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    • CommentRowNumber1.
    • CommentAuthorPaoloPerrone
    • CommentTime5 days ago

    Is there a term to denote collectively all the constructions of “theories” of some sort, without giving rise to ambiguities?

    What I exactly mean is a collective term for monads, operads, clubs, Lawvere theories, and in general all categorical constructions that may have “algebras”, or which describe formal operations in some (possibly generalized) way. In my mind they all encode some sort of “theory”, but the word “theory” seems to be reserved for “algebraic theory”/”Lawvere theory”.

    Which term would you use instead?

    (I hope it’s clear enough what I’m asking.)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTime4 days ago

    I think this class of things is too broad and ill-defined to have any name. Although I think if you needed to, you could use “theories” – I don’t think it’s reserved for finite-product theories.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTime4 days ago

    I’d use something involving the words “universal algebra”.

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTime4 days ago

    Really, Urs? “Universal algebra” usually seems to mean things like the HSP theorem for single-sorted finite product theories, characterizations of their congruence lattices, and so on.

    I don’t think “theory” necessarily means “Lawvere theory”, e.g. a model theorist would probably understand “first order theory”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTime4 days ago
    • (edited 4 days ago)

    It feels inappropriate to point you to Wikipedia. More academic sources include Hyland-Power 07:

    Lawvere theories and monads have been the two main category theoretic formulations of universal algebra

    For operads as universal algebra, see Bai-Guo-Loday 11.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTime2 days ago

    Actually, adding to #3, I think “categorical universal algebra” isn’t a bad approximation.

    • CommentRowNumber7.
    • CommentAuthorSam Staton
    • CommentTime2 days ago

    Thanks. My impression is that the primary meaning of the “universal algebra” is more like the subject of books like the one by Burris and Sankappanavar and for instance in this survey article: An overview of modern universal algebra, which doesn’t mention operads or categories. I mean, focussing on deep and difficult theorems about broad classes of first-order equational theories and their algebras, rather than on making the class of theories as broad as possible. And “universal” only in the sense that it’s not just about groups rings and modules, rather than in the category theoretic sense.

    “Categorical universal algebra” might be ok!

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTime1 day ago

    The “primary” meaning of a phrase often depends on the community one is looking at. Certainly among the people who use it that way, that’s how it’s used. But I expect that among category theorists it has a different emphasis in meaning. So “categorical universal algebra” is a good choice if you want to be unambiguous across communities.