nLab smooth super infinity-groupoid

Context

Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Structures in a cohesive $(\infty,1)$-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 consider one of at least two possible definitions, that differ (only) slightly in some fine technical detail. The other is at super smooth infinity-groupoid.

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

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

for

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

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

Definition

Let

$SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)$

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

Proposition

The (∞,1)-topos $Smooth \infty Grpd$ 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

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

$Smooth Super \infty Grpd \stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} Super \infty Grpd \,.$
Proof

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

Proposition

By the discussion at Super∞Grpd.

Corollary

$SmoothSuper\infty Grpd$ is cohesive and in

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

$\array{ Smooth Super \infty Grpd &\stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} & Super \infty Grpd \\ \downarrow \uparrow && \downarrow \uparrow \\ Smooth \infty Grpd & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }$

where the right vertical adjoints exhibit infinitesimal cohesion.

Structures

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

Exponentiated super $L_\infty$-algebras

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

The Lie integration of $\mathfrak{g}$ 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 July 25, 2016 16:09:48 by Urs Schreiber (62.217.43.218)