Dimensionful Spaces

One things that has bothered me for a while in mathematical physics is that the underlying mathematical formalism of space (i.e. differentiable manifold) has no native notion of unit or dimension. In a first-year physics class, we learn to do dimensional analysis as a sanity check for our equations. Yet when we learn mathematical physics, this is never formalized. Instead, we are told to think about spaces as consisting of a collection of “points”. But what are the units of a point?

This question has confused me for a long time, until I realized that it had no meaningful answer. The right question is to ask is “what are the units of a measurement.”

In the context of geometry, a measurement might be a smooth function \mathcal{X} \to \mathbb{R}, where \mathcal{X} is the space in question. We might ask “what are the units of this measurement?” but such a question has no answer, as posed.

Instead, we should switch to an algebraic context, and think of a space “as something that supports a notion of measurement.” For instance, we can simply define our category of spaces to be the dual of, say, C^\infty-rings. In order to make the distinction between geometry and algebra clear, let \mathsf{Spc} \cong C^\infty\text{-}\mathsf{Ring}^\mathrm{op}, and call this isomorphism M, so that a space \mathcal{X} is defined by its C^\infty-ring of measurements M(\mathcal{X}). The book that has most enlighted me about this way of thinking is Smooth Manifolds and Observables by Jet Nestruev, the pen name of a group of Russian mathematicians.

Having taken an algebraic perspective, we can now simply postulate that our ring of measurements is stratified by dimension. Specifically, if we have D base dimensions (e.g. spatial extent, time, charge, entropy, etc.), we postulate that we have a \mathbb{Z}^D-graded ring of measurements. That is,

R = \bigoplus_{u \in \mathbb{Z}^D} R_u

I talk about dimensions instead of units here, because we are not being particular to meters vs. feet here, only meters vs. seconds. For instance, if D = \{S, T, Q\} (space, time, charge), and u = S^1 T^{-1} Q^0 (we denote the group operation of \mathbb{Z}^D multiplicatively), then R_u is the measurements that have the dimension of a velocity.

A graded ring has the affordances of dimensionful measurements that we would expect. Namely, we can add measurements of like dimension, and we can multiply measurements and get measurements of the right dimension.

We should also be able to multiply measurements by scalars, so we should really have a graded \mathbb{R}-algebra.

However, we started with C^\infty-rings, not just rings, and we have sneakily dropped the C^\infty. What is the right notion of C^\infty-ring in the graded context?

Well, it doesn’t really make sense to apply arbitrary smooth functions to dimensionful measurements. This is downstream of the fact that if 0 \neq A \in R_u for u \neq 1 (where 1 is the unit of \mathbb{Z}^D because we work multiplicatively), then A^2 + A does not fall in any R_u, unless A is nilpotent. However, it does make sense to apply arbitrary smooth functions to dimensionless quantities. This is in complete accordance with the intuition of physicists; we can never take \sin of a length, but we can take \sin of a ratio of two lengths.

So I might propose that a graded C^\infty-ring is a graded \mathbb{R}-algebra R such that R_1 is a C^\infty-ring. I will call this a dimensional ring, and name the corresponding category \mathsf{DimRing}.

We can then simply define \mathsf{DimSpc} (dimensional spaces) to be \mathsf{DimRing}^\mathrm{op} and again for a dimensional space \mathcal{X}, we denote its defining dimensional ring by M(\mathcal{X}).

Let us now investigate the category \mathsf{DimSpc}.

According to our definition, the initial object 1 should have M(1)_1 = \mathbb{R} and M(1)_u = 0 for u \neq 1. This is interesting, because it implies that it is possible for there to be no measurement of a given dimension. If we don’t want this behavior, we could take the coslice of \mathsf{DimSpc} by the space \mathbf{Const}_\mathbb{R}, which has corresponding graded ring that is \mathbb{R} in every dimension. This allows us to assume that we have constants in every dimension.

We leave it to the reader to decide which version of \mathsf{DimSpc} they prefer, and to discover whether or not there is a substantial difference between the two. My suspicion is that after slicing over \mathbf{Const}_\mathbb{R}, we will end up with a category equivalent to \mathsf{Spc} (i.e. the dual to C^\infty\text{-}\mathsf{Ring}), because then each R_u will end up isomorphic to every other R_u via multiplication by constants. This may or may not be desirable; note that even if \mathsf{DimSpc} is equivalent to \mathsf{Spc}, working in \mathsf{DimSpc} is much more natural for applications.

