?? grothendieck topology as special case of limits sketch ??

?? how to get cocone from "covering" ... ??? ....

??saturated sketch ??? ??hmm, well, first, is there maybe some better concept than "cocone" to use here ?? ... ???something like ... ??? "weighted diagram : co-wedge from weighted diagram to object :: weight : ??? from weight to object" ??? .... maybe .... ????element of presheaf ???? ...... ????? new improved "sketch" as maybe something like ... ??collection of presheaf elements to become universal ??? ..... ??saturated such as collection of presheaf elements closed under .... ???? ..... ???or is it maybe somehow ... subobjects rather than elements ???? ..... ??no, that sounded a bit annoying / ugly and now fortunately it also sounds a bit wrong ... ????.....

??? "structure on weight w and object x amounting to cocone from canonical representative diagram for w to x" ???? ??? ???yoneda vs co-yoneda .... ????? .... ??? ..... ??from representable to non-representable functor as easy direction ?? ....

???weight as "virtual object" .... ???? ....

????morphism from presheaf to representable presheaf vs monic such ???? ....

???might it be that .... ????a morphism from a presheaf to a representable presheaf is "colimit-like" precisely in case ..... ??? well, there's two seemingly pretty incompatible things that i was thinking of saying here ... :

1 it's invertible ... ???...

2 its image inclusion is also "colimit-like" .... ????

???... for each object, a full subcategory of it's slice category .... ???? .... ??? ....

???functor taking given formal colimit to given .... ??? .....

???functor promoting given "co-yoneda morphism" to isomorphism ... ????...

???functor promoting given injective co-yoneda morphism to epimorphism .... ??????????

example ... ??? "discrete sum" ... co-yoneda morphism from some given discrete sum of representables to a representable .... ???? as amounting to a cocone for that discrete sum ???? ??moreover sure _looks_ like a cocone ... in category of formal colimits .... but from diagram living in category of actual objects ... ??? ...

???slice topos of presheaf category over representable presheaf ... ???as ... ???... presheaf category over slice category ??? .... ????? .... ???

????? ..... really try to straighten this all out ... "nice, saturated version of colimits sketch described in terms of weights instead of in terms of weighted diagrams or canonically weighted diagrams" ... ??? .... ???? .... ????relationship to ... ????reflective subcategory of presheaf category ??? ..... ????? ..... ????....

????a presheaf "respects" a co-yoneda morphism iff homming into that presheaf takes that co-yoneda morphism to an isomorphism .... ????and this parses ??? ...

??? a presheaf "respects" a pre-sheaf on a slice category iff ... ??? ... ???? ...

???by the way should it be slice or co-slice here ??? .....

??weird how the grothendieck topology analog here seems sort-of clearly more "complicated" than the corresponding (...) sheaf subtopos analog .... ???? ....

???morphism _from_ colimit as nice ... ??but ("yoneda" ... ??as opposed to "co-yoneda", which would be _from_ ...???...) morphism _to_ _formal_ colimit as nice ???? .... ????... ?? relationship to ... "morphism out of colimit is nice, but cocone as made out of morphisms going the other way" ... ???? .....

(???some stuff here vaguely reminding me ... not even that vaguely now ... of ... ??? syntactic vs semantic category of accidental topos ... ??? ... ??? .... manifestation of cone / cocone (!!pun?? ... fan /co-fan ...) as object therein ... ????)

(?? "enriched grothendieck topology" ... ?? over _ab gp_ does this amount to re-invention of concept of "cocomplete abelian category" ??? ... ??? ....... ??? also enriched version of this alleged generalization that we're trying to work out here ...)



??clement berger ... ??? ... ?? maybe sort of mixture of grothendieck topology special case with kleisli category special case ... ???....

???is one or other of these special cases special case of the other?? ...