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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 11th 2015

    In topos of coalgebras over a comonad it is written:

    Also a category of algebras over a commutative finitary algebraic theory in Set has properties very close to the properties of a Grothendieck topos, in fact only one axiom has to be modified. This is one of the themes of the theory of vectoids of Nikolai Durov.

    What is that one axiom?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2015

    This must have been written by Zoran, I suppose. At vectoid the definition is recalled.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 11th 2015

    Thanks; I discovered both those facts you mention, but wasn’t enlightened on reading the definition of vectoid.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 14th 2015
    • (edited Oct 14th 2015)

    Thanks Todd. Essential difference is that the universal disjointness of coproducts is false for vectoids. This axiom is replaced by the totality axiom. I am not an expert here and I am a bit surprised learning that I wrote the line quoted in 1.

    See pages 2 (from def 1.1) and 3 (up to 1.6) in

    • Nikolai Durov, Classifying vectoids and generalisations of operads, arxiv/1105.3114, the translation of “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273
    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 14th 2015

    Thanks, Zoran. I guess in one sense that answer ought to have been obvious (it’s failure of extensivity that prevents any category of algebras of an algebraic theory, not necessarily commutative, from being a pretopos). But then another question I would have is: are vectoids Barr-exact? Staring at the definition, it’s not clear to me. Does pulling back along a morphism preserve all colimits?

    Thinking of “totality” (not exactly the same concept as at total category, but related) as a “modification” of disjointness of coproducts seems somewhat odd to me, at least at first glance.

    I guess I’d have to read the paper to understand the exact role of this totality. I do recall that the category of groups is not total (in his sense), but what it has to do with commutativity of a theory is just a bit murky to me as I write. Normally I understand commutativity in terms of presence of closed category structure, not exactness conditions.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 15th 2015
    • (edited Oct 15th 2015)
    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 15th 2015

    I meant to say the opposite of the category of groups is not total in that sense.