On the fiber product preserving bundle functors
type: Item
instance of: scholarly article
title: On the fiber product preserving bundle functors
zbMATH Open document ID: 0935.58001
DOI: 10.1016/S0926-2245(99)00022-4
published in: Differential Geometry and its Applications
publication date: 2000-05-04T00:00:00Z
review text: Let \(Mf\) be the category of all manifolds and all smooth maps, \(Mf_m\) be the category of \(m\)-dimensional manifolds and their local diffeomorphisms and \(FM_m\) be the category of fibered manifolds with \(m\)-dimensional bases and fiber preserving maps with the base map in \(Mf_m\). The authors deduce a complete description of all fiber product preserving bundle functors on \(FM_m\) in terms of Weil algebras. The main result reads that every such functor has finite order and all such functors of order \(r\) are in bijection with triples \((A,H,t)\), where \(A\) is a Weil algebra of order \(r\), \(H\) is a group homomorphism from the \(r\)-th jet group \(G^r_m\) into the group of all algebra automorphisms of \(A\) and \(t\) is a \(G^r_m\)-invariant algebra homomorphism from the jet algebra \(J^r_0(\mathbb{R}^m,\mathbb{R})\) into \(A\). The proof is based on several properties of the bundle functors on the product category \(Mf_m\times Mf\) that are deduced in the first part of the paper.
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 1367708
zbMATH Keywords: Weil bundle
zbMATH Keywords: jet bundle
zbMATH Keywords: natural transformation
zbMATH Keywords: bundle functors
Wikidata QID: Q115337417
author: Włodzimierz M. Mikulski
author: Ivan Kolář
MaRDI profile type: Publication
cites work: Product-preserving functors on smooth manifolds
cites work: Q5830224
cites work: Q3800762
cites work: On the absolute differentiation of geometric object fields
cites work: Q4693714
cites work: Categories of multiplicative functors and Weil’s infinitely near points
cites work: There exists a prolongation functor of infinite order
cites work: Q4374696
cites work: Prolongation of connections to bundles of infinitely near points
cites work: Natural bundles have finite order
cites work: Q5822224
cites work: Q3831738
Recommended article: Some properties of fiber product preserving bundle functors
Recommended article: Q4374696
Recommended article: Product preserving bundle functors on fibered fibered manifolds
Recommended article: Fiber product preserving bundle functors on fibered-fibered manifolds
Recommended article: On the product preserving bundle functors on \(k\)-fibered manifolds
Recommended article: Q3539840
Recommended article: Q4407743
Recommended article: Iteration of fiber product preserving bundle functors
Recommended article: On the fiber product preserving gauge bundle functors on vector bundles
Recommended article: Product preserving bundle functors on multifibered and multifoliate manifolds
reviewed by: Ivan Kolář
description: scientific article; zbMATH DE number 1367708