?? double-speed dynamical system ... ??? ... ?? "n-fold acceleration" ... ??? ....

?? "half-speed" ... ??? left vs right adjoint ... ??? ....

?? "history" ... ???? ....

?? adjoint pair of bimodules ... ?? essential geometric morphism .... ??? .....

?? adjoint bimodule and absolute weight(-ed colimit) ??? ..... ??? ....

??? absolute weight and essential geometric morphism ??? .... ???? "cauchy completion" .... ???? ..... "morita/karoubi saturation" .... ???? .....

?? "essential geometric morphism as means of reconstructing (?? as best possible ?? ...) functors between sites ...." ... ??? then "ghost" / "virtual" functors .... ??? ....

??? but .... ???? all those adjoint bimodules coming from actual functors ..... ???? .... what sort of alleged "absolute weights" do these correspond to ???? .... ???? .... ??? variable absolute weights ??? ..... ?? plain module (with adjoint ... ???) as "constant" case .... ??? ..... so adjoint module as "virtual object" of site ??? ..... ??? ... ??? seems to fit ??? .... ????? ......

?? maybe also fit with speculation we had about "accidental infinity-topos" and geometric morphism involving such ??? .... ????? ..... ???? .... ??? essential ... ??? .... ???? .... ???? not sure .... ??? .... ???? ....

??? "absolute" .... "stable" .... ???? ....

??? stable paradise .... ????? lots of virtual objects ??? .... ???? .... ????? ..... ?? try to straighten this out ... ??? ... ??? "derived-level cauchy completion of single-object enriched category as including "all" of its modules" .... ?????? ...... ?????? ..... ??? "karoubi ..." .... projectives ..... ???? .... ???? .... ???? .... "projectiveness as vacuous at derived level" ...... ???? ..... ???? ..... ?? familiar ??? .... ???? ... ???? .....

?? "morita-equivalence" .... "morita-map/functor" ... ?? adjoint bi-module ... ???? .... essential geometric morphism ..... ???? .......

?? "ambijunction" ... ???? .... ....... ??????? .......

?? "cartesian bicategory" .... ???? ....

on categorified bialgebras (for todd)

first let me try to give a reasonable definition of "bistable bimonoid in a symmetric monoidal 2-category". i'd guess that this is fairly standard (because it's a straightforward categorification of the concept of "bicommutative bimonoid in a symmetric monoidal 1-category") ...

define wbb ("walking bistable bimonoid") as the symmetric monoidal 2-category where a 0-cell is a finite set, a 1-cell is a finite span, a 2-cell is an isomorphism of parallel spans, and the rest of the structure is hopefully straightforward. a bistable bimonoid in a symmetric monoidal 2-category x is defined to be a symmetric monoidal 2-functor wbb -> x.

define a "bistable categorified bialgebra" to be a bistable bimonoid in the symmetric monoidal 2-category of small-cocomplete categories (with small-cocontinuous functors as 1-cells and natural transformations as 2-cells). define such a bistable categorified bialgebra to be "special" just in case its underlying small-cocomplete category is a presheaf category (so that small-cocontinuous functors are given by bi-presheaves, and natural transformations between them by bi-presheaf morphisms).

given a small symmetric monoidal category m (we'll be especially interested in the case where m has only a single object), the category of presheaves on it is a special bistable categorified bialgebra in a straightforward way (actually in two "opposite" straightforward ways). the corresponding symmetric monoidal 2-functor (finite set, finite span, span isomorphism) -> (small cat, bi-presheaf, bi-presheaf morphism) takes finite set s to the small category m^s, and takes a finite span s <- a -> t to the bi-presheaf given as follows ...

[to be continued]

[note: delineation of the categorified bialgebra structure may seem like conceptual overkill in the case of an affine toric variety (that is, the case of presheaves on a single-object symmetric monoidal category), but my suspicion is that it will turn out to be the right concept in the case of non-affine toric varieties.]

