nLab
super smooth topos

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 $(𝒯,R)$ 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 $𝒯$ 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 $\mathrm{Spec}(W)=R{\mathrm{SAlg}}_{𝒯}(W,R)$ is an infinitesimal object and that $W\to R^{\mathrm{Spec}W}$ 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

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

but there is now also the odd infinitesimal interval

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

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)