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**On the product preserving bundle functors on \(k\)-fibered manifolds.**
*(English)*
Zbl 0994.58001

In [Arch. Math., Brno 32, No. 4, 307-316 (1996; Zbl 0881.58002)] the author showed that there is a bijection between the product preserving bundle functors on the category \(\mathcal {FM}\) of all fibered manifolds with all fibred morphisms and the natural transformations of product preserving bundle functors on \(\mathcal {M}f\). Also, it was shown that the natural transformations between two product preserving bundle functors on \(\mathcal {FM}\) are in bijection with the morphisms between corresponding natural transformations.

In this paper, the author generalizes these results. A \(k\)-fibred manifold is a sequence \(X_k\to X_{k-1}\to\dots\to X _0\) of surjective submersions. The author proves that the product preserving bundle functors on the category \(k\)-\(\mathcal {FM}\) of all \(k\)-fibered manifolds and their morphisms are in bijection with the sequence \(G_k\to G_{k-1}\to\dots\to G _0\) of the natural transformations between product preserving bundle functors on \(\mathcal {M}f\), and that the natural transformations between two product preserving bundle functors on \(k\)-\(\mathcal {FM}\) are in bijection with the morphisms between corresponding sequences of natural transformations.

In this paper, the author generalizes these results. A \(k\)-fibred manifold is a sequence \(X_k\to X_{k-1}\to\dots\to X _0\) of surjective submersions. The author proves that the product preserving bundle functors on the category \(k\)-\(\mathcal {FM}\) of all \(k\)-fibered manifolds and their morphisms are in bijection with the sequence \(G_k\to G_{k-1}\to\dots\to G _0\) of the natural transformations between product preserving bundle functors on \(\mathcal {M}f\), and that the natural transformations between two product preserving bundle functors on \(k\)-\(\mathcal {FM}\) are in bijection with the morphisms between corresponding sequences of natural transformations.

Reviewer: Andrew Bucki (Oklahoma City)

### MSC:

58A05 | Differentiable manifolds, foundations |