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super smooth topos

**superalgebra** and **supergeometry** ## Formal context ## * superpoint * super ∞-groupoid, smooth super ∞-groupoid, synthetic differential super ∞-groupoid * synthetic differential supergeometry ## Superalgebra ## * super vector space, SVect * super algebra * Grassmann algebra * Clifford algebra * superdeterminant * super Lie algebra * super Poincare Lie algebra ## Supergeometry ## * supermanifold, SDiff * super Lie group * super translation group * super Euclidean group * NQ-supermanifold * super vector bundle * complex supermanifold * Euclidean supermanifold * integration over supermanifolds * Berezin integral ## Structures * supersymmetry * division algebra and supersymmetry ## Applications ## * supergravity * supersymmetric quantum mechanics * geometric model for elliptic cohomology

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*** **differential geometry** **synthetic differential geometry** ## Axiomatics ## * smooth topos * infinitesimal space * amazing right adjoint * Kock-Lawvere axiom * integration axiom * microlinear space * synthetic differential supergeometry * super smooth topos * . infinitesimal cohesion * de Rham space * formally smooth morphism, formally unramified morphism, formally étale morphism * jet bundle ## Models ## * Models for Smooth Infinitesimal Analysis * Fermat theory * smooth algebra ($C^\infty$-ring) * smooth locus * smooth manifold, formal smooth manifold, derived smooth manifold * smooth space, diffeological space, Frölicher space * smooth natural numbers * Cahiers topos * smooth ∞-groupoid * synthetic differential ∞-groupoid ## Concepts * tangent bundle, * vector field, tangent Lie algebroid; * differentiation, chain rule * differential forms * differential equation, variational calculus * Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space * connection on a bundle, connection on an ∞-bundle * Riemannian manifold * isometry, Killing vector field ## Theorems * Hadamard lemma * Borel's theorem * Boman's theorem * Whitney extension theorem * Steenrod-Wockel approximation theorem * Poincare lemma * Stokes theorem * de Rham theorem * Chern-Weil theory ## Applications * Lie theory, ∞-Lie theory * Chern-Weil theory, ∞-Chern-Weil theory * gauge theory * ∞-Chern-Simons theory * Klein geometry, Klein 2-geometry, higher Klein geometry * Euclidean geometry, Cartan geometry, higher Cartan geometry * Riemannian geometry * gravity, supergravity

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Contents

Idea

The notion of super smooth topos is to that of smooth topos as supergeometry is to differential geometry.

The objects of a super smooth topos are generalizations of smooth supermanifolds that admit besides the usual odd infinitesimals of supergeometry also the even infinitesimal objects of synthetic differential geometry.

The study of super smooth toposes is the content of synthetic differential supergeometry. (See there for references and details for the moment.)

Definition

A super smooth topos (𝒯,R)(\mathcal{T}, R) is a smooth topos together with the refinement of the kk-algebra structure on RR to that of a kk-superalgebra structure.

So a super smooth topos is a topos 𝒯\mathcal{T} equipped with a superalgebra object (R,+,)(R, +, \cdot) with even part R eR_e and odd part R oR_o etc.

An algebra spectrum object is now an internal object of superalgebra homomorphisms and the condition is that for every super Weil algebra W=RmW = R \oplus m we have that Spec(W)=RSAlg 𝒯(W,R)Spec(W) = R SAlg_{\mathcal{T}}(W,R) is an infinitesimal object and that WR SpecWW \to R^{Spec W} is an isomorphism.

This means that essentially all the standard general theory of smooth toposes goes through literally for super smooth toposes, too. The main difference is that a super smooth topos contains more types of infinitesimal objects.

There is for instance still the standard even infinitesimal interval

D:=D 1|0:={ϵR e|ϵ 2=0} D := D^{1|0} := \{\epsilon \in R_e | \epsilon^2 = 0\}

but there is now also the odd infinitesimal interval

D 0|1:={θR o}. D^{0|1} := \{\theta \in R_o \} \,.

Notice that in the graded commutative algebra AA every odd element θ\theta automatically squares to 0.

Urs Schreiber: I’d think that the cominatorial/simplicial definition of differential forms in synthetic differential geometry applied verbatim in a super smooth topos automatically yields the right/expected notion of differential forms in supergeometry.

Examples

Models for super smooth toposes are constructed in

  • D. N. Yetter, Models for synthetic supergeometry, Cahiers, 29, 2 (1988)

and

  • H. Nishimura, Supersmooth topoi, International Journal of Theoretical Physics, Volume 39, Number 5 (journal)
Revised on October 15, 2009 10:04:53 by Urs Schreiber (87.212.203.135)