?? seems like one key to pulling things together here is .... ??? basic dictionary entries get introduced right aruond time lawvere's approach and competing approach are introduced ....
?? speed of light joke around here ???? ..... section vs morphism ... algebra vs category .... ???? ....
pointing to tableau on whiteboard : "part of what's going on here is that we're introducing an alleged sort of mental/conceptual hygiene; separating quantities of different kinds into separate boxes .... ?? thereby allegedly avoiding certain kinds of mental/conceptual confusion ..." ..... ?????? ??? maybe stuff about related ideas ..... structured programming and so forth .... ???? "typed lambda calculus" ???? .....
?? hmm, maybe torture quiz right after introducing dimensional analysis example .... "we're supposed (according to me) to get a cryptomorphism to projective algebraic geometry .... ?? any opinion as to how that might go ?? .... what might correspond to the dimensions, and to the quantities that live in them ??? ....." ....
??? "i promised you a dictionary ... well this is basically it .... rather short .... you can add lots more entries to it if you want, but all the rest are basically determined by these two" .... ???? ....
??? rant about retrospective obviousness of correspondence here ... terminologies "dimension" vs "line-bundle" .... !!! "line" as "one-dimensional object", aka "dimension" !!!!!! (??? this rant as probably somewhat separate from rant about invertibility under tensor product as hallmark of "one-dimensionality", coming up slightly later ???? ....)
"dimensional analysis is essentially the study of dimensions and quantities that live in those dimensions, projective algebraic geometry is essentially the study of line bundles and sections that live on those bundles, and the two subjects are secretly isomorphic or "cryptomorphic" under this (pointing to 2-entry dictionary on whiteboard) correspondence" ....
(if you doubt the characterization of pag as "all about line bundles and their sections", then that's an additional propaganda point that i can try to rant about if you like .... ???? .....)
"at this point there are two main ways to mathematically formalize [?? what we're doing here ?? ... / ?? this correspondence ... ?? ...] :
1 organize the sections of all the line bundles over a projective variety into a _graded commutative algebra_, with the _grades_ being the line bundles ....
2 .... lawvere approach .... _obects_ being the line bundles .... ???? .....
...............
??speed-of-light joke here somewhere ... ??? but wasn't there something else i wanted here too???? ...
oh yes!!! ..... bit about .... sections of line bundles ..... ???? " ... lowbrow approach involving lowbrow geometric quotienting by re-scaling action on vector space .... before the quotienting (=?= introduction of grading ... g = a ... ??? ....) had functions on a space, but what are they now only functions on some space lying over the space you're interested in .... and if you think about this and pursue it far enough, you can see that what's happening with these former functions (from affine picture ...) is that they're now (in projective geometry) achieving the status of ("twisted functions" or) sections of line bundles ..... ???"
now what makes [this second approach / lawvere's category approach / ... ??? ...] so interesting is that because of the way in which it uses _categories_ it reveals pag as being a part(/aspect ??? ....) of the vast program of _categorical logic_ (?? or "category-theoretic logic") of which lawvere is one of the principal founders ....
?? and in fact that's really my central message here, that projective algebraic geometry can be seen as a part of categorical logic (?? and in fact algebraic geometry more generally can be seen as a part of categorical logic) ... and this message is directed mainly at categorical logicians themselves, to tell them how they can use what they already know in order to understand algebraic geometry and to contribute to its development .... in particular, if you're a categorical logician and you're interested in learning algebraic geometry, then there's a royal road for you to take in order to get there from where you already are ... instead of having to trudge along the commoners's (??) road, which may involve swallowing an awful lot of preliminary introductory material .... and this royal road is pretty different from what algebraic geometers might tell you themselves about how category theory is applied to algebraic geometry ....
(?? "turf wars" ... ?? .... category theory as very naively about getting past hierarchies to more interestingly reciprocal webs of interconnectedness, and that's part of what's going on here .... silly to argue over whose pastime most fundamentally subsumes whose; my message for a particular audience about something they understand as subsuming something else doesn't preclude other ways of looking at it ..... mutual subsumption and so forth .... ???? ...)
the basic idea of category-theoretic logic is (, very roughly,) to study categories with some kind of extra structure and to think of these structured categories as _theories_ of a sort, and to think of (????? "categorical logic" vs "categorical algbera" here ???????? .......) a functor preserving the structure as a _model_ of the domain theory, or as an _interpretation_ of the domain theory inside the codomain theory ....
(?? somewhere .... secondary central message (= not quite central message) directed particularly at attendees of a conference on "category-theoretic methods in representation theory" or whatever .... : that it's important to study categories of representations alongside certain other kinds of categories .... "unification" aspect of tannakian philosophy .... stack concept unifying groups (pure stackiness) with spaces (completely un-stacky stacks) .... ??? .... for example studying interpretation taking g-reps to quasicoherent sheaves over scheme x, as ess g-torsor over x ....... ????? .....)
????!!! get to stuff about .... moduli stack of models ..... as more or less "proj" construction here .... ??? ..... gauge theory and general relativity as prototypical examples of physical theories where "state of affairs" may have automorphism, aka moduli stack of models may be stacky (?? property:structure:stuff::axiom:predicate:type .... ???? ...) .... ???? ..... ??? whole idea of lawvere's "theories" (for example algebraic theories, toposes ....) as inspired (????) both by "logical theories" and by "physical theories" ...... ???? ......
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