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

Luciano, O. O.

Nagoya Math. J. 109, 69-89 (1988).

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

Keywords:

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

Citations:

Zbl 0223.53027

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)