Product preserving gauge bundle functors on vector bundles (Q2773268)

For the category \(\mathcal {VB}\) of all vector bundles with their vector bundles homomorphisms and the category \(\mathcal {FM}\) of fibered manifolds with their fibered maps let \(F: \mathcal {VB}\to\mathcal {FM}\) be a covariant functor with the base functors \(B_{\mathcal {VB}}: \mathcal {VB}\to \mathcal Mf\) and \(B_{\mathcal {FM}}: \mathcal {FM}\to \mathcal Mf\). A gauge bundle functor on \(\mathcal {VB}\) is a functor \(F\) satisfying \(B_{\mathcal {FM}}\circ F=B_{\mathcal {VB}}\) and the localization condition: for every inclusion of an open vector subbundle \(i_{E|U}:E|U\to E\), \(F(E|U)\) is the restriction \(p_E^{-1}(U)\) of \(p_E: FE\to B_{\mathcal{VB}}(E)\) over \(U\) and \(Fi_{E|U}\) is the inclusion \(p_E^{-1}(U)\to FE\). A natural transformation \(\tau :F_1\to F_2\) of two gauge bundle functors \(F_1\) and \(F_2\) is a system of base preserving fibered maps \(\tau_E: F_1 E\to F_2E\) for every vector bundle \(E\) satisfying \(F_2f\circ\tau_E=\tau_G\circ F_1f\) for every vector bundle homomorphism \(f: E\to G\). A gauge bundle functor \(F\) on \(\mathcal {VB}\) is product preserving if for any product projections \(E_1\overset\text{pr}{_1}\longleftarrow E_1\times E_2\overset\text{pr}{_2}\longrightarrow E_2\), in the category \(\mathcal {VB}\), \(FE_1\overset F\text{pr}{_1}\longleftarrow F(E_1\times E_2)\overset F\text{pr}{_2}\longrightarrow FE_2\) are product projections in the category \(\mathcal {FM}\). NEWLINENEWLINENEWLINEIn this interesting paper, the author gives a complete description of all product preserving gauge bundle functors \(F\) on vector bundles in terms of pairs \((A,V)\) consisting of a Weil algebra \(A\) and an \(A\)-module with \(\dim_{\mathbb R}(V)<\infty\). It is shown that the correspondence \(F\mapsto (A^F,V^F)\) induces a bijective correspondence between the equivalence classes of product preserving gauge bundle functors \(F\) on \(\mathcal {VB}\) and the equivalence classes of pairs \((A,V)\). Also, for two product preserving gauge bundle functors \(F_1\) and \(F_2\) on \(\mathcal {VB}\) the correspondence \(\tau\mapsto (\mu^\tau,\nu^\tau)\) is a bijection between the natural transformations \(F_1\to F_2\) and the morphisms \((A^{F_1},V^{F_1})\to (A^{F_2},V^{F_2})\) between corresponding pairs. Some applications of these results are presented.