notes for next discussion with todd

??try to concentrate on "doctrines of toric algebraic geometry" ...

??what about something about?? ... idea that "canonical grothendieck topology" idea here... something about "best topos approximation" or something... ??might be peculiar here because of nature of doctrine ... ??or something ???....

??something about "boilerplate" idea ... ??...

??something about whether "k-linearization" process connects the two doctrine ladders in the intended (or something...) way?? ... ???....

??something about "viable sub-lizard" approach to grothendieck (and/or lawvere-tierney...) topology ... ???....






??puzzle about "minimalistic syntax" ??? ??save for end?? ... ??or something ???....

??so... "toric quasicoherent sheaves over P^n" allegedly form certain topos ... ??and we think that we sort of know what it's the classifying topos for ... but we're mainly interested in it as theory of poorer doctrine, of course... ??so... model of it wrt poorer doctrine in topos t as .... ???or something???

??or something about ... ???model of it wrt poorer doctrine in actions of commutative monoid m in topos t ... ??or something ... ??? ??something about special case m = 1 ... ??... ???maybe very degenerate??? ....or something ... ???...

???given model of it wrt geometric doctrine in topos t, forgetfully get model of it wrt poorer doctrine in t ... ???...

??so consider a slice topos of the object classifier... say over object x ... ??then... ???this is the classifying topos for ... ??algebras of the free "substitution" monoid on x ... ??or something?? ???what about generalizing this somehow to non-free such monoids ?? ... ??or something?? ....

no wait, that's not correct... try some examples ...

x = "the object square", for example ... ???so a model should be... ???an object equipped with ... ???a pair of points??

x = "the object to its own power" ...??"object equipped with endomorphism" ???/ ???or something????

??compare this to some other hopefully straightforward interpretation of "theory of object equipped with endomorphism" ... ??....

??actually, now this (...) whole idea is seeming wrong ... ??simply because of exponentiation not really being part of geometric doctrine ?? ... ??and i sort of almost knew that already or should have... ??but i think that i was influenced by this alleged "minimalistic syntax" idea... about which i'm now a bit puzzled... how can you get "interesting structure" using just (??...) "adding a generic point of a given object" (or something...) ???.....

???for example, how to get classifiyng topos for "dynamical system" using the minimalistic syntax?? ...??maybe ask todd?? ...


??something about ... ??burroni monad ... as _not_ an example of a substitution monoid in object classifier over topos of directed graphs ... ??... ??or something ???...

??so what about "karoubi envelope of an operad" ??? ... or something ...

??suppose we have a morita equivalence (??of operads?? ... or something ??...) e : x -> y ... ??and also a morphism m : y -> z .... ???then ... ??is there a nice concept here of... ??"the x-analog of z" ??? or something?? ....

hmmm ... ???some sort of "distributivity between morita equivalences and morphisms" ?????? or something ???? .....

???what about something about ... ???expressing a morita equivalence as a span of morphisms .... ????or something??? .... ???span of "morita morphisms" ??? ...

a,b : x <- s -> y span of morita morphisms ... ????....

??what about "weak pushout of bm along a" ... ???or something ...

??maybe for operads the morita span apex should be allowed to be a prop ??? ... ??or something ???.... ??or maybe not necessary??? ??? or something ???...

??so... ???karoubi envelope of typed k-linear operad .... ???...

"boilerplate" ...

??"legalese" ... ???

law ... computer programming...

??something about... being told that "the zariski topos (or whatever...) is the classifying topos for local rings" as like having a lawyer tell you the boilerplate without telling you the actual relevant details ... ??or something ...

??so consider forgetful (2,1)-functor from geometric theories to "tag" theories... ???and left adjoint to this??

??vs ... ??classifying topos for models of tag theory over comm monoid m ... ??and so forth ... ??...

so what _about_ "free boolean frame on a frame" and / or "free boolean grothendieck topos on a grothendieck topos", and so forth ?? ... ??_are_ there "divergence" problems here?? ... ??or something?? ...

