The n-Category Café

Are you one of the participants? I am, guessing so, but I have to admit that I am being to dense to figure out who you are. Anyway, thanks for commenting here. Your comments/questions over at the GW-invariants were nice productive comments.

So, tomorrow over tea approach me and let me know who you are. Then I tell you the answer to your last question. :-)

On the other hand, maybe one should look at the embedding of $\mathbb{Z}_2$-graded algebras into $\mathbb{N}$-graded algebras given by the assignment

$$R : (A_{even} \oplus A_{odd}) \mapsto (A_0 \oplus A_1 \oplus A_2 \oplus \cdots) := (A_{even} \oplus A_{odd} \oplus A_{even} \oplus A_{odd} \oplus \cdots)$$

or rather its extension to an embedding of $\mathbb{Z}_2$-graded algebras equipped with an odd square-0 derivation into $\mathbb{N}$-graded differential algebras

$$R : (A_{even} \stackrel{\stackrel{d_{odd}}{\leftarrow}}{\stackrel{d_{ev}}{\to}} A_{odd}) \mapsto (A_0 \stackrel{d_0}{\to} A_1 \stackrel{d_1}{\to} A_2 \stackrel{d_2}{\to} \cdots) := (A_{even} \stackrel{d_{even}}{\to} A_{odd} \stackrel{d_{odd}}{\to} A_{even} \stackrel{d_{even}}{\to} A_{odd} \oplus \cdots)$$

Conversely, there is a map from $\mathbb{N}$-graded cochain algebras to $\mathbb{Z}_2$-graded algebras equipped with an odd square-0 derivation that maps

$$L : (A_0 \stackrel{d_0}{\to} A_1 \stackrel{d_1}{\to} A_2 \stackrel{d_2}{\to} \cdots) \mapsto (A_{even} \stackrel{\stackrel{d_{odd}}{\leftarrow}}{\stackrel{d_{ev}}{\to}} A_{odd}) := (\oplus_k A_{2k} \stackrel{\stackrel{\oplus_k d_{2k+1}}{\leftarrow}}{\stackrel{\oplus_k d_{2k}}{\to}} \oplus_k A_{2k+1})$$

This is the kind of map that is for instance used when turning ordinary integral cohomology into a periodic cohomology by setting

$$A^0(X) := \oplus_k H^{2k}(X,\mathbb{Z})$$

$$A^1(X) := \oplus_k H^{2k+1}(X,\mathbb{Z}) \,.$$

It seems these form an adjoint pair $(L \dashv R)$

$$R : \mathbb{Z}_2 DGA \stackrel{\leftarrow}{\to} \mathbb{N} DGA : L$$

since

$$Hom\left( \array{ A_0 \\ \downarrow^{d_0} \\ A_1 \\ \downarrow^{d_1} \\ A_2 \\ \downarrow^{d_2} \\ \cdots } \,, \;\; \array{ B_even \\ \downarrow^{d_{ev}} \\ B_{odd} \\ \downarrow^{d_{odd}} \\ B_{even} \\ \downarrow^{d_{ev}} \\ \cdots } \right) \simeq Hom\left( \array{ A_0 \oplus A_2 \oplus \cdots \\ \downarrow^{d_0 \oplus d_2 \oplus \cdots} \uparrow^{d_1 \oplus d_3 \oplus \cdots} \\ A_1 \oplus A_3 \oplus \cdots } \,, \;\; \array{ B_{even} \\ \downarrow^{d_{ev}} \uparrow^{d_{odd}} \\ B_{odd} } \right)$$

The structure of an NQ-supermanifold on an ordinary supermanifold would be precisely a lift of its algebra of functions through $L$.

(Well, either I am mixed up or this must be well known…)

One of the crucial new developments in the Stolz-Teichner program is that where previously there were attempts to use various notions of Riemannian metrics on supermanifolds to characterize superconformal or super-Euclidean structure, they are now saying that a much better way to achieve this is to use a generalized Klein’s-Erlanger-program-style-definition along the very lines that is described at $n$Lab:manifold:

instead of specifying extra metric-like structure we imagine the canonical such extra structure on our test spaces (the standard flat or conformal structure on things $\sim \mathbb{R}^{p|q}$) and then just require that all transition functions of charts respect this implicit structure in that they sit in a special subgroup of the generally possible group of transition function. See $n$Lab: Euclidean supermanifold for a bit of details.

I am wondering: this kind of Erlangen-program-style defnition of manifolds, supermanifolds etc with extra structure should have a very general abstract-nonsense formulation.

In particular, it should be applicable to the notion of generalized scheme, where one would require that the cover by generalized affine schemes involved would be required to yield on overlaps transition function in a prescribed $\infty$-group.

Has anyone thought about this?