nLab
smooth super infinity-groupoid

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Super-Geometry

this entry is under construction

Contents

Idea

The notion of smooth super \infty-groupoid or smooth super geometric homotopy type is the combination of of super ∞-groupoid and smooth ∞-groupoid. The cohesive (∞,1)-topos of smooth super-\infty-groupoids is a context that realizes higher supergeometry.

Smooth super \infty-groupoids include supermanifolds, super Lie groups and their deloopings etc. Under Lie differentiation these map to super L-∞ algebras.

Definition

We take a smooth super \infty-groupoid to be a smooth ∞-groupoid but not over the base topos ∞Grpd of bare ∞-groupoids, but over the base topos Super∞Grpd of super ∞-groupoids.

Definition

Write sCartSpsCartSp for the full subcategory of that of supermanifolds on the super Cartesian spaces { p|q} p,q\{\mathbb{R}^{p|q}\}_{p,q \in \mathbb{N}}. Regard this as a site by taking the coverage the product coverage of the good open cover coverage of CartSp and the trivial coverage on superpoints.

Remark

So a covering family of p|q\mathbb{R}^{p|q} is of the form

{U i× 0|q p|q} i \{ U_i \times \mathbb{R}^{0|q} \longrightarrow \mathbb{R}^{p|q} \}_{i}

for

{U i p} i \{ U_i \longrightarrow \mathbb{R}^{p} \}_{i}

a differentiably good open cover of p\mathbb{R}^{p}.

Definition

Let

SmoothSuperGrpd:=Sh (,1)(sCartSp,SuperGrpd) SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)

be the (∞,1)-topos of (∞,1)-sheaves over the site sCartSpsCartSp, def. 1.

Proposition

The (∞,1)-topos SmoothGrpdSmooth \infty Grpd of def. 2 is a cohesive (∞,1)-topos over ∞Grpd.

This and the stronger statement that it is in fact it is actually cohesive over Super∞Grpd is discussed below, see cor. 1.

Properties

Cohesion over smooth \infty-groupoids and over super \infty-groupoids

Proposition

SmoothSuperGrpdSmooth Super \infty Grpd is a cohesive (∞,1)-topos over Super∞Grpd.

SmoothSuperGrpdcoDisc superΓ SuperDisc SuperΠ SuperSuperGrpd. Smooth Super \infty Grpd \stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} Super \infty Grpd \,.
Proof

By definition of the coverage on sCartSpsCartSp in def. 1, the proof of the cohesion of Smooth∞Grpd = Sh (CartSp)Sh_\infty(CartSp) goes through verbatim for each fixed superpoint and that gives precisely the claim.

By the discussion at Super∞Grpd.

Corollary

SmoothSuperGrpdSmoothSuper\infty Grpd is cohesive and in

In fact we have a commutative diagram of cohesive (∞,1)-topos

SmoothSuperGrpd coDisc superΓ SuperDisc SuperΠ Super SuperGrpd SmoothGrpd coDiscΓDiscΠ Grpd \array{ Smooth Super \infty Grpd &\stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} & Super \infty Grpd \\ \downarrow \uparrow && \downarrow \uparrow \\ Smooth \infty Grpd & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }

where the right vertical adjoints exhibit infinitesimal cohesion.

Structures

We discuss realizations of the general abstract structures in a cohesive (∞,1)-topos realized in SmoothSuperGrpdSmooth Super \infty Grpd.

Exponentiated super L L_\infty-algebras

A super L-∞ algebra 𝔤\mathfrak{g} is an L-∞ algebra internal to Sh(SuperPoint)Sh(SuperPoint).

The Lie integration of 𝔤\mathfrak{g} is …

Applications

References

For general references see the references at super ∞-groupoid .

A discussion of smooth super \infty-groupoids is in section 4.5 of

Revised on October 23, 2013 09:50:08 by Urs Schreiber (82.169.114.243)