arXiv subjects

Salil Samant

Publications and source records attributed to Salil Samant.

3 recordsLinked to original sources

arXiv2017

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

In this paper we propose and lay the foundations of a functorial framework for representing signals. By incorporating additional category-theoretic relative and generative perspective alongside the classic set-theoretic measure theory the fundamental concepts of redundancy, compression are formulated in a novel authentic arrow-theoretic way. The existing classic framework representing a signal as a vector of appropriate linear space is shown as a special case of the proposed framework. Next in the context of signal-spaces as a categories we study the various covariant and contravariant forms of $L^0$ and $L^2$ functors using categories of measurable or measure spaces and their opposites involving Boolean and measure algebras along with partial extension. Finally we contribute a novel definition of intra-signal redundancy using general concept of isomorphism arrow in a category covering the translation case and others as special cases. Through category-theory we provide a simple yet precise explanation for the well-known heuristic of lossless differential encoding standards yielding better compressions in image types such as line drawings, iconic image, text etc; as compared to classic representation techniques such as JPEG which choose bases or frames in a global Hilbert space.

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arXiv2017

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

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.

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arXiv2017

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 ${\mathbb{F}}: \mathbf{B} \xrightarrow{F} \mathbf{Man}^{\infty} \xrightarrow{U} \mathbf{Set}$.

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