The n-Category Café

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

Thanks for the question, Jim.

I used to complain, here and elsewhere, that the term “derived xyz-geometry” is unfortunate in its undescriptiveness (and not because of the xyz :-). But it has become so well established and popular that I figured if I want to make myself as well understood as possible in the space that I have for a blog entry headline, then I’d better do it that way.

More concretely concerning your question:

I started writing a reply to your question in the entry higher geometry in the section derived geometry.

You’ll meet old friends there, but now at their home. At their natural home, as one says. ;-)

Could you give us a sense of the difference it might make for synthetic differential geometry working with all finitely generated $C^\infty$-rings rather than just the free ones?

A free $C^\infty$-ring is one of the form $C^\infty(\mathbb{R}^n)$ – smooth functions on a cartesian space.

So ordinary sheaves on the site of formal duals of free $C^\infty$-rings are sheaves on the category of manifolds. These contain for instance diffeological spaces (as the full subcategory of concrete sheaves), but they do not contain the crucial ingredient of a smooth topos: a nontrivial infinitesimal interval $D = \{x \in R| x^2 = 0\}$.

Similarly, $\infty$-sheaves ($\infty$-stacks) on the category of formal duals of free $C^\infty$-rings are $\infty$-sheaves on the category of manifolds. This contains all $\infty$-Lie groupoids, but not for instance $\infty$-Lie algebroids - which are $\infty$-Lie groupoids with “infinitesimal morphisms”.

These infinitesimal objects enter the picture when we test not just on formal duals of free $C^\infty$-rings, but on all of them. Actually, we don’t need to test on all of them, it is sufficient to test on those that are the product of a free ring with a “Weil algebra”: a smooth ring of functions on an infinitesimal neighbourhood of a point.

Similarly when we pass to derived stacks here, by generalizing our test objects to cosimplicial test objects, i.e. our rings to simplicial rings: “$\infty$-$C^\infty$-rings”, if you wish.

For instance we may want to regard the BV-BRST complex as the Chevalley-Eilenberg algebra of a derived $\infty$-Lie algebroid. As such it exists in $\infty$-stacks on formal duals of simplicial $C^\infty$-rings, but not – I think – in $\infty$-stacks on just formal duals of simplicial free $C^\infty$-rings. Again, it is the notion of infinitesimal that is otherwise missing.

So the geometry $\mathcal{G}$ (in the sense of geometry for structured $(\infty,1)$-toposes) for a derived version of synthetic differential geometry should be the formal duals of simplicial $C^\infty$-rings, or similar.

Once the geometry is fixed, the “locally ringed topological spaces” with respect to it will be topological spaces $X$ with a structure sheaf exhibited by a suitable functor $\mathcal{G} \to Sh_{(\infty,1)}(X)$. That’s supposed to be the same as a suitable functor $\mathcal{T} \to Sh_{(\infty,1)}(X)$ for $\mathcal{T}$ the corresponding pregeometry.

Here formal duals of $C^\infty$-rings would form the pregeometry, and formal duals of simplicial (“derived”) $C^\infty$-rings the geometry.

In David Spivak’s work, essentially, up to some technical subtlety, the pregeometry $\mathcal{T}$ is taken to be the category of manifolds or Cartesian spaces, hence formal duals of free $C^\infty$-rings. Again, not that there is any problem with that, I am just making a remark on this from the point of view of what one might call “derived synthetic differential geometry”.

Of course, one has to be careful about a certain iteration of concepts here:

a $C^\infty$-ring it itself not unlike a “0-truncated structured $\infty$-topos” on the pregeometry of duals to free $C^\infty$-rings: namely a product preserving functor

$$CartSp \to Sh(*) = Set \,.$$

And it is this perspective that David Spivak varies, by paassing instead to functors

$$CartSp \to Sh_{(\infty,1)}(*) = sSet$$

($\infty-C^\infty$-rings) and then further to locally $\infty$-$C^\infty$-ringed topological spaces $X$

$$CartSp \to Sh_{(\infty,1)}(X)$$

that constitute the $\infty$-category $dDiff$ (or $dMan$) of “derived smooth manifolds”.

(I am just sketching the rough outline, saying this with all technical qualifiers fully spelled out takes a bit longer, of course.)

Now, from the perspective of derived synthetic differential geometry, what one could do to get the $\infty$-stacks with notions of infinitesimals that I mentioned at the beginning is to take this now as a new category of test spaces , i.e. to pass now from $\infty$-Lie groupoids/$\infty$-stacks on $Diff$ to $\infty$-stacks on $dDiff$!

There is some iteration process going on, where we bootstrap a notion of generalized spaces in terms of $\infty$-stacks on test spaces that are themselved generalized spaces obtained as $\infty$-stacks on other test spaces. It can get a bit confusing. Which is why I thought this is worth making a remark about.