# nLab smooth super infinity-groupoid

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $\left(\infty ,1\right)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

superalgebra

and

supergeometry

## Applications

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 $\mathrm{sCartSp}$ for the full subcategory of that of supermanifolds on the super Cartesian spaces $\left\{{ℝ}^{p\mid q}{\right\}}_{p,q\in ℕ}$. 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 ${ℝ}^{p\mid q}$ is of the form

$\left\{{U}_{i}×{ℝ}^{0\mid q}⟶{ℝ}^{p\mid q}{\right\}}_{i}$\{ U_i \times \mathbb{R}^{0|q} \longrightarrow \mathbb{R}^{p|q} \}_{i}

for

$\left\{{U}_{i}⟶{ℝ}^{p}{\right\}}_{i}$\{ U_i \longrightarrow \mathbb{R}^{p} \}_{i}

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

###### Definition

Let

$\mathrm{SmoothSuper}\infty \mathrm{Grpd}:={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{sCartSp},\mathrm{Super}\infty \mathrm{Grpd}\right)$SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)

be the (∞,1)-topos of (∞,1)-sheaves over the site $\mathrm{sCartSp}$, def. 1.

###### Proposition

The (∞,1)-topos $\mathrm{Smooth}\infty \mathrm{Grpd}$ 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

$\mathrm{Smooth}\mathrm{Super}\infty \mathrm{Grpd}$ is a cohesive (∞,1)-topos over Super∞Grpd.

$\mathrm{Smooth}\mathrm{Super}\infty \mathrm{Grpd}\stackrel{\stackrel{{\Pi }_{\mathrm{Super}}}{⟶}}{\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{Super}}}{←}}{\stackrel{\stackrel{{\Gamma }_{\mathrm{Super}}}{⟶}}{\underset{{\mathrm{coDisc}}_{\mathrm{super}}}{←}}}}\mathrm{Super}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{sCartSp}$ in def. 1, the proof of the cohesion of Smooth∞Grpd = ${\mathrm{Sh}}_{\infty }\left(\mathrm{CartSp}\right)$ goes through verbatim for each fixed superpoint and that gives precisely the claim.

###### Proposition

By the discussion at Super∞Grpd.

###### Corollary

$\mathrm{SmoothSuper}\infty \mathrm{Grpd}$ is cohesive and in

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

$\begin{array}{ccc}\mathrm{Smooth}\mathrm{Super}\infty \mathrm{Grpd}& \stackrel{\stackrel{{\Pi }_{\mathrm{Super}}}{⟶}}{\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{Super}}}{←}}{\stackrel{\stackrel{{\Gamma }_{\mathrm{Super}}}{⟶}}{\underset{{\mathrm{coDisc}}_{\mathrm{super}}}{←}}}}& \mathrm{Super}\infty \mathrm{Grpd}\\ ↓↑& & ↓↑\\ \mathrm{Smooth}\infty \mathrm{Grpd}& \stackrel{\stackrel{\Pi }{⟶}}{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\stackrel{\stackrel{\Gamma }{⟶}}{\underset{\mathrm{coDisc}}{←}}}}& \infty \mathrm{Grpd}\end{array}$\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 $\mathrm{Smooth}\mathrm{Super}\infty \mathrm{Grpd}$.

### Exponentiated super ${L}_{\infty }$-algebras

A super L-∞ algebra $𝔤$ is an L-∞ algebra internal to $\mathrm{Sh}\left(\mathrm{SuperPoint}\right)$.

The Lie integration of $𝔤$ is …

## 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)