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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2011

    There is what I think is an incorrect or misleading statement in the Examples section of total category: that the category of algebras of a monad TT on SetSet is total “because” it is cocomplete (yes), has a generator (yes), and is well-copowered. The last part is incorrect by considering the category of frames as monadic over SetSet. In particular, the free frame on a countable number of generators has a proper class of (isomorphism classes of) epimorphisms coming out of it. See Johnstone’s Stone Spaces, page 57 (corollary of 2.10).

    I do believe it is a theorem that any category which is monadic over SetSet is total, but the proof is perhaps somewhat non-trivial. (The case of monads with rank is, I believe, unproblematic – it’s the unbounded case which is harder.) If anyone knows of a good proof of that, I’d like to hear; otherwise I’ll try to dig it out and write it up for the Lab.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2011

    There is a proof for categories monadic over Set in Tholen’s paper, referenced in the entry, which goes via solid functors. I’ve corrected the statement at total category (I think).

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2011

    Thanks, Mike. I ought to look at that paper.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 2nd 2011
    • (edited Aug 2nd 2011)

    The examples section at total category still looks suspect to me. The edit by Mike looks fine, but the assertion

    Any cocomplete, well-copowered category with a generator is total

    looks dubious to me (even though I don’t have a counterexample). I am guessing that what was intended was a corollary of something from Tholen’s paper (referenced in total category) that I can see is true:

    • Any cocompact category with a generating set is total.

    By definition, a category CC is “cocompact” if any functor G:CAG: C \to A that is co-admissible (i.e., A(a,Gc)A(a, G c) is small for any aOb(A)a \in Ob(A), cOb(C)c \in Ob(C)), and that preserves any limit which exists in CC, has a left adjoint. If CC has a generating set, then it’s not too hard to see that y:CSet C opy: C \to Set^{C^{op}} is co-admissible, and of course yy preserves any limit that exists in CC, so under these conditions yy would have a left adjoint, i.e., CC is total.

    The special adjoint functor theorem gives conditions under which a category is cocompact:

    • A locally small category that is complete, well-powered, and has a cogenerating set is cocompact.

    These conditions are dual to those of the assertion above. So those hypotheses give us that the category is compact. But it is not true that a compact category with a generator need be total.

    If there’s something that I’m missing here, I hope someone can tell me. An example which refutes the iffy assertion would be wonderful. As would a proof of the assertion, but as I say I have my doubts. I’ll wait a bit for a response, before I undertake to rewrite the page (which I am more or less prepared to do), and correct some other errors on the Lab that have emanated from the iffy assertion.

    Edit: The latex is not rendering correctly. To see it, I guess click on Source, because I can’t figure out what’s wrong. Maybe something to do with the server upgrade?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2011

    I believe that assertion is proven in Day’s short paper “Further criteria for totality”, in Cahiers here.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 3rd 2011

    Wow. Thanks! (How did you happen to know of this paper?)

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 3rd 2011

    Well, I think I was the one who originally added that comment to total category based on having read that paper. I should have included the reference then, of course. As for how I found the paper originally, a while back I wanted to know all I could about total categories, so I read all the papers I could find. I have no memory of how I found that particular paper.