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Unified Functorial Signal Representation II: Category action, Base Hierarchy, Geometries as Base structured categories

In this paper we propose and study few applications of the base structured categories $\mathcal{X} \rtimes_{\mathbf{F}} \mathbf{C}$, $\int_{\mathbf{C}} \bar{\mathbf{F}}$, $\mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$ and ${\int_{\mathbf{C}} \bar{\mathbb{F}}}$. First we show classic transformation groupoid $X /\!\!/ G$ simply being a base-structured category ${\int_{\mathbf{G}} \bar{F}}$. Then using permutation action on a finite set, we introduce the notion of a hierarchy of base structured categories $[(\mathcal{X}_{2a} \rtimes_{\mathbf{F_{2a}}} \mathbf{B}_{2a}) \amalg (\mathcal{X}_{2b} \rtimes_{\mathbf{F_{2b}}} \mathbf{B}_{2b}) \amalg ...] \rtimes_{\mathbf{F_{1}}} \mathbf{B}_1$ that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category $(\mathcal{X}_{1} \rtimes_{\mathbf{F_{1}}} \mathbf{B}_{1}) /\!\!/ \mathbf{2G}$, we demonstrate that a hierarchy of bases naturally leads one from 2-groups to n-category theory. Finally we prove that every classic Klein geometry is the Grothendieck completion ($\mathbf{G} = \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{H}$) of ${\mathbb{F}}: \mathbf{H} \xrightarrow{F} \mathbf{Man}^{\infty} \xrightarrow{U} \mathbf{Set}$. This is generalized to propose a set-theoretic definition of a groupoid geometry $(\mathcal{G},\mathcal{B})$ (originally conceived by Ehresmann through transport and later by Leyton using transfer) with a principal groupoid $\mathcal{G} = \mathcal{X} \rtimes \mathcal{B}$ and geometry space $\mathcal{X} = \mathcal{G}/\mathcal{B}$; which is essentially same as $\mathbf{G} = \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{B}$ or precisely the completion of ${\mathbb{F}}: \mathbf{B} \xrightarrow{F} \mathbf{Man}^{\infty} \xrightarrow{U} \mathbf{Set}$.