# What is a super-infinity-groupoid, homotopy-theoretically?

There's a sheafy way to write down the $$\infty$$-category of super-$$\infty$$-groupoids, as detailed on the nlab. Can this notion be reformulated by saying that a super-$$\infty$$-groupoid is an ordinary $$\infty$$-groupoid plus some homotopy-theoretic data?

The sheafy notion is this: let $$sMan$$ be the usual category of supermanifolds, regarded as a site with the usual Grothendieck topology (a cover is a jointly surjective collection of open embeddings). Consider the $$\infty$$-category $$Sh(sMan)$$ of sheaves of $$\infty$$-groupoids on this site, and then localize at the morphisms $$\mathbb R^{p|q} \to \mathbb R^{0|q}$$. If we did this same process with the site $$Man$$ of ordinary manifolds, we'd end up with the $$\infty$$-category of $$\infty$$-groupoids, so when we do it here, the resulting $$\infty$$-category deserves to be called the $$\infty$$-category $$sGpd_\infty$$ of super-$$\infty$$-groupoids. There is a natural forgetful functor $$|-|: sGpd_\infty \to Gpd_\infty$$ induced by the forgetful functor $$sMan \to Man$$.

Question: What additional homotopy-theoretic data is needed to recover a super-$$\infty$$-groupoid $$X$$ from its underlying $$\infty$$-groupoid $$|X|$$? Can this be formulated cleanly enough to give an alternative construction of the $$\infty$$-category $$sGpd_\infty$$?

The kind of data I have in mind is that maybe $$X$$ can be recovered from the data of $$|X|$$ plus maybe a map from $$|X|$$ into the classifying space of some sort of structure group, or something of that sort.

• According to the nLab page super-$\infty$-groupoids are indeed presheaves on the category of $\mathbb R^{0|q}$'s, and $|-|$ is evaluation at $\mathbb R^{0|0}$ (which is a zero object). Since this category is not a groupoid there isn't a simple "classifying space" description. It seems the maximal subgroupoid is the groupoid of real vector spaces, so a super-$\infty$-groupoid has an underlying collection of $\infty$-groupoids $X_n$ with the discrete group $GL_n(\mathbb R)$ acting on $X_n$... Sep 27, 2020 at 18:54
• Actually in your description you only localize wrt $\mathbb R^{p|q}\to \mathbb R^{0|q}$, not all $\mathbb R^{1|0}$-homotopies, so I think it's equivalent to the nLab. Sep 27, 2020 at 19:15
• @TimCampion I don't think there are any interesting homotopies between maps between 'superpoints' Sep 27, 2020 at 19:39
• @TimCampion You're right! Dylan: It seems to me any two maps are homotopic, but perhaps I am not using the definitions correctly. Sep 27, 2020 at 19:42
• If any two maps are made homotopic, then since there are maps in both directions everywhere, all maps are inverted, so any sheaf descends to the classifying space. Since the category of superpoints has a terminal object, the classifying space is contractible. So with the definition I've given, a super-$\infty$-groupoid is just an $\infty$-groupoid. I'm not sure what to make of the nlab's definition. Sep 27, 2020 at 21:18