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,

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.