nLab
super formal smooth 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

Super-Geometry

Contents

Idea

The concept of super smooth \infty-groupoid or super smooth geometric homotopy type is the combination of super ∞-groupoid and smooth ∞-groupoid. The cohesive (∞,1)-topos of smooth super-\infty-groupoids is a context that realizes higher supergeometry.

Super smooth \infty-groupoids include supermanifolds, super Lie groups and their deloopings etc. Under Lie differentiation these map to super L-∞ algebras.

Definition

We consider one of at least two possible definitions, that differ (only) in some fine technical detail. The other is at smooth super infinity-groupoid.

Definition

Write

There are then “semidirect product” sites CartSpInfinPointCartSp \rtimes InfinPoint and CartSpSuperPointCartSp \rtimes SuperPoint (whose objects are Cartesian products of the given form inside synthetic differential supergeometry and whose morphisms are all morphisms in that context (not just the product morphisms)).

Set then

FormalSmoothGrpdSh (CartSpInfPoint) FormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes InfPoint)

for the collection of formal smooth ∞-groupoids (see there) and finally

SuperSmoothGrpdSh (CartSpSuperPoint) SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)

for that of super smooth \infty-groupoid

Properties

Cohesion

Remark

The sites in question are alternatingly (co-)reflective subcategories of each other (we always display left adjoints above their right adjoints)

*CartSpCartSpInfPointCartSpSuperPoint. \ast \stackrel{\longleftarrow}{\hookrightarrow} CartSp \stackrel{\hookrightarrow}{\longleftarrow} CartSp\rtimes InfPoint \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} CartSp \rtimes SuperPoint \,.

Here

Remark

Passing to (∞,1)-categories of (∞,1)-sheaves, this yields, via (∞,1)-Kan extension, a sequence of adjoint quadruples as follows:

Δ: Grpd SmoothGrpd FormalSmoothGrpd SuperFormalSmoothGrpd \array{ & && && &\longleftarrow& \\ & && &\hookrightarrow& &\hookrightarrow& \\ & &\longleftarrow& &\longleftarrow& &\longleftarrow& \\ \Delta \colon & \infty Grpd &\hookrightarrow& Smooth \infty Grpd &\hookrightarrow& FormalSmooth \infty Grpd &\hookrightarrow& SuperFormalSmooth \infty Grpd \\ & &\longleftarrow& &\longleftarrow& \\ & &\hookrightarrow& }
Proposition

Passing to the adjoint triples of idempotent monads and idempotent comonads which this induces, then yields

Hence we get a process of adjoint modalities of the form

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \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{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

where “\vee” denotes inclusion of modal types. The first level is cohesion, the second is differential cohesion (elasticity), the third is a further refinement given by supergeometry, which takes further “square roots” of all infinitesimal generators.

Proof

All the sites are ∞-cohesive sites, which gives that we have an cohesive (infinity,1)-topos. The composite inclusion on the right is an ∞-cohesive neighbourhood site, whence the inclusion SmoothGpdSuperFormalSmoothGrpdSmooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd exhibits differential cohesion.

With this the rightmost adjoint quadruple gives the Aufhebung of \Re \dashv \Im by Rh\rightsquigarrow \dashv Rh and the further opposition \rightrightarrows \dashv \rightsquigarrow.

References

Revised on July 13, 2017 18:10:49 by Urs Schreiber (178.6.113.200)