Unified Functorial Signal Representation I: From Grothendieck fibration to Base structured categories

Abstract

In this paper we study categories $(F,\mathbf{C},\mathbf{D})$ and $(\mathbb{F},\mathbf{C},\mathbf{Set})$ and prove them to be fibred on $\mathbf{C}$. Then we examine Grothendieck construction in the context of an ordinary functor $F: \mathbf{C} \rightarrow \mathbf{D}$ through the concept of trivial categorification, using an appropriate functor $\mathbf{F}: \mathbf{C} \xrightarrow{F} \mathbf{D} \xrightarrow{I} \mathbf{Cat}$ to construct $\int_{\mathbf{C}^{op}} \bar{\mathbf{F}}$. This category characterizes a functor as an abstract right category action while its dual $\mathcal{X} \rtimes_{\mathbf{F}} \mathbf{C}$ or $(\int_{\mathbf{C}^{op}} \bar{\mathbf{F}})^{op}$ characterizes a functor as an abstract left category action. Similarly using $\mathbb{F}: \mathbf{C} \xrightarrow{F} \mathbf{D} \xrightarrow{U} \mathbf{Set}$ we define $\mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$ and ${\int_{\mathbf{C}^{op}} \bar{\mathbb{F}}}$ as categories denoting concrete left and right actions of $\mathbf{C}$ respectively. Collectively referred to as `base structured categories', these are proven abstractly isomorphic to the base category $\mathbf{C}$ but concretely isomorphic to each other or $(\mathbb{F},\mathbf{C},\mathbf{Set}) \cong \int_{\mathbf{C}} \bar{\mathbb{F}} \cong \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$. These are special instances of fibred categories where the base category is $\mathbf{C}$ and fibres are $\mathbf{D}$ objects being treated as trivial categories. The perspective of making only base structure explicit through category theory concealing the vertical structure using identity morphisms enables one to combine intuitions of Grothendieck's relative and Leyton's generative theory. As explored further, it facilitates the application of functors in certain fundamental applications which hitherto have been treating objects of category $\mathbf{D}$ purely in a set theoretic way.