under construction

Contents

Idea

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

Super smooth $\infty$-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) in some fine technical detail. The other is at smooth super infinity-groupoid.

Write

• CartSp for the site of Cartesian spaces;

• $InfPoint \coloneqq WAlg^{op}$ for the category of infinitesimally thickened points, formal duals of commutative algebras over the real numbers of the form $\mathbb{R}\oplus V$ with $V$ a finite-dimensional nilpotent ideal.

• $SuperPoint \coloneqq WAlg_{super}^{op}$ for the formal dual of the category of superalgebras of the above form.

There are then “semidirect product” sites $CartSp \rtimes InfinPoint$ and $CartSp \rtimes SuperPoint$ (whose objects are Cartesian products of the given form inside synthetic differential supergeometry and whose morphisms are all morphisms in that context (not just the product morphisms)).

Set then

for the collection of formal smooth ∞-groupoids (see there) and finally

for that of super smooth $\infty$-groupoid

Properties

Cohesion

Passing to (∞,1)-categories of (∞,1)-sheaves, this yields, via (∞,1)-Kan extension, a sequence of adjoint quadruples as follows:

Observe that the total composite labeled $\Delta$ is indeed the locally constant infinity-stack-functor: let $X$ be any object in its image, and $U \times D_s \in CartSp \rtimes SuperPoint$. Then by adjunction $SuperSmooth\infty Grpd(U\times D_s, X)$ is equivalently homs in $FormalSmooth \infty Grpd$ of out the dual of the Weil algebra which is the quotient of the original one by the ideal generated by its odd part. Hence this, in turn, is equivalently homs in $Smooth\infty Grpd$ out of $U$ and that finally is equivalently homs in $\infty Grpd$ out of $\ast$ into the given $\infty$-groupoid.

Hence passing to the adjoint triples of idempotent monads and idempotent comonads which this induces, then yields

• on the left the shape modality $\int$, flat modality $\flat$ and sharp modality $\sharp$,

• in the middle yields the reduction modality $\Re$, the infinitesimal shape modality $\&$ and the infinitesimal flat modality $\Im$.

• on the right we get an adjoint triple whose whose middle bit $\beta$ is the bosonic modality and whose left piece $\epsilon$ produces super-even components, containing all the “fermion currents” if one wishes , which in this unity of opposites hence deserves to be called the fermionic modality.

Hence we get a process of adjoint modalities of the form

where “$\vee$” denotes inclusion of modal types.