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    • CommentRowNumber1.
    • CommentAuthoreparejatobes
    • CommentTimeJan 31st 2012
    • (edited Jan 31st 2012)

    Hi everyone!

    I’ve created relative adjoint functor, and linked to it from the local definition of adjoint functor (a partially defined adjoint yields an adjoint relative to the inclusion of a full subcategory).

    L JRL {\,\,}_J\!\dashv R and L JRL \dashv_J R is as far as I know nonstandard notation, but I think it’s ok, even if the left subscript feels a bit kludgy. I will add more stuff in the next few days.

    PS: Thanks a lot to all the nLab contributors; in the past few years I’ve learn a lot through here :) I now have the time and a little bit of confidence to contribute, so any pointers, tips, formatting, style suggestions, whatever will be greatly appreciated

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2012


    When I clicked on relative adjoint, I got the message “Unknown web ’show’ “. I don’t know what that means.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJan 31st 2012

    Why not a symmetric notation like LJRL\stackrel{J}\dashv R or alike ?

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2012

    Eduardo : It seems some redirects are needed. Have a look at some other pages and see how redirects are formed.

    • CommentRowNumber5.
    • CommentAuthoreparejatobes
    • CommentTimeJan 31st 2012


    the links were wrong, it should be fixed now, thanks!


    mainly because the notion is not symmetric (see the entry), and also because it works as mnemonic for where to put LL, JJ and RR in the hom-iso:

    • L JRL {\,\,}_J\!\dashv R corresponds to Hom C(L(),)Hom D(J(),R()) Hom_C(L(-),-) \simeq Hom_D(J(-),R(-)) (just like a regular adjunction, with JJ on the left slot of the hom)
    • L JRL \dashv_J R corresponds to Hom D(L(),J())Hom C(,R())Hom_D(L(-), J(-)) \simeq Hom_C(-, R(-)) (now, JJ is on the right)
    • CommentRowNumber6.
    • CommentAuthoreparejatobes
    • CommentTimeJan 31st 2012


    done! thanks

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJan 31st 2012
    • (edited Jan 31st 2012)

    I see.

    Alexander Rosenberg uses these relative adjoints (not in this terminology) a lot in his 1988 Stockholm preprint on Q-categories (where it is used to construct Q-categories). Just a very little piece of that is transferred in the appendix A.1 (page numerated as 61) of more recent

    • M. Kontsevich, A. Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (dvi,ps)
    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2012

    As an aside to Eduardo: can you start a page for yourself on the nLab otherwise someone will just put a stub and that is always less good? By the way one of the links from your homepage gives bizarre results. That to Antonio is fine but the other one leads to a strange page.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2012

    Welcome! A good page to have; I added links back and forth to free object.

    • CommentRowNumber10.
    • CommentAuthoreparejatobes
    • CommentTimeMar 5th 2012

    Well, looks like that of me having more time was off by a month :)

    I’ve added the definition in terms of absolute liftings, and a couple of examples: fully faithful functors and the relative adjoint/absolute lifting reformulation of Yoneda lemma.

    Zoran #7: thanks for the ref, will take a look at it

    Tim #8: will do, thanks. About the link, I guess you’re referring to -that’s where I work! :) maybe we need a somewhat less minimalistic website :)

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 16th 2016

    Since someone pointed me to Terminal semantics for codata types in intensional Martin-Löf type theory, I added it to a new page relative comonad. I hadn’t heard of relative adjoint functors before. All of these pages, with relative monad, could do with some examples.

    Is there anything comparable to the first adjunction one meets – underlying set/free algebra?

    • CommentRowNumber12.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 16th 2016

    All of these pages, with relative monad, could do with some examples.

    I looked at the linked reference:

    Thorsten Altenkirch, James Chapman, Tarmo Uustalu, Monads need not be endofunctors, Logical methods in computer science (pdf)

    and saw they were defining things in terms of skew monoids (which aren’t a thing) but after some reading around realized that they mean objects of a skew monoidal category - one in which the natural families of constraint maps are not necessarily invertible. A definition can be found in

    Mitchell Buckley, Richard Garner, Stephen Lack, Ross Street, The Catalan simplicial set (arXiv)

    and the term skew monoidal originates from

    Kornel Szlachanyi, Skew-monoidal categories and bialgebroids arXiv (2012)

    Stuff at this MO question, What is this operad-like structure called?, indicates maybe that the categories are also

    what Tom Leinster calls in his Higher Operads, Higher Categories (2003) book an unbiased lax monoidal category.

    (since Altenkirch et al. above note that they had been calling them lax monoidal categories)

    As of now, the nLab appears to have no entries on skew or lax monoidal categories.

    [ this post really doesn’t answer any quest for examples. It is just some notes I made while poking around ]

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeSep 16th 2016

    I thought a skew monoidal category was a biased lax monoidal category.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 17th 2016

    Are these ideas in a state of flux still then?

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeSep 18th 2016

    I think it’s pretty consistent: “lax monoidal categories” are unbiased (the terminology is correct because they are the lax algebras for the monad whose strict/strong algebras are monoidal categories), “skew monoidal categories” are biased.

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