?? essentialness of diagonal ... ?? for sober space ....
?? essential inclusion of sober spaces .... ??? essentialness (= extra left adjoint ...) evaluated at total open set x of essential subspace x .... l(x) <= u iff x <= [u intersect x] ..... ???? .... ?? l(t) as smallest open set containing x ??? ....
??? existence of such smallest open set as maybe implying x is itself that open set, given some reasonable "separation property" ???? .... ????? ......
?? maybe converse too ... ???? ...... ???? .... ??? essentialness for inclusion of open subspace as given by .... ??? simply "inclusion", noting that this is totally-defined because relatively open subspace of open suspace is open .... ???? ..... ???? ....
?? maybe explaining to some extent how i've gotten confused about interrelationships among "essentialness" and "openness" and "logicalness" .... ???? .... ??? ... ???? ....
?? open inclusion vs open map .... ???? ....
??? mystical stuff about "open" and "cover" and "grothendieck topology" ...... ????? .....
?? "separation conditions" and compactness .... in topos context ... ?? ....
?? so if (...) essentialness of diagonal is some substantial part of the way towards total distributivity, then what supplementary condition are accidental toposes of non-affine toric varieties missing ?? ... ?? ...
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