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nLab > Latest Changes: stream

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  1. Moved material about elements of the final coalgebra of the endofunctor X↦1+A×XX \mapsto 1 + A \times X from stream to cofree coalgebra

    Thomas Wilson

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeDec 24th 2023
    • (edited Dec 24th 2023)

    The revision is incorrect. While ℕ\mathbb{N}-indexed sequences are indeed streams, finite sequences are streams too.

    The operational property of a stream is that either:

    • It is empty
    • It is nonempty, you can read an element from it, and the remainder of the stream is itself a stream
    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeDec 24th 2023

    Additionally, I’m skeptical that making this an example at cofree coalgebra is a better place for this content than the stream page.

    (the actual transplant that was performed is clearly wrong, but I think that even if it was modified to just make it an example over on the other page, the move didn’t really make sense)

  4. As I asked in the nForum discussion thread

    https://nforum.ncatlab.org/discussion/17554/list-of-notable-initial-algebras-and-terminal-coalgebras

    does anybody have any references in the published literature of the usage of the word “stream” for the terminal coalgebra of either X↦A×XX \mapsto A \times X or X↦1+A×XX \mapsto 1 + A \times X?

    Judging by the reactions of other members of the nLab I highly suspect that both definitions are used in the literature for different purposes, but neither are cited on this page.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 24th 2023
    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 24th 2023

    Re #4: Also this original paper: https://research.vu.nl/files/2409636/219599.pdf.

    • CommentRowNumber7.
    • CommentAuthorNikolajK
    • CommentTimeDec 24th 2023

    Forthcoming? I think it’s many years old, haha

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 25th 2023

    Published in 2016 by the Cambridge University Press:

    B. Jacobs, Introduction to Coalgebra. Towards Mathematics of States and Observations. Cambridge Univ. Press, 2016.

    https://doi.org/10.1017/CBO9781316823187

  9. Adding another reference of streams of AA being used as the terminal coalgebra of X↦A×XX \mapsto A \times X:

    • {#ACS15} Benedikt Ahrens, Paolo Capriotti, Régis Spadotti, Non-wellfounded trees in Homotopy Type Theory, (arXiv:1504.02949)

    Anonymouse

    diff, v7, current

  10. Completing the reference information:

    Anonymouse

    diff, v7, current

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