so is there a good ( =?= left adjoint) concept of "exactization" ??

a -d-> b -e-> c

??graded module of [exterior algebra of 1, with usual (Z-)grading] ... ??as chain complex ... ???

ker(

chain complex condition vs exactness ...

vs ...

??uniqueness vs existence of ... ??? ?????

???hmmm.... ???or are we in fact maybe getting something more like... the exact sequences as being analogous to the "genuinely filtered" things ... ???or something??? ...

hmm... ??might be like we originally wanted ... ???....

???something about ... ???exact sequences where every differential is monic ... ???or something??? ????pretty restrictive???.....

???very restrictive, in fact?? just zero ??....

???"each kernel element is the image of a unique element" ... ???....

chain complex : exact sequence ::

generalized filtered : genuine filtered

?? or something???

so _have_ we thought about it that way before??

...sounds more familiar now...

hmm, so what about "coherent sheaf over projective space of 2d odd vector space" ??...

??so what about "d^2 = 0" vs "d^2 + d^2 = 0" ???? .... ???weird from dold-kan perspective???? or something???....

?? x,y st d1(x)+d1(x) = 0, d1(y)+d2(x) = 0, d2(y)+d2(y) = 0 ....


???what about ... ???associated graded doesn't measure genuineness of generalized filtered... whereas homology measures failure of exactness ... ???or something?? ...

???so what about something about ... ??"bicomplex where every square's a pullback" ???... or something... ???...

??so what about ... ??"line objects l1 and l2, with respective sections s1 and s2, st the hopefully obvious pushout condition holds" ??? or something ... ????


???blecchhh, wait a minute .... ????

with exactness, you're imposing only the existence condition, not the uniqueness...

???whereas with genuineness of filtration, you're imposing only the uniqueness condition, not the existence ... ???....

?????????????? .....

??so maybe exactization is screwed up???? ... doesn't exist ... ????....

??exactness as not a pure colimits property ... ??... ??maybe pretty obvious ... ???... ??hmm, some recent comment somewhere about "revenge of kernels" ... ????...

???so what about chain complexes where the differentials are injective?? or something????....

hmmm... ??well, maybe this misadventure might at least help in future attempts to straighten out relationship between "cohomology" and "associated graded" ... ???and so forth...

also, we could still experiment with "odd" variant of projective space ... and so forth ... ???....

??what about idea of "anti-separated presheaf" ?? .... or something ... ???hmm, so what about two complementary weakenings of "pullback"?? ... ???both somewhat used??? ...