nLab super formal smooth infinity-groupoid

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

• CartSp for the site of Cartesian spaces;

• $InfPoint \coloneqq WAlg^{op}$ for the category of first-order infinitesimally thickened points (i.e. the formal duals of commutative algebras over the real numbers of the form $\mathbb{R}\oplus V$ with $V$ a finite-dimensional square-0 nilpotent ideal).

• $SuperPoint \coloneqq WAlg_{super}^{op}$ for the category of superpoints, by which we here mean the formal duals to commutative superalgebras which are super-Weil algebras.

There are then “semidirect product” sites $CartSp \rtimes InfinPoint$ and $CartSp \rtimes SuperPoint$ (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

$$FormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes InfPoint)$$

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

$$SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)$$

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)

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

Here

• the first inclusion picks the terminal object $\mathbb{R}^0$;

• the second inclusion is that of reduced objects; the coreflection is reduction, sending an algebra to its reduced algebra;

• the third inclusion is that of even-graded algebras, the reflection sends a $\mathbb{Z}_2$-graded algebra to its even-graded part, the co-reflection sends a $\mathbb{Z}_2$-graded algebra to its quotient by the ideal generated by its odd part, see at superalgebra – Adjoints to the inclusion of plain algebras.

Remark

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

$$\array{ & && && &\longleftarrow& \\ & && &\hookrightarrow& &\hookrightarrow& \\ & &\longleftarrow& &\longleftarrow& &\longleftarrow& \\ \Delta \colon & \infty Grpd &\hookrightarrow& Smooth \infty Grpd &\hookrightarrow& FormalSmooth \infty Grpd &\hookrightarrow& SuperFormalSmooth \infty Grpd \\ & &\longleftarrow& &\longleftarrow& \\ & &\hookrightarrow& }$$

Proposition

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

• on the left the shape modality $\int$, flat modality $\flat$ and sharp modality $\sharp$,

• in the middle yields the reduction modality $\Re$, the infinitesimal shape modality $\Im$ and the infinitesimal flat modality $\&$.

• on the right we get an adjoint triple whose whose middle bit $\rightsquigarrow$ is the bosonic modality and whose left piece $\rightrightarrows$ produces super-even components, containing all the “fermion currents” if one wishes , which in this unity of opposites hence deserves to be called the fermionic modality. The further right adjoint $Rh$ is the rheonomy modality.

Hence we get a process of adjoint modalities of the form

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

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 $Smooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd$ exhibits differential cohesion.

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

Related concepts

• super-Cartan geometry

• higher Cartan geometry

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

References

• Urs Schreiber, section 4.5 of: differential cohomology in a cohesive topos

• Hisham Sati, Urs Schreiber, Section 3.1.3 of: Proper Orbifold Cohomology (arXiv:2008.01101)

• Urs Schreiber, Introduction to Higher Supergeometry, lecture at Higher Structures in M-Theory 2018, Durham Symposium

  published in parts as: Higher Structures in M-Theory (with Branislav Jurčo, Christian Saemann, Martin Wolf), Fortschritte der Physik (2019) (arXiv:1903.02807, doi:10.1002/prop.201910001)