There is an adjunction between \mathsf{DimRing} and (\mathsf{Set}^{\mathbb{Z}^D}), the left adjoint of which sends a \mathbb{Z}^D-indexed set to the free dimensional ring on that set of generators. Via this, we can make representing objects for measurements of each dimension, which are the free dimensional rings on one generator of a that dimension.

I’m not quite sure what else to say about this mathematically; it seems to more scratch a philosophical itch, so I will close out with some open questions.

This formalizes measurements which are vectorial in nature, which is to say that they can always be multiplied by scalars. Can we also formalize measurements which are affine in nature, in which the difference between two affine measurements is a vectorial measurement? Lawvere’s paper Grassman’s Dialectics and Category Theory investigates affine quantities in the context of the 16-dimensional real graded algebra associated to \mathbb{R}^3, which contains a linear subspace of vectorial quantities and an affine subspace of affine quantities. Perhaps a similar story could be told if we look at the category of graded algebras with distinguished maps into M(\mathbf{Const}_\mathbb{R}) (i.e. the coslice under \mathbf{Const}_\mathbb{R}) where the vectorial quantities were distinguished by the kernel of this map, and the affine quantities were distinguished by those quantities which were sent to 1 via this map.

Incidentally, this also gives an argument for not slicing over \mathbf{Const}_\mathbb{R}; we might want the 16-dimensional Grassmannian algebra to be the measurements out of a space in this theory.

Another question I have is how do units come into play? Perhaps units are in fact the constants in \mathbf{Const}_\mathbb{R}, and when we speak of something like 5m, we are speaking of the dimensionless quantity 5 multiplied by a dimensional constant m.

Finally, this seems to be a theory only of intensive quantities; i.e. quantities which can be measured at any point in a space. As explained by Lawvere in Categories of Space and Quantity, extensive quantities are those which vary covariantly with maps between spaces (they are summed over the preimage) and intensive quantities are those which vary contravariantly (via precomposition). For instance, measures vary covariantly, while densities vary contravariantly. How can we encompass extensive quantities in this framework?

I’m hoping that posting this will get people thinking about these questions, and I’m really hoping that someone will guide me to other variations on this idea, because I’m sure that I wasn’t the first person to think of using a graded ring for units. In fact, I wouldn’t be surprised if someone told me about this before, but I didn’t have the right intuition for it at the time, so it just rattled around in my subconscious until I wrote it up now, so I apologize in advance if this is very well-known.

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This reminds me a lot of stuff from John Baez and James Dolan, described on the nLab page mysteriously titled “Doctrines of algebraic geometry”:

Another name for the study of categories of line objects is “dimensional analysis”. In dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by the dimension group. We’ll call a physical theory described in this way a dimensional algebra, but the fundamental fact about a dimensional algebra is that it’s equivalent to a dimensional category, which is a symmetric monoidal category where all objects are line objects.

Example. Let G be the abelian group freely generated by the dimensions “mass” and “velocity”. Let R be the G-graded commutative algebra generated by the six quantities “mass of particle #1”, “mass of particle #2”, “initial velocity of particle #1”, “initial velocity of particle #2”, “final velocity of particle #1”, and “final velocity of particle #2”, subject to the two relations “conservation of momentum” and “conservation of energy”.

I don’t know whether it was ever published or what its antecedents might be. @johncarlosbaez could probably say more.

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We never published anything about it. This was written when James Dolan was just getting started on his work using doctrines in algebraic geometry. He was trying to understand why algebraic geometers are so obsessed with line bundles (e.g. all the theorems about ample line bundles) and projective algebraic varieties (which all come with god-given line bundles). He realized that what was going on was this: categorifying the old idea that algebraic geometry is about the duality between commutative rings and affine schemes, algebraic geometry shifted to studying certain kinds of ‘2-rigs’, which are like categorified commutative rings (or rigs), and the geometrical entities dual to these.

A ‘kind of 2-rig’ is an example of a doctrine in Lawvere’s sense. The simplest kind of 2-rig, in a certain sense, is the doctrine of dimensional categories - as defined on that nLab page you quoted.

In that page, we were noting that dimensional categories also naturally show up when you try to understand dimensional analysis using category theory, and AbGp-enriched dimensional categories are essentially the same as commutative rings graded by abelian groups. James had some nice ideas about what’s going on here: why should the same structures show up in dimensional analysis and the study of line bundles in algebraic geometry. It goes back to how the Greeks did geometry using dimensions - e.g., keeping track of the fact that you can’t add an area and a length - while Descartes discarded this.

