added the observation (here) that the push$\dashv$pull-adjunction counit on any object is given by tensoring it with the adjunction counit on the tensor unit:

$\epsilon_{\mathscr{V}} \,\colon\, f_! f^\ast \mathscr{V} \,\simeq\, \mathscr{V} \otimes f_! f^\ast(\mathscr{V}) \overset{ id_{\mathscr{V}} \,\otimes\, \epsilon_{\mathbb{1}} }{\longrightarrow} \mathscr{V} \,\otimes\, \mathbb{1} \,\simeq\, \mathscr{V}$ ]]>added (here ) the remark that when the projection formula is considered in the special case that both its variables are in the image of $f^\ast$, then the following diagram commutes:

$\array{ f_! \big( (f^\ast A) \otimes (f^\ast B) \big) &\overset{ \overline{\pi} }{\longrightarrow}& (f_! f^\ast A) \otimes B \\ \mathllap{{}^{\epsilon_{A \otimes B}}}\Big\downarrow && \Big\downarrow \mathrlap{ {}^{ \epsilon \otimes id } } \\ A \otimes B &=& A \otimes B }$ ]]>I haven’t read the article. But on p. 2 it seems to say that the new idea is to postpone checking the dualizability of the would-be dualizing object, then run with what is now a natural transformation instead of a natural isomorphism, and gain something thereby.

]]>So you’re treating the broader twisted story there, as in 4.6. And May 05 had already covered something of such twistedness.

So is what’s new in this paper the explicit framing of twisted ambidexterity? Presumably there’s a neat linear HoTT way to capture the twisted Wirthmüller context.

]]>Good catch. Yes, the comment about HL’s ambidexterity on p. 3 is along the lines of Rem. 4.7 in the Quantization note (p. 50).

]]>Added a reference

- Bastiaan Cnossen,
*Twisted ambidexterity in equivariant homotopy theory*(arXiv:2303.00736)

Probably deserves to go elsewhere. Seems to make contact with ideas in Quantization via Linear homotopy types (schreiber).

]]>So I have now reverted that one equation (here) to what I had up to rev 45, before “Mike Alex” changed it in rev 46 in a way that I can’t make sense of.

For what it’s worth, the version that I reverted to is verbatim that in Prop. 2.11 of May et al. (p. 6), though the point in question seems completely elementary and shouldn’t need a citation.

If anyone (such as user Mike Alex, who I don’t recall to have met) still thinks that I am missing something and that there is need to modify the entry at this point, please do say so here in the forum comments, so that we can discuss.

]]>Don’t forget what you added for rev 47 as mentioned in #1.

]]>This may need attention:

Those “$f_i$” were introduced in rev 46 by a user signing as Mike Alex.

It looks wrong to me. In your latest version, what’s the very last morphism (here) meant to be?

Maybe I am missing something, as I am looking at this now without leisure, in a stolen minute.

But in my last rev 45, the arrow in question (last one in Def. 3.2) seems evident: precomposition with the $(f_1 \dashv f^\ast)$-unit on $A$.

Right now I don’t see what should be wrong with that, or how the current modification even makes sense.

Okay, let me double check… Yes, my last rev 45 agrees with May et al.’s Prop. 2.11, p. 6.

Have to look into something else now, but unless there is a good reason not to, I feel we need to revert back to rev 45

]]>I changed several occurrences of $f_i$ to $f_!$, because I couldn’t figure out what else it might have meant. If this was wrong, please correct.

]]>re-did the typesetting of the adjoint triple with TikZ

]]>