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A generalized global differential calculus: Application to invariance under a Lie group. I. (English) Zbl 0603.18004
This work is a systematic introduction to X manifolds where X is real analytic, complex algebraic, etc. The approach is novel (with a witty style) making little use of sheaf theory. First, universes (un- Grothendieck) are defined using notions from universal algebra. An additive structure on an universe is introduced. The notion of inverse semigroup plays a key role here. Patching together using cohesive universes one obtains admissible manifolds. In general, properties (completeness, cocompleteness, etc.) of the categories (of universes, of cohesive universes, etc.) encountered are provided. Special elements, phantoms, used in decomposing universes are defined. Local universes and associated local rings appear.
Finally, topological universes forming ”the category of preference” are introduced. The underlying topologies of cohesive universes are seen to be the sober topologies. The author then demonstrates how the above definitions can be used in defining tangent spaces, infinitesimals and integration of one parameter families. Significant applications to Lie groups are promised.
Reviewer: P.Cherenack

MSC:
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
58A05 Differentiable manifolds, foundations
18C05 Equational categories
58H05 Pseudogroups and differentiable groupoids
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