# nLab synthetic differential super infinity-groupoid

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

## Models

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

supersymmetry

# Contents

## Idea

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

## 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.

## References

• Anatoly Konechny and Albert Schwarz,

On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. 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

Created on January 3, 2013 07:10:43 by Urs Schreiber (89.204.139.220)