Thursday, April 7, 2011

??so suppose given a site ... (??not sure whether i want the grothendieck topology to be trivial?? ...)... and also a sheaf of commutative rings over it ... or possibly of commutative monoids of some other nice kind, or something ...

??then hopefully it makes sense to say something like ... ??"pass from the sheaf of commutative rings to the stack of their module categories" .... ????or something like that ???....

(????something about .... ??category-valued vs groupoid-valued stack here????? and so forth ???? .... ??level slip ??? ....)

(??something about "internal ..." here??? .... confusing ... ???.... ....???also something about ... ???topos vs 2-topos here... and so forth ... ???something about vague memory of something from n-category cafe about something like "concept internal (??or something??) to a topos which is like a stack over the (??...) corresponding site" .... and so forth ... ?? ... ??well, so what about something about simply (??...) using giraud's theorem to "cheat" here??? .... canonical (...??...) site ... ??? .... and so forth ... ???also something about ... object vs morphism ... ??and so forth ??? ... ??? .... ??so what about whether category of module objects over ring object (or something...) qualifies as "locally internal category", or something completely different, or what ... ??... ... and so forth ...)

???and then consider ... ???the groupoid of global sections of this stack ???? ... ???or something ??? .... ??any ambiguity in that??? ...???or something ???...

??anyway, somewhere around this point i seem to get very confused about "quasicoherent vs non-quasicoherent sheaves of modules ..." .... ?????and so forth ?????.....

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