# nLab synthetic differential super infinity-groupoid

Contents

### Context

#### 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

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

supersymmetry

## Applications

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# 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.