Infinite-Dimensional Supermanifolds via Multilinear Bundles. (arXiv:1810.05549v1 [math.DG])

In this paper, we provide an accessible introduction to the theory of locally convex supermanifolds in the categorical approach. In this setting, a supermanifold is a functor $\mathcal{M}\colon\mathbf{Gr}\to\mathbf{Man}$ from the category of Grassmann algebras to the category of locally convex manifolds that has certain local models, forming something akin to an atlas. We give a mostly self-contained, concrete definition of supermanifolds along these lines, closing several gaps in the literature on the way. If $\Lambda_n\in\mathbf{Gr}$ is the Grassmann algebra with $n$ generators, we show that $\mathcal{M}_{\Lambda_n}$ has the structure of a so called multilinear bundle over the base manifold $\mathcal{M}_\mathbb{R}$. We use this fact to show that the projective limit $\varprojlim_n\mathcal{M}_{\Lambda_n}$ exists in the category of manifolds. In fact, this gives us a faithful functor $\varprojlim\colon\mathbf{SMan}\to\mathbf{Man}$ from the category of supermanifolds to the category of 查看全文>>