James explains this idea in more detail on that page, and also in the talk videos on that page. James has been working on doctrines in algebraic geometry ever since 2009, and he’s made a lot of progress since then. During the pandemic I started meeting with him every week to talk about this. You can see videos of our conversations here, and also summaries (which are the best way to get a rough idea of what the hell all this stuff is about).

But I’d be happy to answer more questions about anything, since I realize it’s pretty hard to figure out what we’re doing from the available material! Someday I should sit down for 5 years and write a book.

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OK, I definitely remember now you telling me about line bundles a while back, but it didn’t click for me then. But now I see that if you just consider the dual category to the category of \mathbb{Z}^D-graded \mathbb{R}-algebras, then for each d \in \mathbb{Z}^D there is a classifying space for elements of grade d, which corresponds to the graded ring freely generated by an element in that grade. These are all different codomains for observables; unlike in smooth manifolds or affine varieties where there is a single representing object for observables. So the concept of having many different line bundles now makes sense to me.

Also, I never really saw the connection between projective geometry and dimensional analysis until now, but now I see that “dimensionally consistent” equations are precisely those which remain true when you change the numerical values of all of your units, and this invariance under rescaling is the heart of projective geometry.

I’m looking forward to learning more about this and reading/watching the lectures/conversations with James Dolan.

James Dolan has a lot of ideas that sound far-out at first but ultimately make perfect sense. You just mentioned a big one: the algebraic geometer’s obsession with homogeneous polynomial equations is secretly the same as the scientist’s obsession with only adding quantities that have the same units.

In one version of algebraic geometry, people turn any \mathbb{N}-graded commutative ring into a scheme using the Proj construction. This is similar to how we turn a commutative ring into an affine scheme called its spectrum.

The spectrum of a commutative ring is its set of prime ideals, given a certain topology and then equipped with a sheaf of commutative rings. Similarly, Proj of a \mathbb{N}-graded commutative ring is the set of homogeneous prime ideals (except for the biggest one), given a certain topology and then equipped with a sheaf of commutative rings.

The basic example is the ring of polynomials in n+1 variables, with its usual grading by degree. Proj of this is projective n-space. The link explains the details.

If we didn’t put in that annoying thing about “except for the biggest one”, I believe Proj of the polynomial ring would be projective n-space plus one extra point. This may seem bizarre, but maybe it’s a bit less so if you think about this. Take a point in k^{n+1}. This point determines a line through the origin and thus a point in projective n-space… unless this point is the origin itself!

In other words, points in k^{n+1} modulo scaling give projective n-space together with one extra point. You’ve probably heard of ‘points at infinity’ and how projective geometry is so great at dealing with them. Well, this is a ‘point at zero’, and traditional projective geometry is not so great at dealing with that. :smiling_imp: If you read the link you’ll see it says Proj is not a functor. I think this is related.

Anyway, James’ approach is better. It also easily handles commutative algebras graded by an arbitrary abelian group. For dimensional analysis you may decide \mathbb{Z}^n is the most general grading group you want… but then you may run into quantities with units like \mathrm{length}^{1/2}.

Finally, I should say what’s the point of all this algebraic geometry nonsense if all you really care about is dimensional analysis.

Suppose a physical theory is a bunch of equations between dimensionful quantities where you’re only allowed to add quantities when their dimensions match. Then this theory is really a presentation of a graded commutative ring in terms of generators and relations. For example:

Example. Let G be the abelian group freely generated by the dimensions “mass” and “velocity”. Let R be the G-graded commutative algebra generated by the six quantities “mass of particle #1”, “mass of particle #2”, “initial velocity of particle #1”, “initial velocity of particle #2”, “final velocity of particle #1”, and “final velocity of particle #2”, subject to the two relations “conservation of momentum” and “conservation of energy”.

Then, this graded commutative ring gives a scheme! And this scheme is the ‘space of models’ of the physical theory! In algebraic geometry we might call such a space a ‘moduli space’.

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OK, I’m starting to get a handle on the intuition behind this. Here’s my attempt to summarize it.

Affine schemes can be described as those geometric objects which are dual to commutative rings. The traditional move from affine schemes to general schemes has been done by essentially a colimit completion of the category of affine schemes; we think about a general scheme as being the gluing of many affine schemes. The high-brow method for this that I was taught in my algebraic geometry class was that a scheme was a sheaf on the category of affine schemes, given a certain Grothendieck topology.

