?? "relatively free cocompletion of a category c, relative to specified cocones becoming colimit cocones" ...

??vs ... ??? "universally compelling specified cocones in a cocomplete (??up to a point?? ...) category to become colimit cocones" ...

???latter as special case of former, via ... ????.... ??seeing cocomplete category as its own "relatively free cocompletion, relative to the existing cocones becoming colimit cocones" ??? ..... ??? ....

???classification of reflective subcategories of cocomplete categories ... ??? ....

???moore-postnikov-style classification of cocontinuous functors .... ??....

?? "promoting cocones to colimit cocones" ... ??

??limit sketches ??? ....

???small-ly presented presheaves on a large category, vs small-ly valued such ... ??? .... latter doesn't imply former??? ??does former imply latter ??? .... ???maybe yes ???? ..... ??well, only for large categories with nevertheless some reasonable smallness condition ... ????

?? finitely presented presheaf on finite category is finite-valued ...

?? finitely presented presheaf on finitely presented category is not necessarily finite-valued ... ???but maybe nevertheless, situation is "better" for certain higher cardinalities ??? .... ??maybe fairly straightforward ...


?? relatively free co-completion of ... ????....

??for sufficiently nice k, the k-small-ly presented presheaves on a locally-k-small category c form the free k-small cocompletion of c ... ??? hmm, that seems to work even for k = omega ... ??? ... which i didn't intend, i think ... ??have i got something particularly screwed up here ??? ...

?? k-small-ly presented presheaves on locally k'-small category ... ???.... ???with k''-small collection of objects ??? ....

k-small diagram scheme ... ???k-small-ly presented diagram scheme ... ??? ... ???"relations" in diagram scheme as superfluous, _sort of_ ... ???though not to "what it means to have limits of that kind" ... ??? ....




for sufficiently nice k:

1. the free k-small cocompletion of a category c is the category of k-small-ly presented presheaves on c.

2. the relatively free k-small cocompletion of c, subject to the constraint that a class x of k-small cocones in c become colimit cocones, is the k-small-ly presented presheaves on c which take cocones in x to limit cones.

3. as a special case of #2, if c is k-small-ly cocomplete and x includes all of the x-small colimit cocones in c, then this allows us to universally compel a class of k-small cocones in a k-small-ly cocomplete category to become colimit cocones.


??? analogy here between tensor product of ab gps (?? or ... ?? comm semilattices ?? ... ???? ....) via preentations and tensor product of cocomplete categories via sketches .... ???? vague worries about whether there's some sort of duality screw-up in this analogy ??? .... ???? maybe not ??? .....