Abstract
We introduce the concept of modified vertical Weil functors on the category $\F_2\M_{m_1,m_2}$ of fibered-fibered manifolds with $(m_1,m_2)$-dimensional bases and their local fibered-fibered maps with local fibered diffeomorphisms as base maps. We then describe all fiber product preserving bundle functors on $\F_2\M_{m_1,m_2}$ in terms of modified vertical Weil functors.
Highlights
We assume that any manifold considered in this paper is Hausdorff, second countable, finite dimensional, without boundary, and smooth
Let Mf be the category of manifolds and their local maps, Mfm the category of m -dimensional manifolds and their local diffeomorphisms, FM the category of fibered manifolds and their local fibered maps, F Mm1,m2 the category of (m1, m2) -dimensional fibered manifolds and their local fibered diffeomorphisms, FMm the category of fibered manifolds with m-dimensional bases and their local fibered maps with embeddings as base maps, F2M the category of fibered-fibered manifolds and their local fibered-fibered maps, and F2Mm1,m2 the category of fibered-fibered manifolds with (m1, m2) -dimensional bases and their F2M -maps with base maps being F Mm1,m2 -maps
If μ′ : A′ → B′ is another natural transformation between Weil algebra bundle functors on F Mm1,m2 and (φ, ψ) is a morphism μ → μ′ (i.e φ : A → A′ and ψ : B → B′ are natural transformations between Weil algebra bundle functors such that μ′ ◦ φ = ψ ◦ μ ), we have the induced natural transformation (φ, ψ) : T μ → T μ′
Summary
We assume that any manifold considered in this paper is Hausdorff, second countable, finite dimensional, without boundary, and smooth (i.e. of class C∞ ). A bundle functor F on F2Mm1,m2 is fiber product preserving if for any F2Mm1,m2 -objects Y1 = ((Y1 → X1) → (Y → X)) and Y2 = ((Y2 → X2) → (Y → X)) we have F (Y1 ×Y Y2) = F Y1 ×Y F Y2 modulo (F pr, F pr2) , where pri : Y1 ×Y Y2 → Yi are the usual projections. The vertical functor V on F2Mm1,m2 is a fiber product preserving bundle functor. Replacing (in the construction of V ) the tangent functor T by the Weil functor T A corresponding to a Weil algebra A and 0Y by the canonical section eY of T AY , one can define (in the same way) the vertical Weil functor V A on F2Mm1,m2.
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