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at monoidal adjunction the second item says
while the left adjoint is necessarily strong
but should it not say
while the left adjoint is necessarily oplax
?
No, I think it’s correct as stated. It’s lax by assumption, and you’re right that one can also exhibit an oplax structure, but a stronger conclusion holds, that it’s actually strong monoidal (the monoidal constraints are isomomorphisms). (Perhaps a proof should be written out in the article?)
It’s lax by assumption
Oh, I missed that.
But a proof would still deserve to be written out. The entry doctrinal adjunction that is pointed to for reference only knows about oplax structure, not strong structure.
Perhaps one may merely note that the first proposition in the statement section at doctrinal adjunction gives the strength. I’ve gone ahead and pointed to that in monoidal adjunction.
This result seems to be rediscovered now and then, but probably not in Kelly’s generality. :-)
Are there other useful examples to give? How about the one Emily Riehl gives as 13.2.5 in ’Categorical homotopy theory’ between $Mod_R$ and $Ch_{\bullet}(R)$ for a commutive ring $R$?
Thanks, Todd!
At monoidal adjunction I think we should say that for an adjunction to be monoidal in the standard sense that the left adjoint is strong monoidal and the right adjoint is lax monoidal, it is sufficient to demand that the left adjoint is strong monoidal.
And I suppoe there should be a quick self-contained proof that this condition is sufficient for the adjunction unit to preserve monoid structure, without reference to the more general statement of doctrinal adjunction?
That would make the page more usable.
sufficient (…) quick self-contained proof
Yeah, I think this direction is pretty much follow your nose.
But in practice, I find that the most striking and useful consequence is the necessity: that in a lax monoidal adjunction the left adjoint is strong monoidal.
Just a thought from someone who has worked a bit with adjunctions but not with monoidal adjunctions so far: might it be useful to add to the page an “algebraic” point of view?
In the sense that
a usual adjuction can be expressed via
a partial algebra with
2 sorts : 1-cells , 2-cells
2 function symbols :
while for a monoidal adjunction it seems (to me) that one needs
a partial algebra with
3 sorts
3 function symbols
namely, in addition to the aforementioned things,
the sort “0-cell”, and
the function symbol for “monoidal product” (which happens to be total, incidentally, the algebra remaining very partial of course).
This appears no to be emphasized or systematized in this way anywhere, and it helped me, personally, to link the concept to things I already knew. (The increment from (#sort,#func)=(2,2) to (#sort,#fun)=(3,3) and so on…)
I myself wouldn’t consider this POV likely to be that useful for general readers of this page.
Since you so often speak of “systematization”: to me there is the useful systematic idea of internalizing common notions from $Cat$, such as the notion of adjunction, to more general 2-categories. Thus, the notion of adjunction is a simple piece of 2-categorical algebra, and when you apply it to the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations, you get the notion of monoidal adjunction under discussion here.
this direction is pretty much follow your nose.
Yes, but it seems to be tedious. There must be a writeup somewhere, no? (I can’t open Street’s article from where I am here.)
Ah, never mind, I got it.
One considers this commuting diagram here, and the rest is evident:
$\array{ && A \otimes A &\overset{\eta(A \otimes A)}{\longrightarrow}& R L(A \otimes A) &\overset{R(\mu_L^{-1})}{\longrightarrow}& R( L(A) \otimes L(A) ) \\ && \downarrow && \downarrow^{\mathrlap{R L( \eta(A) \otimes \eta(A) )}} && {}^{\mathllap{R( L(\eta(A)) \otimes L(\eta(A)) )}}\downarrow & \searrow^{\mathrlap{id}} \\ \mu_R(L(A)) &\colon& R L (A) \otimes R L ( A) &\overset{\eta(R L (A) \otimes R L (A))}{\longrightarrow}& R L (R L ( A) \otimes R L (A)) &\underoverset{}{ R( {\mu_L^{-1}(R L(A))} ) }{\longrightarrow}& R ( L R L (A) \otimes L R L (A) ) &\overset{R( \epsilon(L(A)) \otimes \epsilon(L(A)) )}{\longrightarrow}& R(L(A) \otimes L(A)) \\ && && && && \downarrow^{\mathrlap{ R( \mu_L(A)) }} \\ && && && && R L( A \otimes A ) \\ && && && && \downarrow^{\mathrlap{ R L (\mu_A) }} \\ && && && && R L (A) }$I am putting this into the entry now.
Okay, I have written out the explicit proof by inspection: here
I have added statement of the example of stabilization at “monoidal adjunction” here and also at “monoidal Quillen adjunction” here
(I had thought that this example had long been stated there, but no)
Thanks. You stated this only for orthogonal spectra and unpointed spaces. Presumably it is true for some other models of spectra and for based spaces as well? Also I don’t remember how this statement figures into Lewis’s no-go theorem about good categories of spectra and how it is avoided by the various models?
The factorization through pointed spaces I took as evident, I can add details on that.
The case for orthogonal spectra happens to be the one for which a fully self-contained proof is spelled out on the $n$Lab. It also works for symmetric spectra, but due to the annyoing issue with the class of stable weak equivalences there not coinciding with the stable weak homotopy equivalences, I didn’t spell that out completely on the $n$Lab. Of course it is in Model categories of diagram spectra.
Once I knew which loophole allowed highly structured spectra to evade Lewis’ conclusion, but now I forget what it was. If we don’t already, then we should discuss this at symmetric monoidal smash product of spectra.
IIRC, EKMM spectra use a different loophole than diagram spectra do. It has something to do with the unit of the EKMM monoidal structure not being cofibrant.
There must be a writeup somewhere, no?
A relevant reference for part of this is (the proof of) 13.7 Lemma in
R. Garner., M. Shulman, Enriched categories as a free cocompletion. Advances in Mathematics 289 (2016)
I do not have sufficient experience with this part of the literature to judge whether one should include this reference into the nLab page and how.
Peter, what you point to is discussion of the mate-business lifted to bicategories. I have added a pointer there.
But what I had been asking for in the line that you quoted is a reference for the fact that for a monoidal adjunction unit and counit on monoid objects are monoid homomorphisms. But never mind, I have written out the proof meanwhile.
I have added to the example of the stabilization adjunction a comment here on how to see that the lax monoidal structure induced on $\Omega^\infty : Ho(Spectra) \to Ho(Top)$ comes out indeed as expected, from the given strong monoidal structure on $\Sigma_{orth}^\infty : Top \to OrthSpectra$.
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