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    • CommentRowNumber1.
    • CommentAuthorKeith Harbaugh
    • CommentTimeOct 5th 2017
    Pardon me for asking a question that may be either stupid or already answered, but would it not be a good idea to have a page for "adjoint strings"?
    Searches using your search engine and at google (using its "site:ncatlab.org") don't produce much that seems relevant.
    I by accident found your page titled "adjoint triples", and then "adjoint quadruples", but that was months after my earlier search.
    If you don't think a page is needed, can something be done so searches on "adjoint string" at least yield the pages mentioned above?
    As to the pages mooted contents,
    both the pages on triples and quadruples could be merged into the page,
    plus the longer strings mentioned in
    http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html
    "Distributive adjoint strings"
    and
    http://www.ams.org/journals/proc/1994-122-02/S0002-9939-1994-1216823-2/
    "An Adjoint Characterization of the Category of Sets".
    Anybody have the time to, and feel the inclination to, create such a page?
    I'd do it myself but I really don't think I have adequate mastery of your style.
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 6th 2017

    Ok, created adjoint string. I think we should keep adjoint triple and adjoint quadruple as separate pages, but I’ve added interlinks.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 6th 2017
    • (edited Oct 6th 2017)

    I added the 5-tuple for SetSet and PSh(Set)PSh(Set), and the 7-tuple for abelian-ish categories and arrow categories. How general is the setting for the latter? Is the characterisation I found somewhere optimal?

    I also recorded the 5-tuple in derivators and 7-tuple in pointed derivators as in Mike’s paper (Generalized stability for abstract homotopy theories, p. 19).

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 6th 2017

    I also added in general examples of odd and even length. Perhaps someone could check I got the conditions right.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2017

    I think you got the conditions right. It looks like example 5 is just example 4 hommed into a category CC. I added another example 6, based on lax idempotent monads, and indicated that example 4 is really the walking version of example 6.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 6th 2017

    This is quite a nice page now, thanks everyone! I merged the examples of pointed categories and pointed derivators, since they are really the same phenomenon. Note that you shouldn’t write “Example 4” since if someone adds more examples or reorganizes the list then the numbering will change.

    I think something is off with the examples 5 and 6 (in current numbering); example 5 says it is a (2n+1)(2n+1)-tuple but example 6 is a (2n+2)(2n+2)-tuple, so if one is just homming out of the other then one of them is wrong. Is the extra adjoint in example 6 obtained from the assumed terminal object?

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2017

    Re #6: yes, it must be. Technically it’s not just homming into CC because of the extra degeneracy, but that degeneracy must be due to the terminal (which doesn’t exist in all categories). But the homming remark seems to me to explain what is essentially going on in that example. (It might even be better to leave out the terminal business, or perhaps even saying what happens if you have an initial.)

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 7th 2017

    I borrowed from Zhen Lin’s Stack Exchange answer, presumably designed to supply strings of even and odd length.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2017

    I added the example of ambidextrous adjunction to the list.

    • CommentRowNumber10.
    • CommentAuthorKeith Harbaugh
    • CommentTimeOct 7th 2017
    Sorry, should have mentioned this in my original post (I remembered it later).
    Shouldn't the most basic example of adjoint strings be that expounded in Section 2, Doctrines, of Street's "Fibrations in Bicategories"?
    http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0
    In particular, his (2.4).
    And of course this leads into the whole semisimplicial theory.
    Anyhow, other than that the page looks great to me!
    Lots of sophisticated examples.
    Thanks.
    • CommentRowNumber11.
    • CommentAuthorKeith Harbaugh
    • CommentTimeOct 7th 2017
    Oops, that's essentially example 5.
    But even so shouldn't that should come first (or zeroth), as the most basic example, needing the least sophistication.
    And maybe FB deserves a citation,
    for the thoroughness of Street's development, and his applications.
    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeOct 7th 2017

    Personally, I like that example where it is; maybe it needs less sophistication in some sense, but it’s also more artificial, and I like having more naturally occurring examples first. But I don’t feel especially strongly.

    Why don’t you dip your toe in the water of editing the nLab by adding a citation to Street yourself? (-: You can just copy the style of the other citations on the page.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2017

    Actually, I don’t think it’s more artificial, as it is the walking lax idempotent monoid (enriching the sense in which the augmented simplex category is the walking monoid). But I don’t feel too strongly either. :-)

    Some of the examples that come before have rich semantic content, which I also like. The adjoint characterization of SetSet is kind of sweet.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2017

    The adjoint characterization of SetSet is kind of sweet.

    Is there something more to be said about why SetSet behaves like this? Is similar behaviour found for the (∞,1)-Yoneda embedding as a way to characterise ∞Grpd?

    Is there anything that can be told about a functor from the fact that beyond an adjoint on one side there are further adjoints?

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 8th 2017
    • (edited Oct 8th 2017)

    If I were to try and condense the Wood-Rosebrugh result in just a few words, it’s a meditation on adjoint strings as imposing tighter and tighter exactness conditions on a category CC. We have CC is a total category if Y=yoneda CY = yoneda_C has a left adjoint XX, lex-total (same as being a Grothendieck topos under a mild size restriction) if that left adjoint XX preserves finite limits, a totally distributive category if XX has itself a left adjoint WW. Totally distributive comes close to being a presheaf topos; the candidate for the small site would be the inverter i:ACi: A \to C (akin to a 2-categorical equalizer) of the canonical 2-cell WYW \to Y that is mated to 1XY1 \cong X Y, which wants to be the category AA of atoms of CC.

