nLab synthetic differential super infinity-groupoid

Idea

The (∞,1)-topos of synthetic differential super $\infty$-groupoids combines the properties of that of

1. smooth super ∞-groupoids

2. synthetic differential ∞-groupoids.

Definition

Let CartSp$_{supersynth}$ be the site which is the full subcategory of that of formal duals of smooth superalgebras on those of the form

$$\mathbb{R}^p \times D \times \mathbb{R}^{0|q} \simeq \mathbb{R}^{p|q} \times D$$

where

• $\mathbb{R}^p$ is the Cartesian space of dimension $p$;

• $\mathbb{R}^{p|q}$ is the super vector space of dimension $(p|q)$ ($\mathbb{R}^{0|1}$ is the odd line);

• $D$ is an infinitesimally thickened point.

If $D$ here is the formal dual of the Artin algebra on $k$ commuting nilpotent elements, then such an object is written $\mathbb{R}^{p \oplus k|q}$ in (Konechny-Schwarz).

Let then

$$SynthDiffSuper\infty Grpd \coloneqq Sh_\infty(CartSp_{supersynth})$$

be the (∞,1)-category of (∞,1)-sheaves over this site.

Related concepts

• cohesive (∞,1)-topos

  • discrete ∞-groupoid

  • Euclidean-topological ∞-groupoid

  • smooth ∞-groupoid

  • synthetic differential ∞-groupoid

  • super ∞-groupoid

  • smooth super ∞-groupoid

References

• Anatoly Konechny and Albert Schwarz,

  On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume) Springer-Verlag (1998) Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)

  Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

• Urs Schreiber, section 4.5 of: differential cohomology in a cohesive topos

• Hisham Sati, Urs Schreiber, Section 3.1.3 of: Proper Orbifold Cohomology (arXiv:2008.01101)

• Urs Schreiber, Introduction to Higher Supergeometry, lecture at Higher Structures in M-Theory 2018, Durham Symposium

  published in parts as: Higher Structures in M-Theory (with Branislav Jurčo, Christian Saemann, Martin Wolf), Fortschritte der Physik (2019) (arXiv:1903.02807, doi:10.1002/prop.201910001)