On Compositionality of Dinatural Transformations

Guy McCusker, Alessio Santamaria

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.
Original languageEnglish
Title of host publication27th EACSL Annual Conference on Computer Science Logic (CSL 2018)
EditorsDan Ghica, Achim Jung
Place of PublicationDagstuhl, Germany
PublisherLeibniz International Proceedings in Informatics
Pages33:1-33:22
Number of pages22
Volume119
ISBN (Electronic)9783959770880
DOIs
Publication statusPublished - 31 Dec 2018

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Volume119
ISSN (Electronic)1868-8969

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Keywords

  • Categorical logic
  • Compositionality
  • Dinatural transformation

ASJC Scopus subject areas

  • Software

Cite this

McCusker, G., & Santamaria, A. (2018). On Compositionality of Dinatural Transformations. In D. Ghica, & A. Jung (Eds.), 27th EACSL Annual Conference on Computer Science Logic (CSL 2018) (Vol. 119, pp. 33:1-33:22). [33] (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 119). Dagstuhl, Germany: Leibniz International Proceedings in Informatics. https://doi.org/10.4230/LIPIcs.CSL.2018.33