# nLab super infinity-groupoid

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $\left(\infty ,1\right)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion

superalgebra

and

supergeometry

# Contents

## Idea

A super $\infty$-groupoid is an ∞-groupoid modeled on super points.

The notion subsumes and generalizes that of bare super groups, but not that of super Lie groups, the latter are instead examples of smooth super ∞-groupoids sitting over the base of super $\infty$-groupoids.

## Definition

Consider the category $\mathrm{SuperPoint}$ of super points as a site with trivial coverage.

We say that the (∞,1)-sheaf (∞,1)-topos over $\mathrm{SuperPoint}$

$\mathrm{Super}\infty \mathrm{Grpd}:={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{SuperPoint}\right)$Super\infty Grpd := Sh_{(\infty,1)}(SuperPoint)

is the (∞,1)-topos of super $\infty$-groupoids.

## Properties

### Cohesion

$\mathrm{Super}\infty \mathrm{Grpd}$ is a cohesive (∞,1)-topos.

Let

$\begin{array}{rl}\mathrm{SmoothSuper}\infty \mathrm{Grpd}& :={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{CartSp},\mathrm{Super}\infty \mathrm{Grpd}\right)\simeq {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{CartSp}×\mathrm{SuperPoint},\infty \mathrm{Grpd}\right)\\ & =:{\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{CartSp}×\mathrm{SuperPoint}\right)\end{array}$\begin{aligned} SmoothSuper\infty Grpd & := Sh_{(\infty,1)}(CartSp, Super\infty Grpd) \simeq Sh_{(\infty,1)}(CartSp\times SuperPoint, \infty Grpd) \\ & =: Sh_{(\infty,1)}(CartSp\times SuperPoint) \end{aligned}

be the (∞,1)-sheaf (∞,1)-topos of smooth super ∞-groupoids. This is cohesive over the base topos $\mathrm{Super}\infty \mathrm{Grpd}$.

### Superalgebra

$\mathrm{Super}\infty \mathrm{Grpd}$ is naturally a ringed topos, with commutative ring-object

$𝕂\in \mathrm{Super}\infty \mathrm{Grpd}$\mathbb{K} \in Super \infty Grpd

which as a presheaf $𝕂:{\mathrm{SuperPoint}}^{\mathrm{op}}\simeq \mathrm{GrAlg}\to \mathrm{Set}↪\mathrm{sSet}$ is given by

$𝕂:\mathrm{Spec}\Lambda ↦{\Lambda }_{\mathrm{even}}$\mathbb{K} : Spec \Lambda \mapsto \Lambda_{even}

with ring structure induced over each super point ${ℝ}^{0\mid q}=\mathrm{Spec}\Lambda =\mathrm{Spec}{\wedge }^{•}{ℝ}^{q}$ from the ring structure of the even part ${\Lambda }_{\mathrm{even}}$ of the Grassmann algebra $\lambda$.

The higher algebra over this ring object is what is called superalgebra. See there for details on this.

For $k$ the ground field and $j\left(k\right)$ its embedding as a super vector space into the topos by the map discussed at superalgebra – In the topos over superpoints – K-modules we have

$𝕂\simeq j\left(k\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{K} \simeq j(k) \,.

### Supergeometry

(…) supergeometry (…)

## Structures in $\mathrm{Super}\infty \mathrm{Grpd}$

We discuss the general abstract structures in a cohesive (∞,1)-topos realized in $\mathrm{Super}\infty \mathrm{Grpd}$.

### Exponentiated $\infty$-Lie algebras

We discuss Exponentiated ∞-Lie algebras in $\mathrm{Super}\infty \mathrm{Grpd}$.

###### Definition

The category of super ${L}_{\infty }$-algebras is

$S{L}_{\infty }\mathrm{Alg}:=\left({\mathrm{ScdgAlg}}_{\mathrm{sf}}^{+}{\right)}^{\mathrm{op}}$S L_\infty Alg := (ScdgAlg^+_{sf})^{op}

the opposite category of semi-free dg-algebras in super vector spaces: commutative monoids in the category of cochain complexes of super vector spaces whose underlying commutative graded algebra is free on generators in positive degree.

For $𝔤$ a super ${L}_{\infty }$-algebra we write $\mathrm{CE}\left(𝔤\right)$ for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

###### Definition

For $𝔤$ a super ${L}_{\infty }$-algebra, its Lie integration is the super $\infty$-groupoid presented by the simplicial presheaf

$\mathrm{exp}\left(𝔤\right)\in \left[{\mathrm{SuperPoint}}^{\mathrm{op}},\mathrm{sSet}\right]$\exp(\mathfrak{g}) \in [SuperPoint^{op}, sSet]

on superpoints given by the assignment

$\mathrm{exp}\left(𝔤\right):\left({ℝ}^{0\mid q},\left[k\right]\right)↦{\mathrm{Hom}}_{{\mathrm{dgsAlg}}_{k}}\left(\mathrm{CE}\left(𝔤,{\Omega }_{\mathrm{vert}}^{•}\left({ℝ}^{0\mid q}×{\Delta }^{k}\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(\mathfrak{g}) : (\mathbb{R}^{0|q}, [k]) \mapsto Hom_{dgsAlg_k}( CE(\mathfrak{g}, \Omega^\bullet_{vert}(\mathbb{R}^{0|q} \times \Delta^k)) ) \,.

Here on the right we have vertical differential forms with respect to the projection of supermanifolds ${ℝ}^{0\mid q}×{\Delta }^{k}\to {ℝ}^{0\mid q}$ and with sitting instants (see Lie integration).