James Dolan is thinking about getting at general schemes (and specifically projective schemes, which are almost never affine) in a different way. Instead of gluing together patches of affine schemes, which are the duals of commutative rings, we simply start with a more advanced algebraic object. Specifically, we might think of an “algebraic geometric theory” as an algebraic object which has the formal properties of the category of sheaves of k-modules over a scheme.

So then does it make sense to think about the 2-category of “algebraic geometric theories” as the dual of some category of spaces? I’ve been reading Joyal and Anel’s Topo-Logie over this break, and they provide good arguments for why the category of toposes should really be thought of as a category of spaces, dual to the category of logoses.

My intuition now for an “algebraic geometric theory” is that it is a space “equipped with many different ways of measuring it”; the morphisms in the theory are all “ways of measuring the space”.

And finally, we can also think about general schemes as embedding into the category of ringed topoi, which @david.jaz showed me the power of in Approximate Morphisms. Is there a connection between algebraic geometric theories and ringed toposes?

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The traditional move from affine schemes to general schemes has been done by essentially a colimit completion of the category of affine schemes; we think about a general scheme as being the gluing of many affine schemes.

You have to be careful here. The category of affine schemes already has all small colimits! After all, it’s the opposite to the category of commutative rings. Since commutative rings are algebras of a Lawvere theory, we know right away that the category of those has all small limits (and colimits). So the category of affine schemes has all small colimits (and limits).

The problem is that colimits of affine schemes aren’t what we want them to be - where “we” means algebraic geometers. We want \mathbb{C}\mathrm{P}^1 to be built as a union of two copies of \mathbb{C}. In the category of sets it’s indeed built as a pushout from two copies of \mathbb{C} in the visibly evident way: everything but the north pole, and everything but the south pole. But you can’t build it that way in the category of affine schemes. It’s good to think about why this fails.

Thus, in the usual story, one really needs to want things like projective spaces to be colimits of affine schemes, to find a category in which that’s true. Purely formal considerations like “colimit completion” won’t get you there, since the category of affine schemes already has all small colimits!

There’s a lot more to say about this, but your description of Dolan’s approach sounds right:

Instead of gluing together patches of affine schemes, which are the duals of commutative rings, we simply start with a more advanced algebraic object.

Yes, I think James doesn’t really like the business of defining an important thing like a scheme as “the result of gluing together lots of things that make good conceptual sense, where gluing is defined in a way that ensures we get the examples we want”.

Specifically, we might think of an “algebraic geometric theory” as an algebraic object which has the formal properties of the category of sheaves of k-modules over a scheme.

Right. There are some important technical issues we could worry about here, but this is a good thing to say.

So then does it make sense to think about the 2-category of “algebraic geometric theories” as the dual of some category of spaces?

Approximately, but you’ll notice you’re saying “2-category” and then “category”, so something fishy is going on.

Really the 2-category of algebraic geometric theories should be the dual (opposite) of some 2-category of spaces! So these spaces are a bit more general than mere schemes. To the extent that a scheme is a glorified set, these spaces are more like glorified groupoids! They are a lot like what people call algebraic stacks. Luckily you don’t need to learn the usual approach to algebraic stacks to understand this stuff. This stuff is supposed to be a more efficient approach.

Is there a connection between algebraic geometric theories and ringed toposes?

Yes! I’m a bit fuzzy about the details but they’re pretty similar and there should be some theorems relating them.

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Approximately, but you’ll notice you’re saying “2-category” and then “category”, so something fishy is going on.

Really the 2-category of algebraic geometric theories should be the dual (opposite) of some 2-category of spaces! So these spaces are a bit more general than mere schemes. To the extent that a scheme is a glorified set, these spaces are more like glorified groupoids! They are a lot like what people call algebraic stacks. Luckily you don’t need to learn the usual approach to algebraic stacks to understand this stuff. This stuff is supposed to be a more efficient approach.

Whoops, I meant to say 2-category of spaces! I’ve been conditioned to think of toposes as spaces “with a category of points”, so I’m perfectly happy thinking of these as schemes “with a groupoid of points”!

And in the context of moduli spaces, I can see this is a very natural thing to want, because we may want to not fully quotient by a symmetry and instead keep that symmetry around as the morphisms of a groupoid. Very cool.

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OK, now I get why people are so into K-theory; it’s just computing properties of the associated algebraic stack to a space.