License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CALCO.2015.130
URN: urn:nbn:de:0030-drops-55317
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5531/
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Kataoka, Toshiki ; Pavlovic, Dusko

Towards Concept Analysis in Categories: Limit Inferior as Algebra, Limit Superior as Coalgebra

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Abstract

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires concept mining and concept analysis to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the limit superior and limit inferior operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.

BibTeX - Entry

@InProceedings{kataoka_et_al:LIPIcs:2015:5531,
  author =	{Toshiki Kataoka and Dusko Pavlovic},
  title =	{{Towards Concept Analysis in Categories: Limit Inferior as Algebra, Limit Superior as Coalgebra}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{130--155},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-84-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{35},
  editor =	{Lawrence S. Moss and Pawel Sobocinski},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5531},
  URN =		{urn:nbn:de:0030-drops-55317},
  doi =		{10.4230/LIPIcs.CALCO.2015.130},
  annote =	{Keywords: concept analysis, semantic indexing, category, completion, algebra}
}

Keywords: concept analysis, semantic indexing, category, completion, algebra
Collection: 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)
Issue Date: 2015
Date of publication: 28.10.2015

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