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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 11th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn [topos of coalgebras over a comonad](http://ncatlab.org/nlab/show/topos+of+coalgebras+over+a+comonad#related_concepts) it is written: > Also a category of [[algebra over an algebraic theory|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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 11th 2015
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   Author: Urs
   Format: MarkdownItexThis must have been written by Zoran, I suppose. At _[[vectoid]]_ the definition is recalled.

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

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 11th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks; I discovered both those facts you mention, but wasn't enlightened on reading the definition of vectoid.

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

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeOct 14th 2015
   • (edited Oct 14th 2015)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThanks 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](http://arxiv.org/abs/1105.3114), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273

   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
5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 14th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks, 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 15th 2015
   • (edited Oct 15th 2015)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNB <http://mathoverflow.net/questions/49175/is-every-saft-category-cocomplete>

   NB http://mathoverflow.net/questions/49175/is-every-saft-category-cocomplete

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 15th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI meant to say the opposite of the category of groups is not total in that sense.

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