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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 12th 2011

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 12th 2011

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 12th 2011
• (edited Apr 12th 2011)

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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 12th 2011
• (edited Apr 12th 2011)

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 ;-)

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeApr 13th 2011

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?)

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeApr 13th 2011

Also, is there a particular reason to use $\eta$ for counits at adjoint quadruple? Usually $\eta$ is a unit.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 13th 2011
• (edited Apr 13th 2011)

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

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 13th 2011

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.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeApr 13th 2011

On the quadruple page you have

$(f_! \dashv f^* \dashv f_* \dashv f^!) : C \to D$

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?

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 14th 2011

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.