nLab super formal smooth infinity-groupoid



Cohesive \infty-Toposes




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.


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.



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




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 \,.



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& }

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.


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.

geometries of physics

A\phantom{A}(higher) geometryA\phantom{A}A\phantom{A}siteA\phantom{A}A\phantom{A}sheaf toposA\phantom{A}A\phantom{A}∞-sheaf ∞-toposA\phantom{A}
A\phantom{A}discrete geometryA\phantom{A}A\phantom{A}PointA\phantom{A}A\phantom{A}SetA\phantom{A}A\phantom{A}Discrete∞GrpdA\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}CartSpA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}A\phantom{A}Smooth∞GrpdA\phantom{A}
A\phantom{A}formal geometryA\phantom{A}A\phantom{A}FormalCartSpA\phantom{A}A\phantom{A}FormalSmoothSetA\phantom{A}A\phantom{A}FormalSmooth∞GrpdA\phantom{A}


Last revised on October 2, 2020 at 07:22:13. See the history of this page for a list of all contributions to it.