super smooth topos

**superalgebra** and (synthetic
) **supergeometry**
## Background
* algebra
* geometry
* graded object
## Introductions
* geometry of physics -- superalgebra
* geometry of physics -- supergeometry
## Superalgebra ##
* super vector space, SVect
* super algebra
* supercommutative superalgebra
* Grassmann algebra
* Clifford algebra
* superdeterminant
* super Lie algebra
* super Poincare Lie algebra
* super L-infinity algebra
## Supergeometry ##
* superpoint
* super Cartesian space
* supermanifold, SDiff
* NQ-supermanifold
* super vector bundle
* complex supermanifold
* Euclidean supermanifold
* super spacetime
* super Minkowski spacetime
* integration over supermanifolds
* Berezin integral
* super Lie group
* super translation group
* super Euclidean group
* super ∞-groupoid
* super formal smooth ∞-groupoid
* super line 2-bundle
## Supersymmetry
supersymmetry
* division algebra and supersymmetry
* super Poincare Lie algebra
* supermultiplet
* BPS state
* M-theory super Lie algebra, type II super Lie algebra
* supergravity Lie 3-algebra, supergravity Lie 6-algebra
## Supersymmetric field theory
* superfield
* supersymmetric quantum mechanics
* superparticle
* adinkra
* super Yang-Mills theory
* supergravity
* gauged supergravity
* superstring theory
* type II string theory
* heterotic string theory
## Applications
* geometric model for elliptic cohomology
***
**synthetic differential geometry**
**Introductions**
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
**Differentials**
* differentiation, chain rule
* differentiable function
* infinitesimal space, infinitesimally thickened point, amazing right adjoint
**V-manifolds**
* differentiable manifold, coordinate chart, atlas
* smooth manifold, smooth structure, exotic smooth structure
* analytic manifold, complex manifold
* formal smooth manifold, derived smooth manifold
**smooth space**
* diffeological space, Frölicher space
* manifold structure of mapping spaces
**Tangency**
* tangent bundle, frame bundle
* vector field, multivector field, tangent Lie algebroid;
* differential forms, de Rham complex, Dolbeault complex
* pullback of differential forms, invariant differential form, Maurer-Cartan form, horizontal differential form,
* cogerm differential form
* integration of differential forms
* local diffeomorphism, formally étale morphism
* submersion, formally smooth morphism,
* immersion, formally unramified morphism,
* de Rham space, crystal
* infinitesimal disk bundle
**The magic algebraic facts**
* embedding of smooth manifolds into formal duals of R-algebras
* smooth Serre-Swan theorem
* derivations of smooth functions are vector fields
**Theorems**
* Hadamard lemma
* Borel's theorem
* Boman's theorem
* Whitney extension theorem
* Steenrod-Wockel approximation theorem
* Whitney embedding theorem
* Poincare lemma
* Stokes theorem
* de Rham theorem
* Hochschild-Kostant-Rosenberg theorem
* differential cohomology hexagon
**Axiomatics**
* Kock-Lawvere axiom
* smooth topos, super smooth topos
* microlinear space
* integration axiom
**cohesion**
* (shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
* discrete object, codiscrete object, concrete object
* points-to-pieces transform
* structures in cohesion
* dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
**tangent cohesion**
* differential cohomology diagram
**differential cohesion**
* (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \)$
* reduced object, coreduced object, formally smooth object
* formally étale map
* [structures in differential cohesion](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion)
**graded differential cohesion**
* fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
$$
\array{
id \dashv id
\\
\vee \vee
\\
\stackrel{fermionic}{} \rightrightarrows \dashv \rightsquigarrow \stackrel{bosonic}{}
\\
\bot \bot
\\
\stackrel{bosonic}{} \rightsquigarrow \dashv Rh \stackrel{rheonomic}{}
\\
\vee \vee
\\
\stackrel{reduced}{} \Re \dashv \Im \stackrel{infinitesimal}{}
\\
\bot \bot
\\
\stackrel{infinitesimal}{} \Im \dashv \ \stackrel{\text{étale}}{}
\\
\vee \vee
\\
\stackrel{cohesive}{} ʃ \dashv \flat \stackrel{discrete}{}
\\
\bot \bot
\\
\stackrel{discrete}{} \flat \dashv \sharp \stackrel{continuous}{}
\\
\vee \vee
\\
\emptyset \dashv \ast
}
$$
{#Diagram}
**Models**
{#models_2}
* Models for Smooth Infinitesimal Analysis
* smooth algebra ($C^\infty$-ring)
* smooth locus
* Fermat theory
* Cahiers topos
* smooth ∞-groupoid
* formal smooth ∞-groupoid
* super formal smooth ∞-groupoid
**Lie theory, ∞-Lie theory**
* Lie algebra, Lie n-algebra, L-∞ algebra
* Lie group, Lie 2-group, smooth ∞-group
**differential equations, variational calculus**
* D-geometry, D-module
* jet bundle
* variational bicomplex, Euler-Lagrange complex
* Euler-Lagrange equation, de Donder-Weyl formalism,
* phase space
**Chern-Weil theory, ∞-Chern-Weil theory**
* connection on a bundle, connection on an ∞-bundle
* differential cohomology
* ordinary differential cohomology, Deligne complex
* differential K-theory
* differential cobordism cohomology
* parallel transport, higher parallel transport, fiber integration in differential cohomology
* holonomy, higher holonomy
* gauge theory, higher gauge theory
* Wilson line, Wilson surface
**Cartan geometry (super, higher)**
* Klein geometry, (higher)
* G-structure, torsion of a G-structure
* Euclidean geometry, hyperbolic geometry, elliptic geometry
* (pseudo-)Riemannian geometry
* orthogonal structure
* isometry, Killing vector field, Killing spinor
* spacetime, super-spacetime
* complex geometry
* symplectic geometry
* conformal geometry

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.)

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

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

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

$D := D^{1|0} := \{\epsilon \in R_e | \epsilon^2 = 0\}$

but there is now also the *odd* infinitesimal interval

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

Notice that in the graded commutative algebra $A$ every odd element $\theta$ 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.

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)