nLab
super infinity-groupoid

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

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

Definition

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

The (∞,1)-sheaf (∞,1)-topos over SuperPointSuperPoint

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

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

Properties

Infinitesimal cohesion

Proposition

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

Proof

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

Disc:GrpdSuperGrpd Disc \;\colon\; \infty Grpd \longrightarrow Super \infty Grpd

has a left adjoint Π\Pi given by forming (∞,1)-colimits and a right adjoint Γ\Gamma given by (∞,1)-limits. Since the category SuperPointsSuperPoints has a terminal object (the point 00\mathbb{R}^{0|0}) its opposite category has an initial object and so Γ\Gamma is just given by evaluation at that object. ΓXX(*)\Gamma X \simeq X(\ast).

It follows that ΓDiscId\Gamma\circ Disc \simeq Id and hence (by the discussion at adjoint (∞,1)-functor) that DiscDisc 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:GrpdSuperGrpdcoDisc \colon \infty Grpd \hookrightarrow Super \infty Grpd. By the (∞,1)-Yoneda lemma and by adjointness this sends an ∞-groupoid XX to the (∞,1)-presheaf given by

coDisc(X): 0qSuperGrpd( 0q,coDisc(X))Grpd(Γ( 0q),X). 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 Γ( 0q)*\Gamma(\mathbb{R}^{0|q}) \simeq \ast for all qq \in \mathbb{N} since every superpoint has a unique global point, this being the archetypical property of infinitesimally thickened points. So it follows that

coDiscDisc coDisc \simeq Disc

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

Relation to smooth super \infty-groupoids

Let

SmoothSuperGrpd :=Sh (,1)(CartSp,SuperGrpd)Sh (,1)(CartSp×SuperPoint,Grpd) =:Sh (,1)(CartSp×SuperPoint) \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 SuperGrpdSuper \infty Grpd.

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

Superalgebra

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

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

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

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

with ring structure induced over each super point 0q=SpecΛ=Spec q\mathbb{R}^{0|q} = Spec \Lambda = Spec \wedge^\bullet \mathbb{R}^q from the ring structure of the even part Λ even\Lambda_{even} of the Grassmann algebra λ\lambda.

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

For kk the ground field and j(k)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

𝕂j(k). \mathbb{K} \simeq j(k) \,.

Supergeometry

(…) supergeometry (…)

Structures in SuperGrpdSuper \infty Grpd

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

Exponentiated \infty-Lie algebras

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

Definition

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

The category of super L L_\infty-algebras is

SL Alg:=(ScdgAlg sf +) 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\mathfrak{g} a super L L_\infty-algebra we write CE(𝔤)CE(\mathfrak{g}) for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

Definition

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

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

on superpoints given by the assignment

exp(𝔤):( 0q,[k])Hom dgsAlg k(CE(𝔤,Ω vert ( 0q×Δ 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 0q×Δ k 0q\mathbb{R}^{0|q} \times \Delta^k \to \mathbb{R}^{0|q} and with sitting instants (see Lie integration).

Note

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

Over 0q\mathbb{R}^{0|q} the super Lie integration from def 3 is the ordinary Lie integration of the ordinary L-∞ algebra (𝔤 kΛ q) even(\mathfrak{g} \otimes_k \Lambda_q)_{even}

exp(𝔮)( 0q)=exp((𝔤 kΛ q) 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 Λ q\Lambda^q for the Grassmann algebra of duals on the generators of Λ q\Lambda_q. Then using that the category sVectsVect of finite-dimensional super vector spaces is a compact closed category, we compute

Hom dgsAlg(CE(𝔤),Ω vert ( 0q×Δ n)) Hom dgsAlg(CE(𝔤),C ( 0q)Ω (Δ n)) Hom dgsAlg(CE(𝔤),Λ qΩ (Δ n)) Hom Ch (sVect)(𝔤 *[1],Λ qΩ (Δ n)) Hom Ch (sVect)(𝔤 *[1](Λ q) *,Ω (Δ n)) Hom Ch (sVect)((𝔤Λ q) *[1],Ω (Δ n)) Hom Ch (sVect)((𝔤Λ q) *[1] even,Ω (Δ n)) Hom dgsAlg(CE((𝔤 kΛ q) even),Ω (Δ n)). \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 CE(𝔤)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 Λ q\Lambda_q-linear, the claim follows.

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

and in

  • V. Molotkov., Infinite-dimensional 2 k\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 (klq)(k \oplus l|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 (klq)(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)

An fairly comprehensive and introductory review is 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 October 23, 2013 10:08:04 by Urs Schreiber (82.169.114.243)