martin e-mailed me about a difficulty in the attempted proof that the coherent sheaves over the projective space of v is the "walking invertible quotient of v" symmetric monoidal finitely cocomplete algebroid... i'll try thinking outloud about it here... i'm not surprised that a difficulty shows up here, but i'm still somewhat optimistic about getting past it...

hmm... been chatting about it with martin a bit... wouldn't be surprised if i've been making some really stupid mistake here...

??to compel the cokernel of f to be zero is to invert the comparison map from...
??well perhaps either from 0 to the cokernel or vice versa, but seems like one of those makes better conceptual sense... the cokernel is always a quotient object of the codomain... ??so maybe it makes better sense to say that the comparison (??if that's an appropriate name for it... ??...) map from the cokernel to 0 is invertible...

??then maybe we should consider inverting all of the morphisms from bounded objects to zero ... ??or something like that??? .... hmmm, maybe not even that's justified ... ???or something?? ... i was going to say though that maybe ... well, some coinverters are easy to "explicitly calculate" and some less so... not sure how easy this (??...) one might be ... ??...

??so suppose that we invert the map from x to 0 in a cocontinuous way, and suppose
also that we have a map f : y -> x st cok(f) = x ... ???then does cok(f) then necessarily become 0 ?? .... hmm, at the moment i seem to be in danger of thinking that that's trivially true... ???or something??? ... ok, yes, that seems trivially true, but it's not the relevant question ... so then what _is_ the relevant question??

well, one question is whether f becomes invertible... but of course the answer to that is pretty obviously no... but perhaps just to nail it into the ground i should describe some universal or at least prototypical example ... or something...

well, so let's consider the "walking map" ... hmmm... this is reminding me now of my discussions with julie bergner about serre subcategories and so forth in the context of quiver representations... but that was pretty certainly before i was clear about what doctrine i should be working in, right?? ... actually i'm even a bit unsure about whether my motivation in those discussions had to do with coherent sheaves at all; i'm not certain that i really knew anything about coherent sheaves by that time... ??...

anyway, let's consider representations of the a2 quiver, and let's cocontinuously compel ....

hmmm... this could be interesting... i'm pretty sure that the discussions with julie were before i'd realized about those nice simple examples of finitely cocomplete algebroids that aren't abelian categories... which are probably showing up here...

so maybe this is hinting to us that there _is_ some pretty easily explicitly
describable localization (but also subcategory...) of the graded modules that has the universal property that i've been (presumably mistakenly) ascribing to the coherent sheaves ... and this localization is not an abelian category... or something ... ??so is there a really obvious guess to make here??

??so 0->1 included into 1->1 is mono ... but it's also the generic morphism in the cocomplete context... or something... and cocontinuously forcing its cokernel to be zero gives you the generic epi... which lives in the walking epi, which is the "fg monos" ... ??so it seems that in the fg monos, the cokernel of the inclusion of 0->1 into 1->1 is, of course not the old cokernel which is 1->0 and not mono, but rather the "monicization" of this... which is perhaps 0->0 ??? or something??

??let's try testing that... ??a morphism from the mono 1->1 to the mono

m : x >-> y

(aka "an element of domain(m)" ??) which has the property that [preceding it by the inclusion from 0->1 to 1->1 gives zero] (aka "it gets taken to zero by m"??) is essentially ... ???just zero?? ??that's sort of the whole point of monos, that the only thing they take to zero is zero ... ???or something??...

right... i think... so 0->1 included into 1->1 is epi in the fg monos... or something...

??but what's the lesson supposed to be here???

??that... ??well, for one thing, 0->1 included into 1->1 certainly isn't invertible in the fg monos ... it's just epi ... ?? or something??

??so what about the graded modules where ... ???????? ....



??so _did_ we miss something about "thick subcategory" being kernel of _exact_ functor ?? ??or something?? ... if so then sounds like pretty stupid mistake ... ???... still not sure about this... ??.... ??something about finite vs arbitrary co/limits?? or something???

??so what about attempted analogy (of _some_thing ... ???...) to grothendieck topology.... ????? ... ???what about tensor products vs cartesian products ? ...
... ???...

??so what about idea that if we think somewhat systematically about all the sorts of semantically (or something...) motivated examples that we've tried to develop in doctrine of symmetric monoidal finitely cocomplete algebroids then we ought to be able to figure out what's wrong... ??or something??? ... ???is it already somewhat near-obvious that what's wrong has something to do with... abelianness vs mere finite cocompleteness... ??what _about_ whether there's some nice way to "get abelianness" (or something...) without demanding (too much...) more than "right-exactness" of the tensor functors and without losing "presentableness" (or something...) ?? .... ????...

hmm... ??if we're not supposed to require exactness, then... ??why _do_ we always seem to mod out by a thick subcategory when passing to a subscheme.... ???or something??... ... ????... ???

??hmmm... "evaluation at a point" isn't flat... ??but then it's not a localization... (??why do you think they call it "localization" ????..... ??...) ...??so what about the "localization part" of such an evaluation?? ... hmm, usual localization/conservative factorization ?? or something??? localization (??or something??) is flat; is conservative "anti-flat" or something??? ....

??any relationship to some puzzle that we had about "artin-wraith glueing" in algebraic geometry?? or something ... ??something about coherent sheaves ... ???...

so what _about_ universal property wrt _exact_ tensor functor?? .... ??despite problematicness of compatibility between tensor product and kernels?? or something?? ... ????..... ... ???....

hmm... ??so consider... the objects that think that a given morphism (or collection thereof) is epi ... ???or something...

??some sort of level slip involving "exact tensor functor" and "flat" ... ??or something?? ... ??something about... ??a line object seems already about as flat as it can get, but .... ???or something???

so what _about_ the universal property of the coherent sheaves over projective space as a smfcca ??... as martin suggests... ??... ??seems somewhat straightforward?? ... but... ??do we at all understand how to look at it systematically?? ... ???.... ??any possibility of expressing it in terms of systematically expressed type of (co-)"embedding"?? ... enhanced epi ... ??or something?? ??or is there some sort of "context-sensitiveness" (or something...) that screws this up?? ... ???hmmm... ???...

??what about sheaf vs separated presheaf here... ??not too clear yet why it might be showing up... ??hmm... ??when we force an otherwise completely generic morphism to become epi, then... ??maybe it makes sense that in that case the corresponding "sheaf condition" (vs "separated presheaf condition" ... ??...) amounts to making the morphism invertible ... ?? ... ??because .... ??? ... ????.....

??is there some nice systematic relationship between universal properties wrt flat ag morphisms and wrt general ag morphisms, allowing maybe to understand the latter in terms of the former?? ??or something like that??...

??relationship between martin's idea about element-wise proofs for tensor categories, and... ??some "doctrine" ideas... ??or something??...

[cut from skype-hat with martin:

one vague idea that i have that i might like to discuss eventually is that because of the importance of abelian categories as compared to mere finitely cocomplete additive categories, it might turn out to be a good idea to place more emphasis on exact tensor functors than on merely right-exact ones, even though right-exact ones ... because of "incocompleteness" issue ... ?? ...]

??so consider ... ???line object L tw sections s,t st ... ?? (s,t) is the cokernel of a certain morphism from ... ?? ...