colimit as representing cocone = diagram in slice category .... ????..... ?? lax case .... ??? showing up .... "quasicoherent = filteredly cocontinuous" .... ????

?? neutral (between abelian and cartesian ...) combined doctrine .... products, coproducts, and tensor products ... ???? ..... ?? morphisms between abelian and cartesian examples here ... ????....

?? proceeding with mirror symmetry program .... ?? drawing graded commutative monoids coming from line bundles over hartog toric varieties .... fano .... anticanonical ...

?? torus action stack of toric variety ... ??? compare to "schubert stack" ... "bi-flagged building" .... ??? ...... ?? "artin-wraith ..." .... ?????....

??? module over local ring of prime ideal, tw module over quotient ring ... ????.... ?? maybe didn't say this (??? ...) quite right yet .... ???? both localizations ????? ...??????

??? vector space equipped with operator x with x-k invertible for all nonzero k ... ????.... ??? algebraic completeness here ??? ..... ???? x^2+1 being invertible ??? .... ......... ??? extensions here ????? ...... ?? ??? "functorial projective cover" (??? ...) and " ... glueing ..." ???? .... ???? .....

?? laurent polynomials vs rational functions with no negative factors of x .... ????? .......

??? explicit linear basis for field of rational functions ??? .....

??? explicit linear basis for k[x,y,z]/xy-1,xz-z-1 .... ?????

1/(x-k) .... ?? case k = 0 ??? .... ????? case k = infinity ???? ..... ??? as morally x ?????? .....

?? multi-variable case .... ????.....

?? vector space over the rational functions as much more concretely understandable than i'd realized ??? ..... invertibility as mere property .... ??? ..... ?? pressley and segal ... ??? "meromorphic vector bundle" .... ???? .....