Torsors, weights, and the scaling torus

The representation-theoretic reading of dimensional analysis: the group of global unit rescalings is a torus (ℝ⁺)ⁿ — one multiplicative factor per base dimension — and a quantity of dimension vector d ∈ ℤⁿ is precisely an object that transforms under the character (weight) d of that torus. Dimensional homogeneity of a physical law becomes equivariance under the torus action; "dimensionless" becomes "weight zero"; and the set of possible units of any one dimension is not a group but a torsor — a set the group acts on freely and transitively, with no distinguished identity. John Baez's "Torsors Made Easy" is the primary source for what a torsor is and why unit-like choices (voltage ground, quantum phase, calendar dates) are torsorial; Terence Tao's 2012 blog post develops the weight-space picture explicitly and connects it to torsors at both ends of the post; and Zapata-Carratala 2021 is the load-bearing published development, rebuilding commutative algebra itself around a partial, slice-wise addition (dimensioned rings) and proving that a choice of units is exactly a trivialization R_D ≅ R₁ × D. Jonsson 2021's scalable monoids give the same "scaling acts on magnitudes" content in a one-sorted universal-algebra idiom and are treated here as the comparison foil.

NOTE

This page covers the group-action layer of the survey: what acts, on what, and what is invariant. Division of labour with the sibling pages: this page owns the torsor concept (Baez), the scaling-torus/weight-space reading of dimensional analysis, Zapata-Carratala's dimensioned algebra, Jonsson's scalable monoids, and Tao's closing torsor variant. Tensor of lines owns Tao's 2012 post as a formalization — the abstract one-dimensional lines composed by ⊗ and duals, the parametric model developed in tandem with them, the dictionary theorem between the two, hybrid quantities and the convex-hull criterion, the structure-group ladder of kinds — together with Janyška–Modugno–Vitolo's positive spaces. Results owned there are cross-referenced below, not restated; how the two pictures relate is set out under Relation to the tensor-of-lines picture. The discrete skeleton ℤⁿ that indexes the weights is the subject of free abelian group of dimensions; and the classical invariance theorem that this picture reframes (every equivariant law factors through weight-zero combinations) is Buckingham π. Where this page states standard representation-theoretic glue that no local source spells out, it is marked [exposition] rather than attributed.

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At a glance

Dimension	Torsors, weights & the scaling torus
Primary structure	A group action: the scaling torus (ℝ⁺)ⁿ acting on parametrised quantities (Tao); algebraically, a dimensioned ring/field R_D fibred over a monoid of dimensions (Zapata-Carratala)
Quantity	A family x = x_{M,L,T} obeying a power-law transformation — a vector of weight d — or an element a_d of the dimension slice R_d
Dimension	A character/weight of the torus (exponent vector d); in dimensioned algebra, a point of the dimension monoid D, i.e. a slice R_d = δ⁻¹(d)
Unit	A point of a torsor: an element of R_d^× (any non-zero element serves); a whole unit system is a multiplicative section u : D → R of the dimension projection
Homogeneity of laws	Equivariance under the torus action; verifiable at a single choice of units (Tao's transfer principle)
Dimensionless	Weight 0 — literally invariant under all rescalings; the dimensionless slice R₁ (an ordinary ring/field)
Central theorem	A choice of units trivializes: R_D ≅ R₁ × D (Prop 3.4) and the power ring of k lines ≅ F × ℤᵏ (Thm 4.1 + its §4 extension) — always non-canonically
Addition across dimensions	Partial (slice-wise) by algebraic necessity — distributivity forces multiplication to act on dimensions (Zapata-Carratala); or total-but-hybrid with no single weight (Tao)
One-sorted alternative	Scalable monoids / quantity spaces: a monoid with a ring scaling action, dimensions as orbits, addition derived from unit elements (Jonsson)
Canonical sources	Baez 2009 (torsor pedagogy); Tao 2012 (weight spaces, structure groups); Zapata-Carratala 2021 (dimensioned algebra); Jonsson 2021 (scalable monoids)

