The n-Category Café

What’s a super $L_\infty$-algebra? (see here for what an ordinary $L_\infty$-algebra is.) Let $SVect$ be the category of $\mathbb{Z}_2$-graded vector spaces equipped with the unique non-trivial symmetric braiding. A super $L_\infty$-algebra is an $L_\infty$-algebra internal to that. So, dually, it’s a $\mathbb{N}$-$\mathbb{Z}_2$-bigraded quasi-free differential graded commutative algebra.

The archetypical example of such a beast is the super tangent Lie algebroid of any smooth superspace $X$: the dual dgc algebra is just that of super-differential forms $\Omega^\bullet(X)$ on $X$. For me here, a smooth superspace is a sheaf on the site $S$ whose objects are superEuclidean spaces and whose morphisms are smooth maps between these $$Obj(S) = \mathbb{N}\times\mathbb{N}$$ $$S(n|n',m|m') = SmoothSupermanifolds(\mathbb{R}^{n|n'},\mathbb{R}^{m|m'}) \,.$$

Given any super $L_\infty$-algebra $g$ we get the smooth superspace $S(CE(g))$ classifying $g$-valued super-forms using the usual adjunction coming from the ambimorphic object $\Omega^\bullet$ (you are hearing my teacher Todd Trimble speak through me, here) $$S(CE(G)) : \mathbb{R}^{n|n'} \mapsto Hom_{superDGCA}(CE(g),\Omega^\bullet(\mathbb{R}^{n|n'})) \,.$$ This classifies (section 6.5) $g$-valued forms in that for any smooth super $X$ $$SmoothSuperspaces(X,S(CE(g))) \simeq \Omega^\bullet(X,g) \,.$$

A thin homotopy class of a superpath in that classifying space $$\gamma : \mathbb{R}^{1|1} \to S(CE(g))$$ is an element in the super $\omega$-group integrating $g$. Continuing this way, we form the super path $\omega$-groupoid

$$\mathbf{B}G := \Pi_\omega^N(S(CE(g)))$$

of the classifying space, for any $N \in \mathbb{N}$, whose $k$-morphisms are thin homotopy classes of $(k|N)$-dimensional paths in the space $$\Sigma : \mathbb{R}^{k|N} \to S(CE(g))$$ (section 2.3)

Voila, the $\omega$-supergroup $G$ integrating $g$.

Now take any (flat for brevity) $g$-valued superform on $X$, represented by a superDGCA morphism $$\Omega^\bullet(Y) \stackrel{A}{\leftarrow} CE(g)$$ and hit that entire morphism with our integration functor $\Pi_\omega^N \circ S : superDGCA \to SuperSmooth\omega Cat$ to produce its corresponding super-parallel transport $$tra_A : \Pi_\omega^N(Y) \to \mathbf{B}G \,.$$ And so on.

Here is a somewhat more philosophical remark. (Experience shows that it’s of the kind that I tend to find deeply puzzling while you are likely to find it at best pointless.)

I keep thinking about what superification really means. For instance, I keep noticing that superification shares many properties with categorification: categorification introduces $\mathbb{N}$-gradings, while superification introduces $\mathbb{Z}_2$-gradings.

I have that funny trait that I feel annoyed by the fact that $\mathbb{R}^{n|n'}$ is defined using graded algebras, somewhat messing with the Zen-like beauty of space and quantity, only for $\Omega^\bullet(\mathbb{R}^{n|n'})$ later to introduce yet another layer of graded algebra.

So I observe this: little to nothing of the above changes if we do a slight repackaging of concepts:

instead of regarding the classifying space $S(CE(g))$ of a super $L_\infty$-algebra $g$ as a smooth superspace, I regard it as an ordinary smooth space probed by ordinary Euclidean spaces by setting $$S(CE(g)) : \mathbb{R}^n \mapsto Hom_{superDGCA}(CE(g),\Omega_N^\bullet(\mathbb{R})) \,,$$ where now $$\Omega^\bullet_N : Euclidean domains \to superDGCAs$$ sends $\mathbb{R}^n$ to the superDGCA of superdifferential forms on $\mathbb{R}^{n|N}$.

The point being that odd superfunctions are so close in concept to differential forms that we can just repackage them with the forms and assign the full package to any ordinary space.

These are lots of words for the simple fact that using the hom-adjunction we can always think $$\begin{aligned} \Omega^\bullet(\mathbb{R}^{n|N}) &:= SuperSmoothSpaces(\mathbb{R}^{n|N},\Omega^\bullet(-)) \\ &= SuperSmoothSpaces(\mathbb{R}^{n|0}\times \mathbb{R}^{0|N},\Omega^\bullet(-)) \\ & \simeq SuperSmoothSpaces(\mathbb{R}^n,hom(\mathbb{R}^{0|N},\Omega^\bullet(-))) \\ & =: SuperSmoothSpaces(\mathbb{R}^n,\Omega^\bullet_N(-)) \\ & =: \Omega^\bullet_N(\mathbb{R}^n) \end{aligned} \,.$$ But still. It makes me wonder if we shouldn’t rethink.