2018/10/12 by Schütt, Jakob
#58A50 (Primary) #58B99 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1810.05549
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 M\colonGr\toMan 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 Λn\inGr is the Grassmann algebra with n generators, we show that MΛn has the structure of a so called multilinear bundle over the base manifold M_ℝ. We use this fact to show that the projective limit \varprojlimnMΛn exists in the category of manifolds. In fact, this gives us a faithful functor \varprojlim\colonSMan\toMan from the category of supermanifolds to the category of manifolds. This functor respects products, commutes with the respective tangent functor and retains the respective Hausdorff property. In this way, supermanifolds can be seen as a particular kind of infinite-dimensional fiber bundles.