nLab
super infinity-groupoid

Idea

A super $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{SuperPoint}$

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

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

Properties

Cohesion

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

Let

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

Superalgebra

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

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

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

$$𝕂:\mathrm{Spec}\Lambda \mapsto {\Lambda }_{\mathrm{even}}$$\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 $j(k)$ 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

$$𝕂\simeq j(k).$$\mathbb{K} \simeq j(k) \,.

Supergeometry

(…) supergeometry (…)

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

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

Exponentiated $\mathrm{\infty }$-Lie algebras

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

Definition

A super L-∞ algebra is an L-∞ algebra internal to super vector spaces.

The category of super $L_{\mathrm{\infty }}$-algebras is

$$S{L}_{\mathrm{\infty }}\mathrm{Alg}:=({\mathrm{ScdgAlg}}_{\mathrm{sf}}^{+}{)}^{\mathrm{op}}$$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 $L_{\mathrm{\infty }}$-algebra we write $\mathrm{CE}(𝔤)$ for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

Definition

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

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

on superpoints given by the assignment

$$\exp (𝔤):({ℝ}^{0\mid q},[k])\mapsto {\mathrm{Hom}}_{{\mathrm{dgsAlg}}_k}(\mathrm{CE}(𝔤,{\Omega }_{\mathrm{vert}}^{•}({ℝ}^{0\mid q}\times {\Delta }^k))).$$\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 ${ℝ}^{0\mid q}\times {\Delta }^k\to {ℝ}^{0\mid q}$ and with sitting instants (see Lie integration).

Note

For $q\in \mathbb{N}$ write ${\Lambda }_q:=C^{\mathrm{\infty }}({ℝ}^{0\mid q})$ 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 $(𝔤{\otimes }_k{\Lambda }_q{)}_{\mathrm{even}}$

$$\exp (𝔮)({ℝ}^{0\mid q})=\exp ((𝔤{\otimes }_k{\Lambda }_q{)}_{\mathrm{even}}).$$\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

$$\begin{array}{rl}{\mathrm{Hom}}_{\mathrm{dgsAlg}}(\mathrm{CE}(𝔤),{\Omega }_{\mathrm{vert}}^{•}({ℝ}^{0\mid q}\times {\Delta }^n))& \simeq {\mathrm{Hom}}_{\mathrm{dgsAlg}}(\mathrm{CE}(𝔤),C^{\mathrm{\infty }}({ℝ}^{0\mid q})\otimes {\Omega }^{•}({\Delta }^n))\\ & \simeq {\mathrm{Hom}}_{\mathrm{dgsAlg}}(\mathrm{CE}(𝔤),{\Lambda }_q\otimes {\Omega }^{•}({\Delta }^n))\\ & \subset {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}(\mathrm{sVect})}({𝔤}^{*}[1],{\Lambda }_q\otimes {\Omega }^{•}({\Delta }^n))\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}(\mathrm{sVect})}({𝔤}^{*}[1]\otimes ({\Lambda }^q{)}^{*},{\Omega }^{•}({\Delta }^n))\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}(\mathrm{sVect})}((𝔤\otimes {\Lambda }_q{)}^{*}[1],{\Omega }^{•}({\Delta }^n))\\ & \simeq {\mathrm{Hom}}_{{\mathrm{Ch}}^{•}(\mathrm{sVect})}((𝔤\otimes {\Lambda }_q{)}^{*}[1{]}_{\mathrm{even}},{\Omega }^{•}({\Delta }^n))\\ & \supset {\mathrm{Hom}}_{\mathrm{dgsAlg}}(\mathrm{CE}((𝔤{\otimes }_k{\Lambda }_q{)}_{\mathrm{even}}),{\Omega }^{•}({\Delta }^n))\end{array}.$$\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 $\mathrm{CE}(𝔤)$ 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.

Related concepts

• cohesive (∞,1)-topos

  • discrete ∞-groupoid

  • Euclidean-topological ∞-groupoid

  • smooth ∞-groupoid

  • synthetic differential ∞-groupoid

  • super ∞-groupoid

  • smooth super ∞-groupoid

  • synthetic differential super ∞-groupoid

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

• Albert Schwarz, On the definition of superspace Teoret. Mat. Fiz., 1984, Volume 60, Number 1, Pages 37–42 (Mi tmf5111), (russian original)

and in

• V. Molotkov., Infinite-dimensional ${\mathbb{Z}}_2^k$-supermanifolds , ICTP preprints, IC/84/183, 1984. (scan)

and in

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

See also

• Anatoly Konechny and Albert Schwarz,

  On $(k\oplus l\mid q)$-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 $(k\oplus l\mid q)$-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

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

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)