title: Constellations and their relationship with categories

authors: Gould, Victoria

subjects: mathematics

abstract: Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category (Formula presented.), its canonical extension, in a simplest possible way, and that P is a quotient of (Formula presented.) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences (Formula presented.) on a given category (Formula presented.) (those for which (Formula presented.), and show that the category of constellations is equivalent to the category of (Formula presented.)-categories, that is, categories equipped with distinguished canonical congruence (Formula presented.). The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are (Formula presented.)-categories) may be subsumed by constellation theory.

text: Constellations and their relationship with categories Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category (Formula presented.), its canonical extension, in a simplest possible way, and that P is a quotient of (Formula presented.) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences (Formula presented.) on a given category (Formula presented.) (those for which (Formula presented.), and show that the category of constellations is equivalent to the category of (Formula presented.)-categories, that is, categories equipped with distinguished canonical congruence (Formula presented.). The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are (Formula presented.)-categories) may be subsumed by constellation theory.

year: 2017

title: Seeking information in realistic books: a user study

authors: Liesaputra, Veronica

subjects: computer science

abstract: There are opposing views on whether readers gain any advantage from using a computer model of a 3D physical book. There is enough evidence, both anecdotal and from formal user studies, to suggest that the usual HTML or PDF presentation of documents is not always the most convenient, or the most comfortable, for the reader. On the other hand it is quite clear that while 3D book models have been prototyped and demonstrated, none are in routine use in today's digital libraries. And how do 3D book models compare with actual books?
This paper reports on a user study designed to compare the performance of a practical Realistic Book implementation with conventional formats (HTML and PDF) and with physical books. It also evaluates the annotation features that the implementation provides.

text: Seeking information in realistic books: a user study There are opposing views on whether readers gain any advantage from using a computer model of a 3D physical book. There is enough evidence, both anecdotal and from formal user studies, to suggest that the usual HTML or PDF presentation of documents is not always the most convenient, or the most comfortable, for the reader. On the other hand it is quite clear that while 3D book models have been prototyped and demonstrated, none are in routine use in today's digital libraries. And how do 3D book models compare with actual books?
This paper reports on a user study designed to compare the performance of a practical Realistic Book implementation with conventional formats (HTML and PDF) and with physical books. It also evaluates the annotation features that the implementation provides.

year: 2008

title: Lake Rotokakahi water quality update 1990-2011

authors: Butterworth, Joseph

subjects: Lake Rotokakahi

abstract: Lake Rotokakahi is an Iwi-owned lake administered by the Lake Rotokakahi Board of Control on Behalf of lake owners who are descendants from the Ngāti Tumatawera and Tūhourangi hapū of Te Arawa. It is mesotrophic (moderate water quality) lake with an area of 4.4 km² comprised of exotic forestry (57.1%), pasture (26.3%) and regenerating indigenous forest/scrub (16.6%).

text: Lake Rotokakahi water quality update 1990-2011 Lake Rotokakahi is an Iwi-owned lake administered by the Lake Rotokakahi Board of Control on Behalf of lake owners who are descendants from the Ngāti Tumatawera and Tūhourangi hapū of Te Arawa. It is mesotrophic (moderate water quality) lake with an area of 4.4 km² comprised of exotic forestry (57.1%), pasture (26.3%) and regenerating indigenous forest/scrub (16.6%).

year: 2012