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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2013

    at total category I have added after the definition and after the first remark these two further remarks:

    +– {: .num_remark}


    Since the Yoneda embedding is a full and faithful functor, a total category CC induces an idempotent monad YLY \circ L on its category of presheaves, hence a modality. One says that CC is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.


    +– {: .num_remark}


    The (LY)(L \dashv Y)-adjunction of a total category is closely related to the (𝒪Spec)(\mathcal{O} \dashv Spec)-adjunction discussed at Isbell duality and at function algebras on ∞-stacks. In that context the LYL Y-modality deserves to be called the affine modality.


    • CommentRowNumber2.
    • CommentAuthorTim Campion
    • CommentTimeDec 21st 2015

    The article currently says something to the effect of “cototal categories are more rare than total categories”. But it occurs to me that TopTop is cototal by Day’s criterion (it’s complete, mono-complete, and has a cogenerator given by the indiscrete space on two elements). In fact, since being a topological functor is self-dual, and since SetSet is cototal, any category which is topological over SetSet is cototal – I’ll add this to the article as a class of examples. I don’t know of a reason to expect categories of a more “algebraic” nature to be cototal, but at least this suggests that many categories of “spaces” might be cototal.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 21st 2015
    • (edited Dec 21st 2015)

    Yes, good observation.

    It’s known for example that GrpGrp is not cototal, and neither is say the category of commutative rings CRingCRing. An easy way to see this is to produce continuous functors CSetC \to Set that are not representable, e.g., for C=GrpC = Grp, the classical example is the class-indexed product of representables hom(G,)hom(G,-) where GG ranges over all simple groups. (For any group HH, Hom(G,H)Hom(G, H) will be trivial once the simple group GG has cardinality greater than HH, so the product of Hom(G,H)Hom(G, H) over all simple GG will still be a set.) A similar example can be cooked up for commutative rings; see e.g. this MO answer. I guess algebraic categories with a plentiful supply of simple objects would be amenable to similar constructions.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 11th 2017

    I added some more examples to total category. (One is that Ab is cototal as well as total.)

    • CommentRowNumber5.
    • CommentAuthorMarc
    • CommentTimeJun 27th 2018

    corrected year of Ross Street’s publication and inserted a link to the article at the AMS journal website

    diff, v30, current

  1. The fact that total implies complete is a categorification of the fact that for partial orders cocomplete implies complete.

    diff, v31, current

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeMay 31st 2021

    I added a reference for proof that categories monadic over Set are total; maybe someone can make the reference into a link to the paper by Kelly at the bottom of the page.

    diff, v32, current

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeMay 31st 2021

    Added link to Kelly reference, and added some other links.

    diff, v33, current

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