2014/06/29 by F. Hanisch, Florian Hanisch, Hanisch, Florian
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1406.7484
51 pages
arxiv created 2014/06/29 · openalex publication_date 2014/06/29 · arxiv updated 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this work is the construction of a "supermanifold of morphisms X → Y", given two finite-dimensional supermanifolds X and Y. More precisely, we will define an object \underlineSC^∞(X,Y) in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula Hom(P × X,Y) ≅ Hom(P,\underlineSC^∞(X,Y)) where P ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the set of certain differential operators and establish a bijective correspondence to the set of sections in certain vector bundles associated to X and Y. Equipping these spaces of sections with infinite-dimensional manifold structures using the convenient setting by Kriegl and Michor, we obtain at a supersmooth structure on \underlineSC^∞(X,Y), i.e. a supermanifold of all morphisms X → Y.