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Zoran has created adjoint triple, I have added adjoint quadruple
This one is nice. We should have more on the duality from the old paper of Eilenberg. I mean the monads having right adjoints and comonads having left adjoints, thsi can be generalized and it is mostly about the dualization of algebraic structures within an adjoint triple.
I have added a remark on how adjoint quadruples induce adjoint triples. Then I expanded the remark on how adjoint triples induce adjoint pairs. Finally I spelled out, for completeness, the proof that for an adjoint triple $F$ is full and faithful precisely of $H$ is.
I have also added still more hyperlinks to adjoint triple. I would like to appeal to you and everybody: please add double square brackets around every single technical term, at least on first occurence.
Because, what do we have a wiki for if, say, the term triangulated functor appears on a page and it is not linked? Which reminds me: the entry (or redirect) triangulated functor is still to be created ;-)
It seems to me there is something missing from the proof of Prop. 1: a priori “being isomorphic to the identity” is a weaker statement than the particular unit or counit map being an isomorphism. Do we need to appeal to A1.1.1 in the Elephant? (and is that fact reproduced anywhere on the nLab yet?)
Also, is there a particular reason to use $\eta$ for counits at adjoint quadruple? Usually $\eta$ is a unit.
Do we need to appeal to A1.1.1 in the Elephant? (and is that fact reproduced anywhere on the nLab yet?)
right, I have added that (and some other basic stuff) to the Properties-section at adjoint functor
Also, is there a particular reason to use $\eta$ for counits at adjoint quadruple?
Not a particular good reason, anyway. I’ll see if I find time to change it.
On the quadruple page you have
Every adjoint quadruple
$(f_! \dashv f^* \dashv f_* \dashv f^!) : C \to D$induces an adjoint triple on $C$
$(f^* f_! \dashv f^* f_* \dashv f^! f_*) : C \to C \,,$
Can we say anything about the situation from $D$’s perspective, i.e., the three endofunctors, $f_! f^*, f_* f^*, f_* f^!$?
And does anything interesting happen if you take the adjoint triple of a quadruple, and then take the adjoint pair of that triple?
Can we say anything about the situation from $D$’s perspective, i.e., the three endofunctors, $f_! f^*, f_* f^*, f_* f^!$?
These also form an adjoint triple, yes. I have added that to the entry.
And does anything interesting happen if you take the adjoint triple of a quadruple, and then take the adjoint pair of that triple?
Good question. I need to think about that. At least I know one example where one has something like an adjoint quadruple and does care about the composite going back-forth-back-forth through the four adjunctions. I once mentioned that here.
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