?? g2 .. idea for getting split quaternion line animation to work ...

?? toric geometry ... ?? way categorified comultiplication (?? ...) issue seems moot in "classical" (=?= unstacky ?? ...) case but possibly annoyingly un-moot in stacky case .... ????? .... ?? "ordinary" vs "toric" approaches to stacky toric case .... ???? ..... ?? but wait, categorified comultiplication here (post-affine toric geometry ... ??? ...), right ??? ... because of inessential diagonal of accidental topos ... ?? so what are we suggesting here ???? ... ??? toric convolution as substitute for categorified comultiplication ??? .... ???? .... ?? or did we have some other idea ??? .... ???? i mean, we did have _some_ ideas, like going to derived level .... ??? .... ?? but was there something more particular here (?? ...) ??? ....

?? are we leaning here towards ... ?? taking plain old smccc doctrine as only good (?? ...) idea for post-affine development ??? ... ?? .... ??? .... ?? property vs structure .... ??? .....

?? project of [trying to understand left universal property of fe projective line in setting uncluding categorified comult] as pretty badly spoiled under this (...) approach ... ??? ...

?? that weird special case where cartesian product of topos coincides with tensor product .... ???? ..... ??? ....

?? grothendieck topology on free finite limit completion (?? .... ?? ....) of small cat .... ??? .... ?? organizing hierarchy of such ... various landmarks and sub-regions .... ???? .... ?? limit vs colimit games on both formula and model side ??? .... ???? ..... ?? "as few/many co-/limit axioms as possible" ... ???? .... ?? "flat" ... "canonical" ... "filteredly cocontinuous" .... ???? .....

?? pair of subgroup inclusions h1 >-> g <-< h2 ... ?? special case where h2 is 1 ... corresponding "restriction" ag morphisms between rep cats .... (??? exactly what sort of reps ??? ..... ?? exactly what sort of group are we / should we be dealing with here ??? .... ???? "algebraic" ??? .... ??? .... ?? ?? with enough reps ... ???? .... ??? ....) .... weak pushout of such .... ?? this as big relationship (discussed with alex to some extent ... ?? also chris rogers ?? ... ??? ... baez ... ??? ...) between consecutive (?? ...) pet obsessions of mine ... : categorified hecke operator (...), and doctrine theory .... ??? .... for example specialize not only h2 = 1 but also g = gl(2) and h1 = borel .... ???? ... projective line as h1,h2 double coset space here .... ???? .... ?? h,1 double cosets as "h-structures" .... ???? "take rep cat of borel and freely adjoin isomorphism from "the big (= 2d ...) object" to actual given concrete 2d vsp ... as acted on trivially by borel .... ???? "universally split the 2d rep" ... ???? ..... ???? effect on "k-theory" ???? .... ???? ....

?? "freely adjoining isomorphism in given hom-space in ag theory" ... ?? "syntactic" approach ??? .... ??? .... ?? ?? analogy to bit about ... ??? adjoining morphism in given hom-space in topos ??? ... ??? .... not sure how accurately i'm remembering that bit ....