?? in toric small zariski topos , for example … ?? applying recent ideas about how the left adjoint parts of lex theory morphisms give essential geometric morphisms, and the right adjoint parts at least ordinary geometric morphisms ….. ???? ….. ?? take free comm ring on toric structure sheaf ….. ???? then take “small zariski spectrum” of that ringed topos …. ???? ….. ???? ….
?? “good virtual embedding between sums of dualizable objects” …. ???? ….
(a,b) : x+y -> z
?? x, y, z invertible …. ??? …
(a,b) epi, and … ???
a* : z* -> x*
b* : z* -> y*
x* + y* -> x* +_z* y* as “virtual cokernel of virtual embedding” …
?? “matrix of morphisms from a1,…,aj to b1,…,bk” … ??? another from b1,…,bk to c1,…,cl … ???? ….
?? pushmi-pushyu diagram ???? …..
?? good virtual embedding of dualizable object into sum of dualizable objects …. ?? and also concept of two such good virtual embeddings with common target “having well-non-intersecting image” …. ???? …..
?? problem with “intersecting” suggesting limits here ???? ….
?? hmmm, with projective line, reasonable (?? …) to expect virtual cokernel of virtual embedding l -> 1+1 to be invertible … ??? but with l -> 1+1+1, for example, we don’t have invertibility to fall back on, it seems …. ???? …..
?? gaussian elimination … a_n building …. flag …. ???? …. ??? …. ?? idempotent object …. ??? …. ?? “vairtual duality matrix” ??? ….
?? the tag theory _exists_, right ?? … so we should be able to describe it ….. ???? ….
?? quasitopos vs topos …… ????? …..
??accidental topos as classifying some sort of “localizations”, but of actions rather than of the monoid … ??? ….
?? stack of (???? “local” … ?? …) ag theories of quasicoherent sheaves over (for example ??? ….) zariski-like site ….. ????
?? “local over local …” … ??? …. ?? … “stack over stack …” …. ??? ….
?? relationship to vacillating over “categorified scheme” concept of “algberaic stack” and “categorified sheaf” concept of “stack” as violently different vs as not so violently different … ?? …
?? relationship of that to …. ??? lawvere’s stuff about “small” (?? corresponding to space … ?? …) vs “big” (?? corresponding to category of spaces … ??? …) toposes ??? ….. ???? …..
?? not sure any parallel whatsoever there …. ????? …..
?? main weirdness in trying to take seriously accidental topos of projective line (for example …) as tag theory as … ?? objects not being closed under sum … ?? especially multiples of 1 … ??? any substitute for dualizableness here ??? …. ??? …
?? “good embedding” … ?? any reasonable sense in which embedding between sums of dualizable objects might be said to be good ?? …. ?? complementation ??? … ??? hmmmm …..
?? so do the n points of n correspond to some interesting n combinatorial features of the projective space of k*n as a toric variety ??? …. ?? anything building-ish here ??? …. ??? highest-dim cones …. ??? …..
?? stalks of structure sheaf over toric small zariski topos … ?? “localness” property ?? … ??? of morphisms too …. ??? ….
?? stalk at highest-dim cone ??? … ??? …..
?? in affine case only one highest-dim cone ….. ???? …..
?? projective line squared …. ???? …… ?? or product in general …. ???? highest-dim cone of product as choice of highest-dim cone from each factor …. ???? ……
?? maybe hints here about … ??? how to generalize fan concept to stacky case ???? ….. ????? ……
?? highest-dim cone in fan as “sector” … ??? ….
?? so … ?? sector of fan corresponds to “classical model” of corresponding tag theory ??? ….. ?? is this really true ??? ….. ?? “trivially” thought perhaps nevertheless very “interestingly” true in affine case ??? …. ???? …
?? “classical model” = “(tag, here …) theory morphism to theory of nothing” ….
?? philosophy (mentioned in discussion with roberts …) that classical model = theory morphism to theory of nothing … ?? pretty simple-minded philosophy … ??? categorification of “terminal object as walking point …” … ??? “un-parameterized …” …. ???? ….
?? kaleidoscope as fan case here ???? …..
?? “classical model of toric dimensional theory” ……. ????? ….. ?? paradox ????? ….. ?? non- / absoluteness of sums and / or scalar multiples here ?? ….
?? so are the two classical models that we’re imagining for the tag theory corresponding to the projective line really there ??? …..
