### Derived Synthetic Differential Geometry

#### Posted by Urs Schreiber

Ordinary synthetic differential geometry – at least the well adapted models – is concerned with 0-truncated generalized spaces that are modeled on smooth loci: the formal duals of finitely generated $C^\infty$-rings.

Under **derived synthetic differential geometry** I suppose we should want to understand the study of the notions of space that are induced from the geometry (in the sense of geometry for structured $(\infty,1)$-toposes) that is the geometric envelope of the pregeometry constituted by $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, with one of its familiar site structures.

This would seem to be an excellent candidate for the ambient geometry in which most of fundamental physics, as presently conceived, takes place.

Maybe we can chat a bit about it here.

For convenience, I repeat the above paragraph, with hyperlinks to background information included:

Ordinary synthetic differential geometry – at least the well adapted models – is concerned with 0-truncated generalized spaces that are modeled on smooth loci: the formal duals of finitely generated $C^\infty$-rings. Under

derived synthetic differential geometryI suppose we should want to understand the study of the notions of space that are induced from the geometry (in the sense of geometry for structured $(\infty,1)$-toposes) that is the geometric envelope of the pregeometry constituted by $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, with one of its familiar site structures.

There are various things to talk about, I want to start with this aspect here:

the approach sketched in the very last paragraph of *Structured Spaces* and then carried out in some detail in

goes in the direction of *derived synthetic differential geometry* . But it is different from what is indicated above:

following Lurie, David Spivak considers,

*essentially*, the pregeometry given by $\mathcal{T} :=$ CartSp. This is the formal dual of free finitely generated $C^\infty$-rings.the above suggests to use $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, the formal dual of

*all*finitely generated $C^\infty$-rings.

I understand well how the choice of pregeometry in the above article is motivated, and have no quarrals with that, I also understand that $CartSp$ is not in fact quite a pregeometry, as discussed in the article, and I understand that in the original version of the thesis instead Diff is used, which is a pregeometry and is conjectured to yield an equivalent result as that discussed in the arXiv version. Still, $Diff$ is just a full subcategory $Diff \hookrightarrow (C^\infty Ring^{fin})^{op}$. I just think for the record this is a point that deserves some highlighting, obvious as it may be.

In fact, it would seem that we could go through sections 4.2 and 4.3 of Structured Spaces, and pretty much systematically replace the sites of ordinary rings appearing there by the corresponding sites of $C^\infty$-rings. Can’t we?

## Re: Derived ???

Is this `derived’ in a new sense or are notions of `up to homotopy’ or resolution by a suitably free object hiding here somewhere?