??so _is_ it true that ... ???the left adjoint to the forgetful (2,1)-functor from symmetric monoidal cocomplete k-linear categories to symmetric monoidal cocomplete categories takes the "toric quasicoherent sheaves" over P^n to the quasicoherent sheaves over P^n ?? ... ??or something ... ??...

so consider "the free symmetric monoidal category on one invertible object with trivial self-braiding" ... martin asks about whether the inverse object here has the same property ...???

??something about... ???taking inverse as contravariant symmetric monoidal equivalence on the invertible objects?? ...??or something?? ... ??something about mates??

hmmm... ???mate of identity morphism as identity morphism ... ??only if... you're careful to "use the same inverse" on both domain and co-domain ?? ... ??or something ???...

??what about something about ... "adjoint equivalence" here, or something... ??was that supposed to be different somehow from an ordinary equivalence??? ... sounds weird... ??maybe that issue is a level-slip away???...

??or maybe it's _not_ a level-slip away???

??inverse objects vs adjoint-inverse objects??

??so why poset of forcing _"conditions"_ ??? .... ... and so forth ... ???

hmm...

??well, there's the vague idea about how the double-negation topology "causes everything that can happen to happen", or something like that... ??"as much as can happen to happen" ... or something...

??so consider the "toric ag theory" of... ???

??well, consider the graded actions of the free Z-graded commutative monoid on n+1 generators in grade 1 ...

??which we can think of as forming a pre-sheaf topos?? ... objects of site category = integers .... morphisms = ... ???..

??but then consider the sheaves for a certain topology here ??...


??so what about something about... ??the per-sheaf topos here as a slice topos, and some conceptual interpretation of that ... ???and so forth ... ???

???something about ... ??torsor of group completion of commutative monoid ... ???something about with frame for certain associated torsor ... ???or something ???.... and so forth ... ????

??maybe reminding me of something about toric varieties here, in fact ??? ....

my experiences with john baez

i met john baez via the medium of "usenet newsgroups", particularly the newgroups "sci.math" and "sci.physics"...

he was clearly very articulate and knowledgeable... he also struck me as apt to take the "safe", "establishment" side in any dispute, or at least in any scientific or mathematical dispute... even in cases where i had good reason to disagree with the establishment side...

i remember that at some point he posted a message saying that he wouldn't mind hearing from people who had what they thought was some brilliant new theory of physics ...

(i wonder if i can find this message somewhere...)

i had ideas that i wanted to tell someone about, but i didn't think of them as constituting a "brilliant new theory of physics", exactly... it was more that i had found an amazingly simple way to understand some of the brilliant old theories of physics... a way that i thought was probably already more or less understood by everyone who really knew what they were talking about, but which for some reason seemed to be kept secret from beginning students... this is pretty much always the way it is with me; it's what i do... try to find the amazingly simple ways of understanding things that are usually kept secret from the beginning students, so that i can try to teach them... to beginning students...

so the kind of ideas that i wanted to tell someone about didn't exactly match the kind of ideas that he seemed to be looking for, but they seemed close enough... as an unemployed drop-out from a mathematics doctoral program i found it difficult to get anyone in the academic world interested in my ideas, so my standards as to what constitutes a sufficiently receptive audience were set very low...



??something about ... "peculiar early work" / "i want to be famous" ... ??or something... ???....


??something about ... ??being pretty honest about not wanting to (intensively...) work with me... at some points ... ???

[?? a "go-between", apparently, is someone who lies to you about what the other fellow said and then goes back and lies to him about what you said ...

?? butch cassidy ... ?? "the fall will probably kill you" ... ??? ....]

??so let x be a dimensional category, and then let x' be the dimensional category of x-objects equipped with actions by abelian group g ... ???or something?? ... ???what's this like??....

??is an x'-model maybe esentially an x-model together with a g-torsor?? ???or something?? ...??? ...

??maybe in some cases, but ... ???shouldn't it be a bit different from that, conceptually???....??? or aomething ?? ....

??well, so what about sort of same idea with ag theory or g theory, for example??

hmm... geometric theory case seems like it should be straightforward, no ?? ...???...

