##
**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.

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

### Keywords:

multiplicative functors; category; covariant endofunctors; generalized prolongation functors; covariant functor; Morimoto’s conjecture### Citations:

Zbl 0223.53027
Full Text:
DOI

### 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) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.