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In this paper we propose and study few applications of the base structuredcategories $\mathcal{X} \rtimes_{\mathbf{F}} \mathbf{C}$, $\int_{\mathbf{C}}\bar{\mathbf{F}}$, $\mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$ and${\int_{\mathbf{C}} \bar{\mathbb{F}}}$. First we show classic transformationgroupoid $X /\!\!/ G$ 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 structuresas a special case of composite Grothendieck fibration. Further utilizing theexisting notion of transformation double category $(\mathcal{X}_{1}\rtimes_{\mathbf{F_{1}}} \mathbf{B}_{1}) /\!\!/ \mathbf{2G}$, we demonstratethat a hierarchy of bases naturally leads one from 2-groups to n-categorytheory. Finally we prove that every classic Klein geometry is the Grothendieckcompletion ($\mathbf{G} = \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{H}$) of${\mathbb{F}}: \mathbf{H} \xrightarrow{{F}} \mathbf{Man}^{\infty}\xrightarrow{U} \mathbf{Set}$. This is generalized to propose a set-theoreticdefinition of a groupoid geometry $(\mathcal{G},\mathcal{B})$ (originallyconceived by Ehresmann through transport and later by Leyton using transfer)with a principal groupoid $\mathcal{G} = \mathcal{X} \rtimes \mathcal{B}$ andgeometry space $\mathcal{X} = \mathcal{G}/\mathcal{B}$; which is essentiallysame as $\mathbf{G} = \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{B}$ or preciselythe completion of ${\mathbb{F}}: \mathbf{B} \xrightarrow{{F}}\mathbf{Man}^{\infty} \xrightarrow{U} \mathbf{Set}$.

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