?? "allegory" .... ?? with factorization system of relation into co-monic followed by monic .... ??? ... ?? getting allegory (?? ...) from lawvere theory (?? ...) .... ???? .... ?? niceness (?? ...) of such process ?? ...

no wait ... confusion about "joint monic" vs "placewise monic" ... ?? factorization of relation into co-map followed by map .... ???? .... ....

relation as span ... ?? ...

?? matrix treated sesquivariantly : bistable bimonoid : matrix treated bicovariantly : "double stable monoid" .... ??? .... ???? .... just stupid guess ... ?? ...

?? "quantifier sequence" .... ??? arising when both adjoints are available ....
?? when only one is available, then maybe much simpler .... ???? ...

?? "whole as sum of parts" ... "big jump as sum of little jumps" ... ??? "jump" ?? ..."step" ?? .... ??? ..... "journey of 1000 miles begins with a single step" ... ?? ?? big steps vs little steps .... ???? .... ?? ....??? .... "longitudinal vs transverse" ??? .... ??? contrast as disappearing in infinitesimal limit ???? ..... ??? vs "wave equations" where it doesn't ?? ... ???? .... .... telescope ... longitudinal .... line-painting truck ... ???? .....

?? cohomology of toric variety and "toric variety as built out of toruses" .... fan combinatorics ... ??? ....

?? "toric variety as built out of toruses" and toric blow-up .... ??? .....

?? forms over various commutative monoids ..... ??? ..... .... ??? ....

?? so ... comm monoid as one-object sym mon cat ... ?? functorial operation associated to span s under sym mon structure as suspiciously similar to ordinary operation associated to s under comm monoid structure ... ?? ...

?? bi-module manifestation f# of functor f : x -> y ... f#(x1,y1) = [f(x1),y1]_y ... ??? ....

?? s from a to b ... comm monoid m .... f : m^a -> m^b associated to s under comm monoid / sym mon structure of m .... f#(*,*)=[f(*),*]_m^b .... ???? so m^b as m^a,m^b-bimodule .... vanilla (= "cayley" ?? ...) m^b-module self-structure, m^a-module structure given by matrix / span s ... ???

??? so ... ???? "cartesian" structure here ????? ..... ????? ......

?? adjoint bimodules here ???? ...... affine vs non-affine case ..... ???? ....

??? complex / symplectic // commutative / symmetric .... ???? ....

?? at some point(s ??) thought of "toric category" as just sym mon cat with properties ... vs now thinking of it as categorified bi-alg with properties .... ??? ... ?? compatible ??? .... ?? relationship to "automatic comonoid structure in cartesian environment" etc .... ??? .... ???? ....

?? translating hemingway (or hemingway parody) into german ... ??? .... and/or having hemingway parody contest in german ... ?? ...

?? wiles's description of doing mathematics ... ?? previous room as speck in next room ... ?? ...

?? writing with blue ink and yellow anti-ink on green paper ... ?? or maybe green-tinted glass ... ?? sunglasses ... emerald city ... ??? ... precision-erasing ... ?? ... deconvolution .... ????.....

?? or photographic negative ... superimposed with positive .... ??? .... issue of "neutral" here ??? ... ??? ....

?? positive and negative superimposed with slight shift ... ??? bishop berkeley ... ghost of departed quantity ... ?? cheshire cat ??? ... ??? .....

?? hmmm, gradual / continuous fadeout if you do this with spatially continuous light intensity ... ???? .... "renormalization" .... ??? ..... ?? (somewhat ??) avoiding this renormalization issue with discrete shifts instead of continuous ... ???? ...

?? superimposing sequentially shifted positive copies of superimposition of shifted positive and negative copies ... ?? in effect magnifying the shift ... fundamental theorem of calculus / summing geometric series ... bank account ... sine, cosine ... ??? excessive deposits and withdrawals ... hedge fund .... ??? ....

"big shift as sum of little shifts" ....

?? cross-country skiing on high-relief nazca-like writing ??? .... (?? actual existing examples ?? ...) ...?? "slope" ... positive vs negative ... ??? ... blue and yellow pens yoked together horizontally ... horizontal partial derivative ... or vertically, etc ... horizontal vs vertical bars of various parts of various letters ... in effect more or less disappearing or not disappearing .... ??? .....