    The candidate actually works (i.e., CC is an atomic category on top of being totally distributive – this forces CC to be a presheaf topos) if WW has itself a left adjoint. Then the situation is this: C=P(A)=Set A opC = P(A) = Set^{A^{op}}, the category of presheaves on AA, and the adjoint string looks like

    X=Set y A op:Set (Set A op) opSet A op,X = Set^{y_A^{op}}: Set^{(Set^{A^{op}})^{op}} \to Set^{A^{op}},

    and W=P(y A)W = P(y_A), the left Kan extension along y A opy_A^{op} that is left adjoint to Set y A opSet^{y_A^{op}}. (It’s almost as if the presheaf construction PP wants to be a lax idempotent monad, except there are size restrictions getting in the way. The multiplication for the monad would be m A=Set y A opm_A = Set^{y_A^{op}}.) But now we have a further left adjoint VP(y A)V \dashv P(y_A), and now the argument is that VV must look like P(ξ)P(\xi) where ξy A\xi \dashv y_A. In other words, now the (small) category AA is itself a total category!

    We are near the end: a small totally cocomplete category AA must be a poset (a sup-lattice), by a famous argument of Freyd. If further V=P(ξ)V = P(\xi) has a left adjoint UU, then UU is of the form P(η)P(\eta) where ηξ\eta \dashv \xi. In other words, now the small category AA is now a small posetal totally distributive category, forcing it in particular to be a posetal Grothendieck topos, and the only candidate there would be the degenerate topos A=Set 01A = Set^0 \simeq 1. So then CSet 1C \simeq Set^1.

    I’m not sure how much of the argument carries over to the (,1)(\infty, 1)-world, but I expect a lot, and maybe even all of it. I’m inclined to think that’s true.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2017

    Note that you shouldn’t write “Example 4” since if someone adds more examples or reorganizes the list then the numbering will change.

    To nevertheless refer to examples by numbers, as is good practice, enclose each example in

      +-- {: .num_example #ExampleName}
      ###### Example
    
      (Example text)
    
      =--
    

    and then refer to it from elsewhere by

    see example \ref{ExampleName}
    

    This then makes sure that correct numbering is automatically taken care of.

    (Most of you know this, of course.)

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeOct 8th 2017

    Re #16: of course, that doesn’t work if the examples are an enumerated list, and it would be weird to have one example in an environment but the rest in a list. One could break them all out into Example environments, but that seems overkill and visually distracting when they are all fairly short. It would be nice if we had a way to refer to the items in an enumerated list by number.

    • CommentRowNumber18.
    • CommentAuthorKeith Harbaugh
    • CommentTimeOct 8th 2017
    Okay, I screwed up my courage and made a little edit to example 5, linking the "adj string" claim to the relevant part of the simplex category page.
    Let me know if there's a problem, or correct it :-)
    By the way, is there any way to get a preview of changes to the pages before they are submitted?
    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2017

    Thanks Keith! I fixed up your link a little: before linking to a subsection of another page, it’s better to give that section an anchor name manually that is more stable in case of renaming.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2017

    Re: #15, it’s interesting that Freyd’s non-constructive theorem comes in there. I wonder whether constructively there can be categories other than SetSet admitting such an adjoint string.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2017

    that doesn’t work if the examples are an enumerated list

    It would be better not to have them in an enumerated list. Giving them each their environment, whether they are short or not, is more gentle on the reader and is more robust against future edits.

    • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 9th 2017

    We ought to find a home for what Todd writes at #15. Perhaps at the end of this page, or a page for itself.

    • CommentRowNumber23.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 9th 2017

    It probably needs a little cleaning up, but sure. The apparent reference to a nonconstructive result might be spurious, or my recollection unnecessarily coarse.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2017

    It would be better not to have them in an enumerated list.

    I guess we have to agree to disagree about that.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2017

    Perhaps a logical place for #15 would be at Set? Or maybe totally distributive category?

    Is there a name for totally distributive categories with one further left adjoint? “Atomically distributive”?

    • CommentRowNumber26.
    • CommentAuthorKeith Harbaugh
    • CommentTimeOct 9th 2017
    Is there a name for totally distributive categories with one further left adjoint? “Atomically distributive”?
    (How do you do those quotes?)
    Sounds like a question for Rosebrugh and Wood :-)
    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeOct 10th 2017

    How do you do those quotes?

    Select the radio button “Format comments as Markdown+Itex”. (I thought that was supposed to be the default these days.) That way you also get linking, math syntax, etc.; I don’t think there is any good reason to use “Text”.

    is there any way to get a preview of changes to the pages before they are submitted?

    No, instiki’s philosophy is that “save is preview”. If something comes out wrong in your edit, just edit again; multiple edits from the same person within a short span of time are recorded in the database as only one edit.

    • CommentRowNumber28.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 10th 2017

    And you need to preface the block of text with a greater than character, with a blank like after the quoted text.

    • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2017
    • (edited Oct 10th 2017)

    Re #28, typo: …blank line…