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super smooth topos

superalgebra and (synthetic ) supergeometry

Background

Introductions

Superalgebra

Supergeometry

Supersymmetry

supersymmetry

Supersymmetric field theory

Applications

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synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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)

Last revised on October 15, 2009 at 10:04:53. See the history of this page for a list of all contributions to it.