###### Note

For $q\in ℕ$ write ${\Lambda }_{q}:={C}^{\infty }\left({ℝ}^{0\mid q}\right)$ for the Grassmann algebra on $q$-generators, being the global functions on the super point ${ℝ}^{0\mid q}$.

Over ${ℝ}^{0\mid q}$ the super Lie integration from def 2 is the ordinary Lie integration of the ordinary L-∞ algebra $\left(𝔤{\otimes }_{k}{\Lambda }_{q}{\right)}_{\mathrm{even}}$

$\mathrm{exp}\left(𝔮\right)\left({ℝ}^{0\mid q}\right)=\mathrm{exp}\left(\left(𝔤{\otimes }_{k}{\Lambda }_{q}{\right)}_{\mathrm{even}}\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(\mathfrak{q})(\mathbb{R}^{0|q}) = \exp( (\mathfrak{g}\otimes_k \Lambda_q)_{even} ) \,.
###### Proof

This is the standard even rules mechanism: write ${\Lambda }^{q}$ for the Grassmann algebra of duals on the generators of ${\Lambda }_{q}$. Then using that the category $\mathrm{sVect}$ of finite-dimensional super vector spaces is a compact closed category, we compute

$\begin{array}{rl}{\mathrm{Hom}}_{\mathrm{dgsAlg}}\left(\mathrm{CE}\left(𝔤\right),{\Omega }_{\mathrm{vert}}^{•}\left({ℝ}^{0\mid q}×{\Delta }^{n}\right)\right)& \simeq {\mathrm{Hom}}_{\mathrm{dgsAlg}}\left(\mathrm{CE}\left(𝔤\right),{C}^{\infty }\left({ℝ}^{0\mid q}\right)\otimes {\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \simeq {\mathrm{Hom}}_{\mathrm{dgsAlg}}\left(\mathrm{CE}\left(𝔤\right),{\Lambda }_{q}\otimes {\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \subset {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}\left(\mathrm{sVect}\right)}\left({𝔤}^{*}\left[1\right],{\Lambda }_{q}\otimes {\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}\left(\mathrm{sVect}\right)}\left({𝔤}^{*}\left[1\right]\otimes \left({\Lambda }^{q}{\right)}^{*},{\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}\left(\mathrm{sVect}\right)}\left(\left(𝔤\otimes {\Lambda }_{q}{\right)}^{*}\left[1\right],{\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}\left(\mathrm{sVect}\right)}\left(\left(𝔤\otimes {\Lambda }_{q}{\right)}^{*}\left[1{\right]}_{\mathrm{even}},{\Omega }^{•}\left({\Delta }^{n}\right)\right)\\ & \supset {\mathrm{Hom}}_{\mathrm{dgsAlg}}\left(\mathrm{CE}\left(\left(𝔤{\otimes }_{k}{\Lambda }_{q}{\right)}_{\mathrm{even}}\right),{\Omega }^{•}\left({\Delta }^{n}\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} Hom_{dgsAlg}(CE(\mathfrak{g}), \Omega^\bullet_{vert}(\mathbb{R}^{0|q} \times \Delta^n)) & \simeq Hom_{dgsAlg}( CE(\mathfrak{g}), C^\infty(\mathbb{R}^{0|q}) \otimes \Omega^\bullet( \Delta^n) ) \\ & \simeq Hom_{dgsAlg}( CE(\mathfrak{g}), \Lambda_q \otimes \Omega^\bullet( \Delta^n) ) \\ & \subset Hom_{Ch^\bullet(sVect)}(\mathfrak{g}^*[1] , \Lambda_q \otimes \Omega^\bullet( \Delta^n)) \\ & \simeq Hom_{Ch^\bullet(sVect)}(\mathfrak{g}^*[1]\otimes (\Lambda^q)^* , \Omega^\bullet( \Delta^n)) \\ & \simeq Hom_{Ch^\bullet(sVect)}((\mathfrak{g} \otimes \Lambda_q)^*[1] , \Omega^\bullet( \Delta^n)) \\ & \simeq Hom_{Ch^\bullet(sVect)}((\mathfrak{g} \otimes \Lambda_q)^*[1]_{even} , \Omega^\bullet( \Delta^n)) \\ & \supset Hom_{dgsAlg}( CE((\mathfrak{g}\otimes_k \Lambda_q)_{even}), \Omega^\bullet( \Delta^n)) \end{aligned} \,.

Here in the third step we used that the underlying dg-algebra of $\mathrm{CE}\left(𝔤\right)$ is free to find the space of morphisms of dg-algebras inside that of super-vector spaces (of generators) as indicated. Since the differential on both sides is ${\Lambda }_{q}$-linear, the claim follows.

## References

The proposal that the study of super-structures in mathematics should be regarded as taking place over the base topos on the site of super points has been made around 1984 in

and in

• V. Molotkov., Infinite-dimensional ${ℤ}_{2}^{k}$-supermanifolds , ICTP preprints, IC/84/183, 1984. (scan)

and in

• Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz., 60(1):43–48, 1984.

• Anatoly Konechny and Albert Schwarz,

On $\left(k\oplus l\mid q\right)$-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 $\left(k\oplus l\mid q\right)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 { 486

• Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

A comprehensive discussion of the situation over the site of superpoints is given in

The site of formal duals not just to Grassmann algebras but to all super-infinitesimally thickened points is discussed in

• L. Balduzzi, C. Carmeli, R. Fioresi, The local functors of points of Supermanifolds (arXiv:0908.1872)

Revised on February 22, 2013 02:33:58 by Urs Schreiber (80.81.16.253)