Singular superspaces

Abstract

We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.Comment: v3: final version prior to publication, examples added, corrections, numbering adapted, 47 pages, Math. Z. (2014

Representability in supergeometry

“…4.1). This broadens the range of application of the criterion, first published in [CCF11], to include, not only superschemes or supermanifolds, but also some less trivial categories like Leites regular supermanifolds and locally finitely generated superspaces, introduced by Alldridge et al in [All13]. Our hope is that more general objects can be studied using this criterion, which formalizes the ideas of Grothendieck, adapting them to the supergeometric context.…”

“…We say that a morphism f : X → Y between superspaces is an embedding if f = g • h where h is a closed embedding and g is an open embedding. See [All13] for more details.…”

“…Furthermore, it is possible to give a definition of Leites superspaces which comprehends not only differentiable or holomorphic supermanifolds, but also the real analytic ones. We do not introduce them here, referring the reader to [All13] for more details; their treatment is similar to the other ones.…”

“…On the other hand in the categories (SSch) and (SSpaces) lfg k , fibered products always exist, hence the morphism f : h X → h Z is always representable, even in the case in which it is not an open embedding. We shall not need this fact in the sequel, see [CCF11] and [All13] for more details.…”

“…In[All13] these are called presupermanifolds. In many references the definition of supermanifold requires more hypotheses (e.g.…”