nLab super infinity-groupoid

This entry is about ∞-groupoids parameterized over superpoints. Hence discrete ∞-groupoids equipped with super-structure. For smoothly supergeometric ∞-groupoids, parameterized over supermanifolds, see super smooth ∞-groupoid and super formal smooth ∞-groupoid.

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

Definition

Let $SuperPoints$ be the category of super points, regarded as a site with trivial coverage.

The (∞,1)-sheaf (∞,1)-topos over $SuperPoint$

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

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

Properties

Infinitesimal cohesion

Proposition

The (∞,1)-topos $Super\infty Grpd$, def. , is an infinitesimal cohesive (∞,1)-topos over ∞Grpd.

Proof

Being an (∞,1)-category of (∞,1)-presheaves the constant $\infty$-presheaf (∞,1)-functor

$$Disc \;\colon\; \infty Grpd \longrightarrow Super \infty Grpd$$

has a left adjoint $\Pi$ given by forming (∞,1)-colimits and a right adjoint $\Gamma$ given by (∞,1)-limits. Since the category $SuperPoints$ has a terminal object (the point $\mathbb{R}^{0|0}$) its opposite category has an initial object and so $\Gamma$ is just given by evaluation at that object. $\Gamma X \simeq X(\ast)$.

It follows that $\Gamma\circ Disc \simeq Id$ and hence (by the discussion at adjoint (∞,1)-functor) that $Disc$ is a full and faithful (∞,1)-functor.

Moreover, evaluation preserves (∞,1)-limits (since for (∞,1)-presheaves there are computed objectwise for each object of the site) and so by the adjoint (∞,1)-functor theorem there does exist a further right adjoint $coDisc \colon \infty Grpd \hookrightarrow Super \infty Grpd$. By the (∞,1)-Yoneda lemma and by adjointness this sends an ∞-groupoid $X$ to the (∞,1)-presheaf given by

$$coDisc(X) \;\colon\; \mathbb{R}^{0|q} \;\mapsto\; Super\infty Grpd(\mathbb{R}^{0|q}, coDisc(X)) \simeq \infty Grpd(\Gamma(\mathbb{R}^{0|q}), X) \,.$$

Now the crucial aspect is that $\Gamma(\mathbb{R}^{0|q}) \simeq \ast$ for all $q \in \mathbb{N}$ since every superpoint has a unique global point, this being the archetypical property of infinitesimally thickened points. So it follows that

$$coDisc \simeq Disc$$

and hence that $\Pi \simeq \Gamma$. Therefore $\Pi$ preserves in particular finite products so that $Super \infty Grpd$ is cohesive, but of course this shows now that it is in fact infinitesimally cohesive.

Relation to smooth super $\infty$-groupoids

Let

$$\begin{aligned} SmoothSuper\infty Grpd & := Sh_{(\infty,1)}(CartSp, Super\infty Grpd) \simeq Sh_{(\infty,1)}(CartSp\times SuperPoint, \infty Grpd) \\ & =: Sh_{(\infty,1)}(CartSp\times SuperPoint) \end{aligned}$$

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

For more on this see at smooth super ∞-groupoid.

Superalgebra

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

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

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

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

with ring structure induced over each super point $\mathbb{R}^{0|q} = Spec \Lambda = Spec \wedge^\bullet \mathbb{R}^q$ from the ring structure of the even part $\Lambda_{even}$ 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

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

Supergeometry

(…) supergeometry (…)

Structures in $Super \infty Grpd$

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

Exponentiated $\infty$-Lie algebras

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

Definition

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

The category of super $L_\infty$-algebras is

$$S L_\infty Alg := (sdgcAlg^+_{sf})^{op}$$

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 $\mathfrak{g}$ a super $L_\infty$-algebra we write $CE(\mathfrak{g})$ for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

Definition

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

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

on superpoints given by the assignment

$$\exp(\mathfrak{g}) \;\colon\; (\mathbb{R}^{0|q}, [k]) \mapsto Hom_{sdgcAlg_k}\big( CE(\mathfrak{g}), \Omega^\bullet_{vert,si}(\mathbb{R}^{0|q} \times \Delta^k) \big) \,.$$

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

Lemma

For $q \in \mathbb{N}$ write $\Lambda_q \coloneqq C^\infty(\mathbb{R}^{0|q})$ for the Grassmann algebra on $q$-generators, being the global functions on the super point $\mathbb{R}^{0|q}$.

Over $\mathbb{R}^{0|q}$ the super Lie integration from def is the ordinary Lie integration of the ordinary L-∞ algebra $(\mathfrak{g} \otimes_k \Lambda_q)_{even}$

$$\exp\left( \mathfrak{g} \right)\left(\mathbb{R}^{0|q}\right) = \exp\left( (\mathfrak{g}\otimes_k \Lambda_q)_{even} \right) \,.$$

Proof

This is the standard even rules mechanism. Using that the category $sVect$ of finite-dimensional super vector spaces is a compact closed category, we compute

$$\begin{aligned} Hom_{sdgcAlg} \left( CE(\mathfrak{g}), \Omega^\bullet_{vert,si}(\mathbb{R}^{0|q} \times \Delta^n) \right) & \simeq Hom_{sdgcAlg}\left( CE(\mathfrak{g}), C^\infty(\mathbb{R}^{0|q}) \otimes \Omega^\bullet( \Delta^n) \right) \\ & \simeq Hom_{sdgcAlg}\left( CE(\mathfrak{g}), \Lambda_q \otimes \Omega^\bullet( \Delta^n) \right) \\ & \subset Hom_{Ch^\bullet(sVect)}\left( \mathfrak{g}^*[1], \Lambda_q \otimes \Omega^\bullet( \Delta^n) \right) \\ & \simeq Hom_{Ch^\bullet(sVect)}\left( \mathfrak{g}^*[1]\otimes (\Lambda_q)^*, \Omega^\bullet( \Delta^n) \right) \\ & \simeq Hom_{Ch^\bullet(sVect)}\left( (\mathfrak{g} \otimes \Lambda_q)^*[1], \Omega^\bullet( \Delta^n) \right) \\ & \simeq Hom_{Ch^\bullet(sVect)}\left( (\mathfrak{g} \otimes \Lambda_q)^*[1]_{even}, \Omega^\bullet( \Delta^n) \right) \\ & \supset Hom_{sdgcAlg}\left( CE((\mathfrak{g}\otimes_k \Lambda_q)_{even}), \Omega^\bullet( \Delta^n) \right) \end{aligned} \,.$$

Here in the third step we used that the underlying dg-algebra of $CE(\mathfrak{g})$ 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 observation that the study of super-structures in mathematics is usefully 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, (russian original pdf)

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

and in

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

A summary/review is in the appendix of

• Anatoly Konechny and Albert Schwarz,

  On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)

  Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471-486

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

A fairly comprehensive and introductory review is 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)

Discussion of smooth super ∞-groupoids as such:

• 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)