This article defines a word and clarifies its application and usefulness in scale studies.

Overview

Categories are a formal mathematical language dealing with objects and sub-objects. It is possible that this formal concept may help address some issues of scale, such as the difficult to describe relationship between the sequence that enables one to move from scale A to scale B, the succession of those scales, and the part-to-whole pasthood relationships that relate sub-objects within a scale.

This research is pending. Below, some notes that may be helpful in future.

Etymology

Category 1580s, from M.Fr. catégorie, from L.L. categoria, from Gk. kategoria, from kategorein "to speak against; to accuse, assert, predicate," from kata "down to" (or perhaps "against") + agoreuein "to declaim (in the assembly)," from agora "public assembly," from PIE base *ger- "to gather" (see gregarious). Original sense of "accuse" weakened to "assert, name" by the time Aristotle applied kategoria to his 10 classes of things that can be named.

Taxonomy

Taxonomy is a hierarchical classification scheme that features the types of order that we see in scale surveys. A taxon may be within/without, intensive/extensive in relation to other taxons.

Category Theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category.
Consider the following example. The class Grp of groups consists of all objects having a "group structure". One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms that the identity element of a group is unique.

Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a precise sense – it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.

The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However it is important to note that the objects of a category need not be sets nor the arrows functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.

One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid, defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with algebraic topology.

Subobject

A subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory. Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a morphism which describes how one object sits inside another, rather than relying on the use of elements.
It is extremely difficult to communicate the vast scale of physical phenomena (super-scale) using any graphical technique. In this presentation the scalometer is proposed as a solution. This super-scale survey’s roots are traced to 1960’s work by USA artists Eames and McHale, whose works inspire many imitators to this date. The surveys enable viewers to survey huge amounts data in order to perceive and participate in the macro- and micro-scales (e.g. manipulate single atoms and navigate distant spacecraft). All simple renderings of super-scale involve distortions of space and meaning and these inaccuracies are usually not made explicit. Hence they often mislead people as to their ability to measure, report and act upon such information. In this project, scale rendering nomenclature and mathematical models are clarified and distortions are enumerated, along with reference to classic myths and stories that communicate these challenges. Circular and sigmoid scalometers are rendered over various data sets and applications to problem-solving and risk-analysis are proposed.

## Category

## Table of Contents

## Overview

Categories are a formal mathematical language dealing with objects and sub-objects. It is possible that this formal concept may help address some issues of scale, such as the difficult to describe relationship between the sequence that enables one to move from scale A to scale B, the succession of those scales, and the part-to-whole pasthood relationships that relate sub-objects within a scale.

This research is pending. Below, some notes that may be helpful in future.

## Etymology

Category 1580s, from M.Fr. catégorie, from L.L. categoria, from Gk. kategoria, from kategorein "to speak against; to accuse, assert, predicate," from kata "down to" (or perhaps "against") + agoreuein "to declaim (in the assembly)," from agora "public assembly," from PIE base *ger- "to gather" (see gregarious). Original sense of "accuse" weakened to "assert, name" by the time Aristotle applied kategoria to his 10 classes of things that can be named.

## Taxonomy

Taxonomy is a hierarchical classification scheme that features the types of order that we see in scale surveys. A taxon may be within/without, intensive/extensive in relation to other taxons.

## Category Theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category.

Consider the following example. The class Grp of groups consists of all objects having a "group structure". One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms that the identity element of a group is unique.

Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a precise sense – it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.

The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However it is important to note that the objects of a category need not be sets nor the arrows functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.

One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid, defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with algebraic topology.

## Subobject

A subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory. Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a morphism which describes how one object sits inside another, rather than relying on the use of elements.It is extremely difficult to communicate the vast scale of physical phenomena (super-scale) using any graphical technique. In this presentation the scalometer is proposed as a solution. This super-scale survey’s roots are traced to 1960’s work by USA artists Eames and McHale, whose works inspire many imitators to this date. The surveys enable viewers to survey huge amounts data in order to perceive and participate in the macro- and micro-scales (e.g. manipulate single atoms and navigate distant spacecraft). All simple renderings of super-scale involve distortions of space and meaning and these inaccuracies are usually not made explicit. Hence they often mislead people as to their ability to measure, report and act upon such information. In this project, scale rendering nomenclature and mathematical models are clarified and distortions are enumerated, along with reference to classic myths and stories that communicate these challenges. Circular and sigmoid scalometers are rendered over various data sets and applications to problem-solving and risk-analysis are proposed.