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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).
$L {\,\,}_J\!\dashv R$ and $L \dashv_J R$ 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
Welcome!
When I clicked on relative adjoint, I got the message “Unknown web ’show’ “. I don’t know what that means.
Why not a symmetric notation like $L\stackrel{J}\dashv R$ or alike ?
Eduardo : It seems some redirects are needed. Have a look at some other pages and see how redirects are formed.
@Todd
the links were wrong, it should be fixed now, thanks!
@Zoran
mainly because the notion is not symmetric (see the entry), and also because it works as mnemonic for where to put $L$, $J$ and $R$ in the hom-iso:
@Tim
done! thanks
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
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.
Welcome! A good page to have; I added links back and forth to free object.
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 http://ohnosequences.com -that’s where I work! :) maybe we need a somewhat less minimalistic website :)
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?
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 ]
I thought a skew monoidal category was a biased lax monoidal category.
Are these ideas in a state of flux still then?
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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