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

Abstract

In this paper we propose and study few applications of the base structured categories $\mathcal{X}{⋊}_{\mathbf{F}}\mathbf{C}$, ${\int }_{\mathbf{C}}\bar{\mathbf{F}}$, $\mathcal{X}{⋊}_{\mathbb{F}}\mathbf{C}$ 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 $[({\mathcal{X}}_{2a}{⋊}_{{\mathbf{F}}_{2a}}{\mathbf{B}}_{2a})⨿({\mathcal{X}}_{2b}{⋊}_{{\mathbf{F}}_{2b}}{\mathbf{B}}_{2b})⨿...]{⋊}_{{\mathbf{F}}_1}{\mathbf{B}}_1$ that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category $({\mathcal{X}}_1{⋊}_{{\mathbf{F}}_1}{\mathbf{B}}_1)//2G$, 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 ($\mathbf{G}=\mathcal{X}{⋊}_{\mathbb{F}}\mathbf{H}$) of $\mathbb{F}:\mathbf{H}\stackrel{F}{\to }{\mathbf{Man}}^{\infty }\stackrel{U}{\to }\mathbf{Set}$. 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 $\mathcal{G}=\mathcal{X}⋊\mathcal{B}$ and geometry space $\mathcal{X}=\mathcal{G}/\mathcal{B}$; which is essentially same as $\mathbf{G}=\mathcal{X}{⋊}_{\mathbb{F}}\mathbf{B}$ or precisely the completion of $\mathbb{F}:\mathbf{B}\stackrel{F}{\to }{\mathbf{Man}}^{\infty }\stackrel{U}{\to }\mathbf{Set}$.

1.Notations revised to reflect the difference in abstract and concrete category actions. 2.Both Category and set-theoretic versions of the definition of groupoid geometries made explicit

References in corpus (2)

• Higher Operads, Higher Categories
• In Ehresmann's footsteps: from Group Geometries to Groupoid Geometries

Cited by in corpus (1)

• Unified Functorial Signal Representation III: Foundations, Redundancy, $L^0$ and $L^2$ functors