## Categories of multiplicative functors and Weil’s infinitely near points.(English)Zbl 0661.58007

Let $${\mathcal S}{\mathcal L}$$ be the category that has as objects the $${\mathbb{R}}$$- algebras which split as $${\mathbb{R}}$$-algebras in a finite sum of local algebras and as morphisms the $${\mathbb{R}}$$-algebra morphisms, and $${\mathcal L}$$ the full subcategory of $${\mathcal S}{\mathcal L}$$ that has as objects local algebras. Let $${\mathfrak M}$$ be the category whose objects are $$C^{\infty}$$, Hausdorff, second countable manifolds and whose morphisms are $$C^{\infty}$$-maps between such manifolds and $${\mathcal G}{\mathcal P}$$ the category that has as objects the product preserving covariant endofunctors $${\mathfrak M}\to^{T}{\mathfrak M}$$ for which $${\mathbb{R}}\to^{^ i{\mathbb{R}}}T({\mathbb{R}})$$ $$[i_{{\mathbb{R}}}(x)$$ is the value of the constant map $$T(\{x\})\to T({\mathbb{R}})$$ obtained by application of T on the inclusion $$\{x\}$$ $$\hookrightarrow {\mathbb{R}}]$$ is injective and continuous, and as morphisms the natural transformations. The objects of $${\mathcal G}{\mathcal P}$$ are called generalized prolongation functors. One can define a covariant functor $${\mathcal A}: {\mathcal G}{\mathcal P}\to {\mathcal S}{\mathcal L}$$. If $${\mathcal I}{\mathcal P}$$ denotes the full subcategory of $${\mathcal G}{\mathcal P}$$ that has as objects those T of $${\mathcal G}{\mathcal P}$$ satisfying algebraic locality, the property that $${\mathcal A}(T)$$ is a local algebra and local determination, the author proves that there exists an equivalence between the categories $${\mathcal I}{\mathcal P}$$ and $${\mathcal L}.$$
This result gives an answer to Morimoto’s conjecture [see the introduction of A. Morimoto’s Prolongations of geometric structures, Nagoya, Japan: Math. Inst., Nagoya Univ. (1969; Zbl 0223.53027)]. Certain properties of the objects of $${\mathcal I}{\mathcal P}$$ are established. A full subcategory of $${\mathcal G}{\mathcal P}$$ which includes $${\mathcal I}{\mathcal P}$$ and has an endofunctor R with the property $$R\circ R=R$$ and having $${\mathcal I}{\mathcal P}$$ as its image is also considered.
Reviewer: M.Craioveanu

### MSC:

 58A99 General theory of differentiable manifolds 18B99 Special categories

Zbl 0223.53027
Full Text:

### References:

 [1] Works of the Dept. of Math 85 (1985) [2] Math. Inst (1969) [3] Colloque de Topologie et Géométrie Différentielle, Strasbourg pp 111– (1953)
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