nLab smooth super infinity-groupoid



Cohesive \infty-Toposes


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 sCartSpsCartSp for the full subcategory of that of supermanifolds on the super Cartesian spaces { p|q} p,q\{\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 p|q\mathbb{R}^{p|q} is of the form

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


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

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



SmoothSuperGrpd:=Sh (,1)(sCartSp,SuperGrpd) SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)

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


The (∞,1)-topos SmoothGrpdSmooth \infty Grpd of def. 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. .


Cohesion over smooth \infty-groupoids and over super \infty-groupoids


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

SmoothSuperGrpdcoDisc superΓ SuperDisc SuperΠ SuperSuperGrpd. 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 \,.

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

By the discussion at Super∞Grpd.


SmoothSuperGrpdSmoothSuper\infty Grpd is cohesive and in

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

SmoothSuperGrpd coDisc superΓ SuperDisc SuperΠ Super SuperGrpd SmoothGrpd coDiscΓDiscΠ Grpd \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.


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

Exponentiated super L L_\infty-algebras

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

The Lie integration of 𝔤\mathfrak{g} is …



For general references see the references at super ∞-groupoid .

Discussion of smooth super \infty-groupoids:

Last revised on April 14, 2024 at 15:22:22. See the history of this page for a list of all contributions to it.