In his recent investigation of a super Teichm\"uller space, Sachse (2007), based on work of Molotkov (1984), has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin-Kostant-Leites supermanifolds is equivalent to the category of finite-dimensional Molotkov-Sachse supermanifolds. Simultaneously, using the differential calculus of Bertram-Gl\"ockner-Neeb (2004), we extend Molotkov-Sachse's approach to supermanifolds modeled on Hausdorff topological super-vector spaces over an arbitrary non-discrete Hausdorff topological base field of characteristic zero. We also extend to locally k-omega base fields the `DeWitt' supermanifolds considered by Tuynman in his monograph (2004), and prove that this leads to a category which is isomorphic to the full subcategory of Molokov-Sachse supermanifolds modeled on locally k-omega spaces.Comment: 36 pages; minor corrections, expanded introductio

We define diffeological spaces and give some examples. The diffeological category contains the category of smooth manifolds as full subcategory. We prove that diffeological spaces and smooth maps form a cartesian closed category. The concepts of differential form and tangent functor are extended from smooth manifolds to diffeological spaces. Subjclass : 57R55 Differentiable Structures. See www.ams.org/msc Now we proceed to define diffeological spaces and smooth maps between diffeological spaces as in [Sou85]. Definition 1.3 (Diffeology). Given a set X consider a collection P(X) = ∪ n P n (X) of maps into X, such that P n (X) ⊂ L n (X). We call P(X) a diffeology if it satisfies the following axioms: (D1) Every constant map x : U → X is in P(X). (D2) Given a compatible family of maps in P n (X), its smallest extension is again in P n (X). (D3) If h ∈ L pn and α ∈ P n (X) are composable, then α • h ∈ P p (X). The elements of P n (X) are called n-plots or simply plots of the diffeology. The pair (X, P(X)) is called a diffeological space. Definition 1.4 (D-topology). Given a diffeological space (X, P(X)) we define the-topology to be the initial topology with respect to the plots. That is, a subset U ⊂ X is open if and only if for each plot α ∈ P(X), the inverse image α −1 (U) is open in U α. We can immediately give two examples. Example 1.5 (Indiscrete Diffeology). Given any set X, the collection L(X) of all possible maps into X with open domain is a diffeology. The only open sets are X and the empty set, so the-topology is the indiscrete topology. Example 1.6 (Discrete Diffeology). Let P(X) denote the collection of all locally constant maps. This is the smallest possible diffeology on the set X. Note that we cannot restrict to all constant maps! This would violate (D2): If we take two maps that are constant, taking distinct values on disjoint domains, their smallest extension is merely locally constant. Every subset of X is open in the-topology, so X is a discrete topological space.

We define a Fourier transform and a convolution product for functions and distributions on Heisenberg-Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley-Wiener-Schwartz theorem, and define a convolution Banach algebra.

Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C ∞ (M, G) or Diff(M ), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category. We call groups with this property Frölicher groups. One can define tangent spaces to Frölicher spaces, and in the present article we prove that, under a certain additional assumption, the tangent space at the identity of a Frölicher group can be equipped with a Lie bracket. We discuss an example which satisfies the additional assumption.

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