?? universal property of "toric topos" ... ?? how much embodied (?? ...) by "category of classical models" ?? ... ?? or by models in other toric toposes ... which we think we know something about .... ??? .... ???? ...

?? "concept of cartesian abstract category as making sense in cartesian 2-category because [abstract category in cartesian 2-category is automatically stably comonoidal, so if also stably monoidal then bistably bimonoidal, and then ....] ..." .... ???? ....

??? level slip ??? .... ????? ..... ???? ......

?? microcosm ... ??? ....

?? ask todd about etymology of "accessible" ?? ....

?? pixellated vs pixillated ... ??

?? golem vs gollum ... ??

?? idempotent monad and "localization" ... ??? ..... ?? right-exact idempotent monad ... right-exactness vs idmepotence here .... ????? .....

?? frame rate .... james cameron ....

Peter Jackson is shooting “The Hobbit” at 48 ...

douglas trumbull ...

article about trumbull here ...

?? "electron gun" ... ?? ....

?? record maximum frame rate ?? read about scientific applications somewhere recently ??? ....

?? collapsible cup ... telescope ... ??? ....

?? stock market ...

?? grand unified theory of mathematical politics ...

3 kinds of mathematicians :

1 lack social intelligence

2 normal social intelligence but separate from mathematical intelligence

3 incorporate normal social intelligence into mathematical intelligence

kind 2 use social intelligence for career politics and dominate the field career-wise. kind 2 doesn't know about kind 3, often mistaking it for kind 1, which is the worst possible outcome.

?? bracket of observable x with hamiltonian ..... ???? ....

?? heisenberg vs schroedinger picture here .... ???? ....

???? .....

??? vanilla base hamiltonian vs "actual" perturbation .... ??? ....

???? .....

??? ....

??? also "interaction picture" ..... ???? ..... ????? .......

???? .....

?? "momentum" .... ??? symplectic .... ??? ....

?? commutative monoids in a symmetric monoidal poset ... as forming ... ???? ... ??? ... ?? "bi-idempotence" ??? ... ???? ....

?? "walking idempotent monoid" monoidal cat =?= "walking monoid" monoidal poset ??? .... ??? ....

?? right adjoint to forgetful 2-fr from _cartesian sym mon poset_ to _sym mon poset_ .... ??? ....

?? ... ??? ....

?? growth, change, _movement_, variation ... ??? ....

?? important to remember here (...) .... be very careful around farm machinery .... ???

?? how many of you have ever found yourself doing something like this ??? .... (row of differences ...) ....

?? "constant of integration" as "differential galois ambiguity" ?? .... ??? ...

?? "solveableness" ... ??? ....

?? "product integral" ... ????? ......

?? analog for [equation with multiple solutions] of idea that possibility of having to solve [equation with no solutions] is in some sense (and to some extent ...) "protected against" by ... ?? fact that "if it never happens then no one will ever try to reverse it" .... ???? ..... ??? analog as maybe that it's _anti_-protected against ?? ... ???? ....

?? was going to mention "maurer-cartan" here but maybe what i was really trying to get at is simply diff eq for translations on given lie group ... ?? ....

?? analog (wrt discrete : continuous analogy ...) of "lagrange extrapolation" here ?? ....

?? naive idea of "reducing" integration to taking area under curve .... ??? explaining what's wrong with it ... ?? relatively bizzaro formal properties of taking area vs very simple formal properties of integration ... ?? .....

?? relaitonship to "projection-valued measure" idea ?? ..... ????? .... ??? ...

?? microcosm principle as tirivial for multicategories, less so for .... ??? ....

"1-sided vs 2-sided" .... ??? ....

?? given a "cocontinuous monad" (?? ...), and a simplicial set, get .... ???? ....

?? kock-zoeberlein analog ??? .... ??? ....

?? "derived" analog ??? ... ???? .... ???? ....