this entry is under construction
The notion of smooth super -groupoid or smooth super geometric homotopy type is the combination of of super ∞-groupoid and smooth ∞-groupoid. The cohesive (∞,1)-topos of smooth super--groupoids is a context that realizes higher supergeometry.
Smooth super -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 -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.
be the (∞,1)-topos of (∞,1)-sheaves over the site , def. 1.
This and the stronger statement that it is in fact it is actually cohesive over Super∞Grpd is discussed below, see cor. 1.
Cohesion over smooth -groupoids and over super -groupoids
is a cohesive (∞,1)-topos over Super∞Grpd.
By definition of the coverage on in def. 1, the proof of the cohesion of Smooth∞Grpd = goes through verbatim for each fixed superpoint and that gives precisely the claim.
By the discussion at Super∞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 .
Exponentiated super -algebras
A super L-∞ algebra is an L-∞ algebra internal to .
The Lie integration of is …
For general references see the references at super ∞-groupoid .
A discussion of smooth super -groupoids is in section 4.5 of