lawvere

something i've wondered about, about whether lawvere explicitly meant to draw certain connections between ideas that i remember him talking about...

i have a pretty isolated and vague memory of lawvere talking once about a certain category-theoretic interpretation of "dimensional analysis" that he'd developed. i think that this was in the middle of a course of lectures or a seminar on other stuff, and was presented as somewhat of a digression. (i think that i remember him writing it down on the side blackboard as though apart from the main action on the front blackboard.) i think that he also presented it somewhat laconically as an idea that he'd been somewhat excited about without managing to get too many other people excited about it.

i pretty much forgot the details of his idea, but later came to suspect that it was more or less the same as an idea that i rediscovered semi-recently, which is that a symmetric monoidal algebroid with all objects invertible and all braidings trivial is on the one hand equivalent to a sort of graded commutative algebra, while on the other hand can also be interpreted as a "dimensional theory", meaning a theory (in a certain sense) built according to the rules of dimensional analysis. the objects of the algebroid correspond to the grades of the graded commutative algebra and also to the "dimensions" in which the quantities of the theory live.

moreover, the usual technique (what algebraic geometers sometimes abbreviate as "proj") for extracting a projective variety from a certain kind of graded commutative algebra can be applied in this setting (in a slightly generalized way) to show that dimensional analysis is secretly essentially equivalent to slightly generalized projective algebraic geometry. in retrospect this is "obvious", in that both are about the study of "homogeneous quantities" and of covariance of quantities with respect to rescaling. it is similarly "obvious" that the concept of "dimension" in dimensional analysis manifests itself in algebraic geometry as the concept of "line bundle" (which is an abstract way of looking at a projective embedding); after all a "line bundle" is really just a _line_ from an "internal" viewpoint, and a "line" is a pure 1-dimensional object, aka a "dimension".

despite this "obviousness" i don't remember hearing people (not even lawvere)
talk about this. maybe it's one of those things that's so "obvious" that people don't talk about it, but i generally favor talking about such things. or maybe people do talk about it but i just haven't been listening carefully enough.

anyway, one of the things that i wonder about lawvere is whether he explicitly intended to connect dimensional analysis with projective algebraic geometry in this way. but furthermore, i wonder whether his use of the word "theory" in certain contexts is explicitly intended to emphasize this kind of connection.

thus my own usage of "dimensional theory" to describe a certain kind of graded commutative algebra or equivalently a certain kind of symmetric monoidal algebroid is itself intended as a sort of pun, in that a dimensional theory is a "theory" in what i perceive of as two ways (though i wonder whether lawvere himself perceives it as just a single way). first, it's a "theory" in a sort of lowbrow old-fashioned "physical" sense; it contains some quantities (such as perhaps "x-component of momentum of particle 1" or "y-component of angular momentum of particle 2") living in various dimensions (such as "momentum" or "angular momentum") and obeying some homogeneous (aka "dimensionally consistent") algebraic constraints (such as "conservation of momentum" or "conservation of angular momentum"). but second, it's a "theory" in the sense developed (according to lawvere) by jon beck and later explored in more detail by lawvere, of being a category equipped with some sort of extra structure amounting to a so-called "doctrine"; in this case the doctrine of "symmetric monoidal algebroids with all objects invertible and all braidings trivial".

thus i wonder whether this was lawvere's (and/or even beck's) intent all along- that when we hear of a "theory" we're supposed to think about it in both of these ways.

(perhaps the concept of "logical theory" is a third strand here, again problematic to decide how entangled it is with these other two.)

another synonym for "line bundle" is "invertible sheaf". invertible sheaves take their place among the "coherent sheaves", which in turn live in a ringed topos. the puns turning on the meaning of "theory" become only sharper in the context of these enrichenings.

anyway i tried writing to lawvere about this, to try to find out some of the history of this, about how explicitly he intended various connections to be drawn, but i haven't heard back from him. maybe he's written about this somewhere without my having read it or understood it properly.