generalized / "based" toric geometry ... some kind of 3-place chain complex ... ??confusion about ptd set vs action of walking idempotent ... ?? "zero" as somewhat ambiguously point vs constant map .... ??? .... ?????..... (??ptd set vs family of ptd sets ?? ...)

?? 2-place complex : 3-place complex :: topos : ?? .... ..... ???? ....

??hybrid (??? ...) of topos and abelian category ??? .... ???ptd objects in topos ... ?? .... ???recovering original topos from it ???? ...... ?????? ......

??? topos over (??? ...) topos of m-sets ... m commutative monoid .... ???? ..... ?? .... ???? "topos equipped with m-torsor ..." ... ????? ......