On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds

The Weil bundle $T^{\mathbb A}M_n$ of an $n$-dimensional smooth manifold $M_n$ determined by a local algebra $\mathbb A$ in the sense of A. Weil carries a natural structure of an $n$-dimensional $\mathbb A$-smooth manifold. This allows ones to associate with $T^{\mathbb A}M_n$ the series $B^r(\mathbb A)T^{\mathbb A}M_n$, $r=1,\dots,\infty$, of $\mathbb A$-smooth $r$-frame bundles. As a set, $B^r(\mathbb A)T^{\mathbb A}M_n$ consists of $r$-jets of $\mathbb A$-smooth germs of diffeomorphisms $(\mathbb A^n,0)\to T^{\mathbb A}M_n$. We study the structure of $\mathbb A$-smooth $r$-frame bundles. In particular, we introduce the structure form of $B^r(\mathbb A)T^{\mathbb A}M_n$ and study its properties.
Next we consider some categories of $m$-parameter-dependent manifolds whose objects are trivial bundles $M_n\times\mathbb R^m\to\mathbb R^m$, define (generalized) Weil bundles and higher order frame bundles of $m$-parameter-dependent manifolds and study the structure of these bundles. We also show that product preserving bundle functors on the introduced categories of $m$-parameter-dependent manifolds are equivalent to generalized Weil functors.