structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
superalgebra and (synthetic ) supergeometry
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.
Let $SuperPoints$ be the category of super points, regarded as a site with trivial coverage.
The (∞,1)-sheaf (∞,1)-topos over $SuperPoint$
we call the (∞,1)-topos of super $\infty$-groupoids.
The (∞,1)-topos $Super\infty Grpd$, def. 1, is an infinitesimal cohesive (∞,1)-topos over ∞Grpd.
Being an (∞,1)-category of (∞,1)-presheaves the constant $\infty$-presheaf (∞,1)-functor
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
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
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.
Let
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.
$Super \infty Grpd$ is naturally a ringed topos, with commutative ring-object
which as a presheaf $\mathbb{K} : SuperPoint^{op} \simeq GrAlg \to Set \hookrightarrow sSet$ is given by
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
(…) supergeometry (…)
We discuss the general abstract structures in a cohesive (∞,1)-topos realized in $Super \infty Grpd$.
We discuss Exponentiated ∞-Lie algebras in $Super \infty Grpd$.
A super L-∞ algebra is an L-∞ algebra internal to super vector spaces.
The category of super $L_\infty$-algebras is
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.
For $\mathfrak{g}$ a super $L_\infty$-algebra, its Lie integration is the super $\infty$-groupoid presented by the simplicial presheaf
on superpoints given by the assignment
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).
For $q \in \mathbb{N}$ write $\Lambda_q := C^\infty(\mathbb{R}^{0|q})$ 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 3 is the ordinary Lie integration of the ordinary L-∞ algebra $(\mathfrak{g} \otimes_k \Lambda_q)_{even}$
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 $sVect$ of finite-dimensional super vector spaces is a compact closed category, we compute
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.
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
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 (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)
Theory of $(k \oplus l|q)$-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