?? for example, _z_ as co-group object in _monoid_ ... ?? ... first, forgetting the co-group structure, just think of it as a monoid .... and consider the functor _monoid_ -> _set_ given by homming from it ... ?? but then notice that this can be lifted to land in _group_ instead of _set_ ... ?? precisely by making use of the co-group structure ?? ... ?? and then observe that this lifted functor is right adjoint to something ... ??? namely to ... ??? inclusion _group_ -> _monoid_ ??? ....

??? also right adjointness of the unlifted functor ???? ..... ?? special case of co-set object ??? .....

?? now try parallel example ... ?? model of accidental topos of p^1 ?? ... ?? (lex) left adjoint from accidental topos to _set_ ... ??? corresponding to (special ?? ...) toric quasiherent (co)sheaf ... ?? "first, forget the co-sheaf structure ..." ... ??? "just think of it as a set" ... ?? hmm, or perhaps a triple of sets ... and consider the functor _set_ -> _set_^3 given by homming from it ... ??? but then notice that this can be lifted to land in _toric quasicoherent sheaf over p^1_ instead of _set_^3 ... ???? ....

?? so for example, consider alleged toric quasiherent cosheaf given by .... ???... ?? "z+z with the positive halves identified" ... ???? .... i mean, that as the first n-set ... ??? and for the other n-set ... ??? well, how about with the negative halves identified ??? .... ?? hmmm, maybe i need to reverse those, according to convention i'm trying to stick to ... ?? ...


?? "pairs of histories" ... ???

?? "past-agreeing pairs of histories" ....

?? "future-agreeing pairs of histories" .... ?????? .....

??? ..... hmmmmm ..... ?????? past-agreeing pairs of histories have predecessors that are also past-agreeing ... and future-agreeing pairs of histories have successors that are also future-agreeing ... ?? right ?? ... ... ??? but ... ??? there are pairs of histories that don't become past-agreeing no matter how far you shift them ... ??? so doesn't that mean that we're not getting a toric quasicoherent sheaf from homming from this alleged toric quasicoherent cosheaf ?? .... ?? so what's wrong ??? ....

?? well, we knew that the reasoning was sloppy ... ??? ....