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nLab > Latest Changes: generalized algebraic theory

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    • CommentRowNumber1.
    • CommentAuthorKevin Watkins
    • CommentTimeOct 13th 2018

    I’m having trouble understanding the relationship between what is described in the page generalized algebraic theory and what is in the referenced article by Cartmell.

    For example, the nLab page seems to contemplate three levels of symbols—“supersort”, “sort”, and “operation”—while if I understand correctly, Cartmell’s GATs only have symbols at two levels, for sorts and operations. Also, I would only expect a GAT’s sort symbols to be applied to terms, not types as the nLab page seems to contemplate. (The nLab page speaks of “derived operations in the theory of sorts” rather than types, but I believe the same concept is intended.) My intuition is that the world of GATs in Cartmell’s sense more or less corresponds to a certain sublanguage of LF (rather than F ωF_\omega), so there shouldn’t be anything like a symbol that is applied to arguments that are types and yields a type.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 13th 2018

    The author of those remarks apparently hasn’t been around for some time, so I don’t know that an explanation of that description would be forthcoming.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 25th 2021

    One way to compare internalized/enriched categories here in a page that could use some attention.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 26th 2021

    Added in the reference.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 9th 2021

    For what it’s worth, the “definition” generalized algebraic theory is a bit lackluster (I realise the onus is on me to do something about it, but in case anyone out there has the info at their fingertips…)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    I have added pointer to

    • John Cartmell, Generalised Algebraic Theories and Contextual Categories, PhD thesis, Oxford University (1978) [pdf]

    Also, I am removing this old query box discussion:

    +– {: .standout} Might there be such a thing as an nn-GAT, where a 00-GAT is an algebraic theory and an (n+1)(n+1)-GAT is defined as above except that the sort algebra is an nn-GAT rather than an algebra? – Adam =–

    +– {: .standout} I feel like there must be some sort of way to eliminate the notion of “arity” and put in its place an arbitrary (G)AT, recovering the original notion using the single-sorted Peano algebra (one constant “0”, one unary operation “S”, and no equations) or binary tree algebra (one constant “0”, one binary operation “B”, and no equations). But I can’t quite put my finger on how to do it. – Adam =–

    diff, v9, current

  7. As mentioned on the discussion page, this page was largely an attempt at an original presentation of the concept, which was not in fact coextensive with the correct definition. I’ve made a first attempt at correcting and cleaning it up.

    karlin

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2023

    I have touched the wording in the idea section and added plenty more hyperlinks to technical terms (such as to initial model).

    diff, v12, current

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