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A product preserving functor is a covariant functor ${\cal F}$ 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: ${\cal F} (M_1 \times M_2) = {\cal F} (M_1) \times {\cal F} (M_2)$. It is known that any product preserving functor ${\cal F}$ is equivalent to a Weil functor $T^A$. Here $T^A (M)$ is the set of equivalence classes of smooth maps $\varphi : \bbfR^n \to M$ and $\varphi, \varphi'$ are equivalent if and only if for every smooth function $f : M \to \bbfR$ the formal Taylor series at 0 of $f \circ \varphi$ and $f \circ \varphi'$ are equal in $A = \bbfR [[x_1, \dots, x_n]]/{\germ a}$. In this paper all known properties of product preserving functors are derived from the axioms without using Weil functors. (English) |
