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Properties of product preserving functors. (English) Zbl 0848.55010
Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 69-86 (1994).
A product preserving functor is a covariant functor $${\mathcal F}$$ from the category of all manifolds and smooth mappings into the category of fibered manifolds satisfying a list of axioms the main of which is product preserving: $${\mathcal F} (M_1 \times M_2) = {\mathcal F} (M_1) \times {\mathcal F} (M_2)$$. It is known that any product preserving functor $${\mathcal F}$$ is equivalent to a Weil functor $$T^A$$. Here $$T^A (M)$$ is the set of equivalence classes of smooth maps $$\varphi : \mathbb{R}^n \to M$$ and $$\varphi, \varphi'$$ are equivalent if and only if for every smooth function $$f : M \to \mathbb{R}$$ the formal Taylor series at 0 of $$f \circ \varphi$$ and $$f \circ \varphi'$$ are equal in $$A = \mathbb{R} [[x_1, \dots, x_n]]/{\mathfrak a}$$. In this paper all known properties of product preserving functors are derived from the axioms without using Weil functors.
For the entire collection see [Zbl 0823.00015].
##### MSC:
 55R10 Fiber bundles in algebraic topology 58A10 Differential forms in global analysis