?? preservation of projectiveness by blow-up ??? …. toric and non-toric cases … ??? …. ?? segre embedding ??? …. survives to toric case ??? ….. ???? …. ??? ….
????? cartesian of classical toric universe ??????? ….. ?????? …. hmmmmm …….. ???? …..
?? “z-graded set with pair of degree 1 operators” …. ???? ….
?? could it be that with the toric analog of inclusion of projective space of v into full orbit stack of action of (??? toric ????? …..) gl(1) on v, instead of getting a full and faithful inclusion on classical models we’re getting a “surjection” ??? …. ???? ….. ???? …. ???? ……
?? “cohomology” …. ????? ….. ???? …. as boring on “affine” stuff …. ???
?? toric classical models of tag theory corresponding to punctured plane …. ???? ….. ?? “sector”s here ?? …. ??? case where sectors are different dimensions ?? … “closed point” of toric small zariski locale …. ??? …
?? “toric big / small zariski …” … ??? terry bisson …. ??? …… toric classical model ….. ??? ….
?? hmmm, confusion between embedding l -> 1*n, and co-embedding 1*n -> l …. ??? so was this all just a complete phantom ??? …. ?? or was there _some_thing there somewhere ?? …. ??? … … ?? closed point vs open point … ??? …. 1*n -> 1 as giving “unit point” of toric variety …. ???? …..
??? models over (1,0) …. ?????…..
?? co-walking ideal ??? ….. ?? …..
?? “spectrum” as maybe some sort of pullback-ish thing ?? … ?? to what extent did we already think about that ??? …. ???? ….
?? spectrum of (possibly variable …) pair (a,b) of truth values wrt requirement “a or b” … ??? ….
??? automatic quasicoherence ….. ????? …. ?? ag morphism between quasicoherent and non-quasicoherent stuff …. ??? …
?? math as like some kind of climbing … finding the easy handholds … ??? …..
?? seems like (?? using some hopefully not too non-obvious convention …) left adjoint geometric morphism is automatically essential ?? …. ???? …
?? geometric morphism from “ab gp pair” to ” biadditive map” … ??? whether it’s clear that this is an injection … ??? …. ?? seems yes ?? …
?? …. automatic quasicoherence … ??? ….
?? decategorification of idea of “doctrine of monads on k” … ?? … ?? roberts ?? …
?? “2-valued …” and … ?? “property holding locally but not globally” …. ??? …. ?? “propostional” (?? …) case of “disjunction, holding in given form only locally” … ??? …
? “unease” bit … ?? idea that with geometric doctrine, formula category seems “foundationally” tamer than model category (in general …), but “conceptually” bigger … over the self-duality threshold … … ?? … ?? one side getitng the directed stuff and the other side the finitary stuff …. ???? …. ??? …
?? seems like our “doctrines” often have operations of too high arity (?? for example doctrine of small-cocomplete categories … ??? …) for “doctrine” to be special case of “2-topos” … if 2-topos generally only has finite (?? …) 2-limits …. ??? …. ?? anyway, might affect discussion for lurie … ??? …
?? topos arising from pure colimits theory … ??? …. ??? vague feeling of relationship to “topos” from “formula as model” viewpoint ??? …. ??? …. ?? “tensoring with theory of totally distributive boolean algebra” ???? … ???? … ??? ….
?? flat presheaf on poset as something like truth-value-valued …. ??? … subobject of 1 …. ??? …. ??? due to extra magical duality (?? or unality ??? … ??? …) for truth-values ??? …. ??? … nevertheless, any interesting generalization of this beyond poet case ??? …. ???? …..
?? arbitrary commutative square in topos (for example accidental topos of affine plane …) … ??? as geometric morphism to theory of “family of spans” … ??? …. another geometric morphism to same theory, from theory of pair of truth-values … ?? pushing subtopos forward along latter and then pulling back along former …. ???? trying to keep track of essentialness and so forth here …. ???? ….
?? adjoint pair of geometric morphisms between theory of set and theory of ab gp, for exaple … “free” and “underlying” … ??? the left adjoint here as essential, right as non-essential ??? …. ??? is the essentialness of the left adjoint automatic or something ??? …..
?? various approaches to understanding tensor product of ab gp objects in topos …. ??? ….
?? forgetful geometric morphism from “biadditive map” to “pair of abelian groups” … ???? …. ?? as retraction …. ??? ….
notebook361 