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.CSL.2018.33
URN: urn:nbn:de:0030-drops-97006
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9700/
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McCusker, Guy ; Santamaria, Alessio

On Compositionality of Dinatural Transformations

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LIPIcs-CSL-2018-33.pdf (0.6 MB)


Abstract

Natural transformations are ubiquitous in mathematics, logic and computer science. For operations of mixed variance, such as currying and evaluation in the lambda-calculus, Eilenberg and Kelly's notion of extranatural transformation, and often the even more general dinatural transformation, is required. Unfortunately dinaturals are not closed under composition except in special circumstances. This paper presents a new sufficient condition for composability. We propose a generalised notion of dinatural transformation in many variables, and extend the Eilenberg-Kelly account of composition for extranaturals to these transformations. Our main result is that a composition of dinatural transformations which creates no cyclic connections between arguments yields a dinatural transformation. We also extend the classical notion of horizontal composition to our generalized dinaturals and demonstrate that it is associative and has identities.

BibTeX - Entry

@InProceedings{mccusker_et_al:LIPIcs:2018:9700,
  author =	{Guy McCusker and Alessio Santamaria},
  title =	{{On Compositionality of Dinatural Transformations}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic  (CSL 2018)},
  pages =	{33:1--33:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Dan Ghica and Achim Jung},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9700},
  URN =		{urn:nbn:de:0030-drops-97006},
  doi =		{10.4230/LIPIcs.CSL.2018.33},
  annote =	{Keywords: Dinatural transformation, categorical logic, compositionality}
}

Keywords: Dinatural transformation, categorical logic, compositionality
Collection: 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)
Issue Date: 2018
Date of publication: 29.08.2018


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