???something about ... ????a g-rep equipped with an action of h as a gXh-rep ... ???? ... ??or somethihng??? ...

??so consider ... ???the dimensional theory of "a t-model together with an a-torsor" where a is, for example, some finite abelian group ?? .... or something .... ???.....

???somethihg about ??"using gabriel-ulmer duality" (or something....) to simulate tneosr product via homming" ???? .... ???or something??? ... ???....

??so what _about_ pro-simplicial set allegedly coming from topos?? ... ?and what about "galois shapeshifter" here, or something ??...

hmm... johnstone p 205 ...

"t-modeled topos" ...

???but also something about "spectrum" .... ???...

johnstone p 204 :

"in the suggestive terminology of tierney, we say that the topology j _forces_ f to be an interpretation of l"

hmm, l here is just a "language", but they go on to consider the case of a theory t as well ...

f here is... ???something like... ??the generic model of a "stuff-level" theory, to which forcing conditions / axioms / coverings are being added ... ??or something??

??something about ... maybe the "l" bit involves shoe-horning "structure-level" in with "property-level" to some extent ... ??or something ...

(for martin)

hi...

i'd like to try to state some questions here that i'm interested in ...

we've already talked about many questions of the general form:

for some specific algebraic-geometric theory t, can we give a nice description of its universal property; that is, of what it's the "classifying space" for, or of what it's the "moduli space" of?

(where "algebraic-geometric theory" is my terminology for "symmetric monoidal cocomplete k-linear category"; sometimes i use "finitely cocomplete" instead of "cocomplete" but for now i'll stick with "cocomplete".)

thus for example we've talked a lot about the case of t = quasicoherent sheaves over P^n, and we've explored possible answers in that case such as "t is the theory of a line object L equipped with a good embedding into the direct sum of n+1 copies of the unit object".

but the new questions that i'm interested in (actually i've been thinking about them for a while, but i don't think that i've gotten a chance to explain them to you very well yet) are the same kind of questions, except dealing with so-called "geometric theories" instead of "algebraic-geometric theories". and just as "algebraic-geometric theory" is a synonym for "symmetric monoidal cocomplete k-linear category", "geometric theory" is a synonym for "grothendieck topos".

(roughly speaking, a grothendieck topos is a category which has all finite limits and all small colimits, and where the finite limits "distribute over" the colimits in the same way that they do in the category of sets.)

thus for example, at there's a brief discussion of many different ways of associating a topos to a scheme:

"More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos..."

my main idea here is that when we create a topos from a scheme (or stack) in this way, the universal property of the resulting topos (or "geometric theory") should be very directly related to the universal property of the algebraic-geometric theory of quasicoherent sheaves over x.

thus for example consider ...

??something about "strictly local ring" and/or "henselian ..." or something ?? ...

hmm, from wpa:

The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.

...

??so consider artin-wraith glueing approach to sheaves over 1-point compactification of N ...

??something about quantifiers here?? .. and so forth ....

??so what _does_ it mean to have a model of a classical first-order theory over the stone space given by 1-point compactification of N? ... ???.... ... somewhat concretely and explicitly ... ??.....

??hmm, so what about something about ... ???stone space (or something...) of subquotients (or something) of a set x ???? .... ???... ???hmm, what _about_ something about "orthogonality of partitions" (or something...) here??? .... ???what about something about "connection information" ?????? .... ????......

??maybe just something about "in-/efficiency" here??

?? ... more generally (...), what about something about ... ??stone space of t-structures on some stuff ... ???or something ... ??...

???hmm, so maybe the subquotients of a set really do form a stone space??? ??or something???... ....solutions of a system of boolean equations ... ??...

???something about ... ??"continuous map from stone space x to stone space of subquotients of s, with union of all the subs equal to s and coarsest mutual refinement of all the quotients equal to actual equality on s" ??? .... ???o something ???....

???but what about some "inefficiency" here ????.....

hmmmm...... ????.....

for example something about if the continuous map is constant ... ???....