structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
superalgebra and (synthetic ) supergeometry
this entry is under construction
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.
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.
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.
So a covering family of $\mathbb{R}^{p|q}$ is of the form
for
a differentiably good open cover of $\mathbb{R}^{p}$.
Let
be the (∞,1)-topos of (∞,1)-sheaves over the site $sCartSp$, def. 1.
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.
$Smooth Super \infty Grpd$ is a cohesive (∞,1)-topos over Super∞Grpd.
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.
Super∞Grpd is infinitesimally cohesive over ∞Grpd.
By the discussion at Super∞Grpd.
$SmoothSuper\infty Grpd$ is cohesive and in
In fact we have a commutative diagram of cohesive (∞,1)-topos
where the right vertical adjoints exhibit infinitesimal cohesion.
We discuss realizations of the general abstract structures in a cohesive (∞,1)-topos realized in $Smooth Super \infty Grpd$.
A super L-∞ algebra $\mathfrak{g}$ is an L-∞ algebra internal to $Sh(SuperPoint)$.
The Lie integration of $\mathfrak{g}$ is …
The brane bouquet of Green-Schwarz action functionals for super $p$-brane sigma-models.
For general references see the references at super ∞-groupoid .
A discussion of smooth super $\infty$-groupoids is in section 4.5 of