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superalgebra and (synthetic ) supergeometry
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.
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
for the collection of formal smooth ∞-groupoids (see there) and finally
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)
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.
Passing to (∞,1)-categories of (∞,1)-sheaves, this yields, via (∞,1)-Kan extension, a sequence of adjoint quadruples as follows:
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
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 $Smooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd$ exhibits differential cohesion.
With this the rightmost adjoint quadruple gives the Aufhebung of $\Re \dashv \Im$ by $\rightsquigarrow \dashv Rh$ and the further opposition $\rightrightarrows \dashv \rightsquigarrow$.
$\phantom{A}$(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$\phantom{A}$ |
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)
Last revised on October 2, 2020 at 07:22:13. See the history of this page for a list of all contributions to it.