still write it this way, yes.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAh, so I wonder how things should be said in a model independent way. We could have $loc_{\mathbb{A}^1}$ and $loc_{\mathbb{R}^{0|1}}$, but isn't it possible that in some other nontrivial model of solid cohesion that we wouldn't have $Rh$ as $loc_{\mathbb{R}^{0|1}}$?

   Ah, so I wonder how things should be said in a model independent way.

   We could have $loc_{\mathbb{A}^1}$ and $loc_{\mathbb{R}^{0|1}}$, but isn’t it possible that in some other nontrivial model of solid cohesion that we wouldn’t have $Rh$ as $loc_{\mathbb{R}^{0|1}}$?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2015
   • (edited May 13th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, there could be models of solid cohesion where $Rh$ is not $loc_{\mathbb{R}^{0\vert 1}}$. We discussed this before, that it seems unclear and in fact unlikely that solid cohesion as presently set up is sufficient to fully characterize supergeometry. But slight variants of this localization business will work more generally. For instance I suppose it follows readily that in complex analytic cohesion the shape modality is given by localization at the collection of all all [[Stein spaces]]. Similarly, reduction will maybe not in general be localization at $Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$, but should fairly generally be localization at the collection at all infinitesimally thickened points.

   Yes, there could be models of solid cohesion where $Rh$ is not $loc_{\mathbb{R}^{0\vert 1}}$. We discussed this before, that it seems unclear and in fact unlikely that solid cohesion as presently set up is sufficient to fully characterize supergeometry.

   But slight variants of this localization business will work more generally. For instance I suppose it follows readily that in complex analytic cohesion the shape modality is given by localization at the collection of all all Stein spaces.

   Similarly, reduction will maybe not in general be localization at $Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$, but should fairly generally be localization at the collection at all infinitesimally thickened points.