nLab synthetic differential super infinity-groupoid

Contents

Context

Cohesive \infty-Toposes

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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)

Super-Geometry

\infty-Lie theory

∞-Lie theory (higher geometry)

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

  1. smooth super ∞-groupoids

  2. synthetic differential ∞-groupoids.

Definition

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

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

where

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

Let then

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

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

References

Last revised on October 2, 2020 at 07:23:46. See the history of this page for a list of all contributions to it.