so let's consider certain doctrines of the rough flavor "tensor categories with adjoints for all objects" ...

one motivation for this is coming from learning or perhaps being reminded about "coherent" (and/or quasicoherent...) meaning something at least vaguely like "cokernel of morphism between dualizable objects" ...

??so what _about_ relationship between "dualizable" and "flat" ?? ??hmm, also something about variants of "flat" ... something about finitary vs small limits... ??or something?? ???maybe somewhat more to it than that???

??at some point i should worry a lot here about issues like "bosonicness" or something, but for the moment i'm going to try not worrying about it too much...

??what about whether there's any particular nice kind of cokernel (or something...) that the class of dualizable objects is guaranteed to be closed under?? ???or something ... ???...

??so what _about_ doctrine of symmetric monoidal finitely cocomplete k-linear categories where all objects have adjoints ????....

but also...

symmetric monoidal k-linear category with all small sums and all adjoints ...

symmetric monoidal k-linear category with all finite sums and all adjoints ...

???... other possibilities??

??what about relationship to "derived category" ideas ???

??so what _about_ universal property of dualizable module-sheaves over nice scheme or stack (or something...), wrt some such doctrine ??

???hmmm, so what about something about splitting of idempotents??? .... or something .... ????....

??symmetric monoidal "absolutely cocomplete" k-linear category ...

??same as "absolutely cocomplete" symmetric monoidal k-linear category??? ??or something??

??so _is_ it true that retracts of dualizable objects are dualizable?? ??or something?? ??and ... ??is this in some vague sense what we really want, in the sense that .... ??? ??maybe what i'm groping for here is something like... ??whenever it's possible to construct one dualizable module sheaf from some others via finite colimits, it's possible to do so via absolute colimits ... ???or something like that???

??hmm, maybe case of line bundles over elliptic curve might help in thinking about this question?? ??? ...

??so what about whether embeddings (or something) in theories of various of these doctrines are automatically good?? ??seems like we had an example that almost seems to suggest the answer is no... but perhaps not?? ... i think that all we had an example of was... a non-good embedding between dualizable objects, with dualizable cokernel ... ???or something... ??which might not be good enough because of lack of duals in larger context?? ???or something??? ...try checking up on this...

??what about relationship between good embeddings and retractions???? or something?? ... ??? seems problematic because of non-canonicalness of retraction in general ... ???or something??? ....

??what about something about direct sum of object and its dual?? ... ???...

??still a big problem here?? ... ??we pretty much know, from case ofprojective line for example, that the symmetric monoidal k-linear category of the dualizable objects doesn't know about the whole category, right??? ...

???what _about_ something about "map of constant rank" or something?? ... ??something about trying to have "intrinsic concept" of such ... ???or something?? ... and so forth ... ???....