Monday, May 16, 2011

??? me : them (= those who emphasize "stacks over (2,1)-sites" ... ??? ....) :: doctrine meta-doctrine : (2,1)-topos meta-doctrine ??? .... ????.....

(some other meta-doctrines in the fray as well ... some just minor (?) variants/variations, perhaps...

?? "environment" as twice-categorified "point" ??? environment of 2-topos ... model of 1-topos ... point of locale ... ??? does "locale" here deserve to count precisely as "0-topos" ??? ..... ??"theory" as _not_ twice-categorified point ... ??rather twice-categorified "propositional formula" ? ... ??? .... "theory" here as ambiguous between "object in syntactic (2,1)-category of doctrine d" and "object in syntactic (2,1)-category of (2,1)-topos coming from d" ... this latter syntactic (2,1)-category being the pre-stacks on the former ...

?? "doctrine of toposes" .... (???clarify theories vs environments here... (???also for all our other doctrines???? ....)) also of t-modeled toposes ... ???but then glorify these doctrines into 2-toposes ... ??try to get explicit about what's going on here ... ???


??? size issues ... ??? ......

?? lex theory has "big" moduli stack ... ??? "unbounded" ?? ...

?? whereas topos or AG theory (??for example??) may have more "manageable" moduli stack ... ???

?? doctrine has big moduli 2-stack ...

??whereas 2-topos (or ... ??? ....) may have more "manageable" moduli 2-stack ...

??what about "backing up to get a running start" here?? ... ??? .....

?? doctrine as determined by it's (2,1)-category of environments, but not so for (2,1)-topos ??? ... ??? .... ??? ... ??need whole 2-stack of environments ??? ... ??? .... ??? ... ???...

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