超代数与超几何的范畴表述
A Categorical Formulation of Superalgebra and Supergeometry
摘要：
我们重新制定超代数和超几何在完全范畴的条款，随之而来的使用函子的点。这种方法增加了抽象性，带来了许多巨大的好处。首先，我们表明，可以扩展超几何完全自然的无限维的情况下。其次，根据这些结果，超几何的一些微妙的、有时看起来模糊的点变得清晰起来，例如，Berezin-Leites-Kostant方法与de Witt-Rogers方法之间的关系，以及奇参数在超几何中的精确几何意义。此外，这种方法允许我们以简单的方式构造超对象之间的态射的超空间，即，与这些态射的集合相关联的内部Hom对象。我们的工作成果在很大程度上依赖于V. Molotkov的思想，并受到其启发，他首先概述了这里广泛阐述的方法。
英文摘要：
We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can extend supergeometry completely naturally to infinite-dimensional contexts. Secondly, some subtle and sometimes obscure-seeming points of supergeometry become clear in light of these results, e.g., the relation between the Berezin-Leites-Kostant and the de Witt-Rogers approaches, and the precise geometric meaning of odd parameters in supergeometry. In addition, this method allows us to construct in an easy manner superspaces of morphisms between superobjects, i.e., the inner Hom objects associated with the sets of such morphisms. The results of our work rely heavily and were inspired by the ideas of V. Molotkov, who first outlined the approach extensively expounded here.
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