orbit … fan ….

?? confusion about … ?? idea of toric variety, for example … “closure of single orbit” viewpoint … ?? the single orbit as “generic” (?? …) in some sense … ?? but then how could it have degenerate orbits in its closure …. ??? …. ?? or wait, _is_ that typical ?? …. ??? ….. flag geometry ideas lurking in background here … ??? ….

?? apply “fan as tangent space of 1” idea to kaleidoscope as fan ??? …. ????? …..

?? (simple-ish …) algebraic group itself as … having its coordinate algebra become multihomogeneous wrt weight (??? ….) lattice, yielding multihomogeneous coordinate alg of generic flag pair space ….. ???? …. ?? lots of games to play here … ?? ….

?? idea of clash between “alg geom as kind of categorical logic” and something like “derived / htpy-theoretic alg geom as fancier kind of logic …” as … ??? analog of prior clash in “geometric” case …. “locale / topos as kind of logical theory,  vs htpy theory of such (??? …) as fancier kind of logic … confusingly related …” … ??? …. ???? ….

small sub, big quotient, big sub, small quotient … ??? to try to help organize “projective space of v” theory …. ??? …. kernel …. ???  …

 

?? taking serio…

?? taking seriously possibility of compatibility between ab cat structure and tensor cat structure as being essentially doctrinal ??? ….. ??? … hmmmm …. dualizable objects …. ??? we did think about this fairly long ago, but … ?? maybe not so much in connection with idea of “taking full compatibility relationship existing  “in nature”” …. ???? ….

 

?? “if you hav…

?? “if you have theories then you don’t need stacks …” … ??

?? but if decategorify then “if you have algebras then you don’t need sheaves”, whereas sheaves actually are good for some stuff … ?? but generally (??? ….) just in how they form a topos … ???? …. ?? so recategorifying, stacks as interesting mainly in how they form a 2-topos ??? …. ???? ….

 

 

?? so _does_ cu…

?? so _does_ curing no-headed monster somehow automatically cure two-headed monster too ??? …. ??? ….

?? no-headed monster as cokernel (or coequalizer, if you prefer …. ???? ….) of _something_ … ?? namely of … ?? the actual virtual embedding ??? ….. ?? maybe some confusion here between subspace and quotient space ??? … ??? and / or their duals …. ???? ….

?? formal limit in formal colimit category …. ???? ….

?? regular epi as cokernel of its kernel (pair … ??? …) …. ???? …… ????? …..

?? “generic fl…

?? “generic flag pair in building b” as pure dimensional theory ?? … ?? for b any standard building ??? …. ?? simply mod out by cartan ??? … ??? … ???… ?? mention in letter ?? …

?? special case of pairing generic (?? a-series ??? …) flag pair with other (?? prop-like ??? …) structure ?? … ?? prop for iso to constant v …. ??? …. ?? “coordinate system” ??? ….. ???? …..

?? was originally trying to get this to be a toric dimensional theory, but ran into problem with path sum … ?? though in case of 2-by-2 matrix multiplication the path sums are binary and setting them to 0 can be simulated without addition perhaps by somehow conceptually negativizing one of the summands and putting it on the other side of the equation; might this somewhat explain (?? …) why case of projective line seems anomalously simple in some ways ???? ….. ??? … ?? hmm, but what about setting path sum to 1 ??? ….. ???? ….

?? (?? possibly bad, or at least annoying … ?? …) idea of trying to use this as step on way to developing theory of single flag in b …. ??? …. ?? hmm, to what extent is this equivalent to other approaches that we thought about ??? …. ?? pairing coordinate system with feature to get space of features …. ??? …. ?? maybe somewhat different … ??? … ??? different features at least ??? …. ???? …. ??? ….

?? any relation…

?? any relationship of “book-balancing” interpretation of ftc to archimedes’s “method” ??? ….

?? “in the limit the telescope is packed up so tight that it disappears (?? shrinks to a single point ?? …) … making it difficult to see …” … ??? …

?? with mixture…

?? with mixture of “small” and “large” topos who gets custody ?? … ??? either or both as “positive” vs “negative” … ??? …. ?? “one drop …” ?? ….

 

?? “monsters” … ?? quasi-sheafification …. ????

?? two-headed monster … ?? self-pushout of … ??? “cure” map to one-headed monster as epi …. ???? …

?? no-headed monster … ?? cokernel of … ??? …. ??? “cure” map to one-headed monster as mono …… ???? …..

?? two-headed monster as self-pushout of cure map from no-headed monster ??? ….

??bit about “[n+1]d hole as redundant filling of nd hole” ??? …..

?? switch from lizard’s tail to monster’s head …. ????? …..

?? given lawvere-tierney topology, consider collection of morphisms consisting of monos “in” the topology, and their self-pushouts …. ???? … ?? saturate wrt … ???? …. ??? invertibilization under cocontinuous functor …. ???? …..

?? boundary of disk …. double of disk …. ??? …..

?? trying to understand example of toric projective plane … ??image factorization of 1+1 -> l …. ??? confusion about which half of facotrization is causing the problem …. ???? ….. trying to identify extra axiom needed …. ??? …..

?? roberts … 2-group … generalize …. other stuff …. main theme … ??? ….response …

?? co/homology …. ??? derived functors … of …. ??? ?? operation vs homomorphism … ???   … leray-serre …. ??? … product along for the ride …. ??? ….

?? co-yoneda morphism …. slice yoga …. ?? of presheaf topos over representable … ??? … ?? level slip ??? … “potential coverings” (?? …) …. ?? then grothendieck topology on site category of such …. ??? …. ?? …. formula / model confusion ….. ????? …..non-/distributivity … and / or no limits in doctrine at all …. ???? …..

?? but …  when colimits already exist then co-yoneda morphism reduces to just plain morphism … invertiblization of morphisms, in cocomplete world …. ??? …..

?? morphisms vs monomorphisms …. vs cocomplete cat vs topos …. ??? …. ?? is it “x gets inverted if its monic part does” or is it “getting inverted implies monicness”, or …. ???? ….

?? sums, and pullbacks along summands … ??? weird to have colimits before limits ??? ….

?? being 2-group as mere property of toric dimensional theory ?? … ??? invertibleness of morphisms giving covariance of inverse …. ???? ….

?? “formula ob…

?? “formula object which believes given cocone is universal” …. ??? vs model which “believes” it …. ??? …. ???? …..  ??? “morphism st its target believes it to be an isomorphism” …. ???? …..

??? other (??? ……) formula / model confusion near here …. ???? …..

?? can we just “take some graded module that we know is supposed to be corrected to something else but st that correction isn’t forced by the axioms that we have so far, and make a new axiom out of it …” … ?? simply by saying that that morphism should be invertibilized ?? …. ???? …

“conservative” as complementary to “does nothing but invert morphisms …” … ??? analogs of this … ?? “does nothing but universally promote certain cocones to universal” … ???? …. ??? ….

??? “reflect …” … ??? pun ?? … ??? ….

?? notes from discussions with brandenburg …. ???? ….

??  toric universe of semilattices ??? …. ….