nLab
synthetic differential super infinity-groupoid

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\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

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

tangent cohesion

differential cohesion

graded differential 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& 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)

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

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

  • Anatoly Konechny and Albert Schwarz,

    On (kl|q)(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 (kl|q)(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)