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Primary sources

• John Baez, "Torsors Made Easy", web essay, December 27, 2009. The primary source for the torsor concept as used throughout this page: the definition (free + transitive action, unique "ratio"), the physics examples (energy zero, voltage ground, quantum phase), the everyday examples (calendar dates, musical notes, antiderivatives' +C), and the affine-space moral. Inspected in full — the local capture $REPOS/papers/units-of-measure/baez-torsors-made-easy-web.html is the complete essay; all quotes below are transcribed from it.
• Terence Tao, "A mathematical formalisation of dimensional analysis", What's New (blog), December 29, 2012. Used here for the group-action half of the post: dimensional parameters M, L, T ranging over ℝ⁺, the structure group (ℝ⁺)³, quantities-of-dimension as power-law families, the explicit weight space terminology, the equivariance reading of dimensional consistency, hybrid quantities (in brief), non-toral structure groups (GL₃(ℝ), E(2), Poincaré, diffeomorphism and gauge groups), and the two torsor remarks bracketing the post. The post's other half — the abstract tensor-of-lines model, the dictionary theorem between the two pictures, and the convex-hull criterion for hybrid inequalities — is developed on tensor of lines and only cross-referenced here. Inspected in full — the local capture $REPOS/papers/units-of-measure/tao-2012-formalisation-dimensional-analysis-blog.html includes the comment thread; formulas were recovered from the WordPress LaTeX image alt attributes. Every Tao claim on this page was verified to appear in the capture.
• Carlos Zapata-Carratala, "Dimensioned Algebra: the mathematics of physical quantities", arXiv:2108.08703, August 2021. The modern algebraic development and this page's load-bearing published treatment: dimensioned sets, dimensional vs dimensioned binary operations, dimensioned rings/fields/modules/algebras, units as monoid sections, the trivialization propositions, and the power functor from the category Line of 1-dimensional vector spaces to dimensioned rings. Inspected in full via pdftotext -layout of $REPOS/papers/units-of-measure/zapata-carratala-2021-dimensioned-algebra-arxiv.pdf.
• Dan Jonsson, "Magnitudes, scalable monoids and quantity spaces", arXiv:2108.02106 (v6, January 2023). The alternative algebraicization of "scaling acts on magnitudes": a scalable monoid is a monoid with a compatible ring action, dimensions are commensurability orbits, and addition is a derived operation. Inspected (the definitional core §§2–3) via pdftotext -layout of $REPOS/papers/units-of-measure/jonsson-2021-magnitudes-scalable-monoids-quantity-spaces-arxiv.pdf.
• Cited from within the above but not inspected here (with one exception): Janyška–Modugno–Vitolo's semi-vector spaces — the 2007 preprint is pinned locally and is a primary source of tensor of lines; for this page only Note 2.3 (the ℝ⁺-torsor structure of positive spaces) and the §3.1 definition of a unit as a semi-basis were verified directly against the local capture, and the 2010 Acta Appl. Math. journal version is cited via Zapata-Carratala's introduction [unverified]; Hart's 1980s–90s work, which Zapata-Carratala credits as "the first efforts in developing a general mathematical theory of physical quantities" (see Hart's multidimensional analysis); the Baez–Dolan "Doctrines of Algebraic Geometry" notes characterising physical quantities as line objects [unverified]; and Vysoký's graded-manifold formalism (arXiv:2105.02534 [unverified]), which Zapata-Carratala's §8 compares to dimensioned algebra.

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Formal core

Torsors (Baez)

For a group G, a G-torsor is a set X with an action of G — 1x = x and (g₁g₂)x = g₁(g₂x) — satisfying one extra axiom that separates torsors from arbitrary actions:

"The special thing that makes a group action be a torsor is this: for any two elements x₁ and x₂ of our torsor there exists a unique group element g with g x₁ = x₂." — Baez, "Torsors Made Easy" (local capture, definition section)

The unique g is the ratio x₂/x₁ (or, additively, the difference x₂ − x₁). The action is thus free (uniqueness) and transitive (existence) [exposition: terminology]. The operational consequence is Baez's additive-language summary:

"But you can't add elements of a G-torsor X. Instead, you can add an element of G to an element of X and get another element of X. You can also subtract two elements of X and get an element of G." — Baez, "Torsors Made Easy"

and the two slogans: "An affine space is like a vector space that has forgotten its origin" and "A torsor is like a group that has forgotten its identity." Baez's physics examples are all unit-like or origin-like choices: energies and voltages live in ℝ-torsors (only differences are real numbers — the electrical ground convention is an arbitrary base-point choice); quantum phases live in a U(1)-torsor (only relative phases are unit complex numbers); calendar dates and musical notes are ℤ-torsors; antiderivatives of a fixed function form an ℝ-torsor (the +C); positions form a torsor of the vector space of displacements. The structure theorem — trivial to prove, load-bearing everywhere below — is:

"Any group G is a G-torsor, and every other G-torsor is isomorphic to G - but not canonically!" — Baez, "Torsors Made Easy"

Proof sketch (Baez's own): pick any x₁ ∈ X and "declare it to be the identity"; the map X → G sending x₂ to the unique g with g x₁ = x₂ is well defined (torsor axiom), injective (freeness) and surjective (transitivity), hence a bijection intertwining the actions — but it depends on the arbitrary choice of x₁. ∎

The scaling torus and weight spaces (Tao)

Tao's parametric model postulates dimensional parameters — for the M, L, T system, three of them — ranging "freely and independently among the positive real numbers ℝ⁺, thus the parameter space (or structure group) here is given by the multiplicative group (ℝ⁺)³" (Tao, §1). A dimensionful quantity is a family x = x_{M,L,T} indexed by the parameters; it has a dimension precisely when the dependence is a power law:

Structure group (the scaling torus):    G = (ℝ⁺)ⁿ        one ℝ⁺ factor per base dimension
                                        (Tao: (ℝ⁺)³ with coordinates M, L, T)

x has dimension MᵃLᵇTᶜ  ⟺   x_{M,L,T} = x̃ · M⁻ᵃ L⁻ᵇ T⁻ᶜ   for some number x̃    (Tao (1))

(negative exponents because (1) is a PASSIVE change of units, not an active change of x)

(The worked examples, the model's extension to dimensionful sets, functions, and integrals, and the transfer principle's proof sketch are developed on tensor of lines.) Dimensionless quantities are the constant families — the weight-zero case a = b = c = 0. The representation-theoretic identification is explicit and verbatim:

"In the language of representation theory, the collection of dimensionful quantities of dimension MᵃLᵇTᶜ is a weight space of the structure group (ℝ⁺)³ = {(M,L,T): M,L,T ∈ ℝ⁺} of weight (a,b,c). One can indeed view dimensional analysis as being the representation theory of groups such as (ℝ⁺)³ …" — Tao, §1 (formulas restored from the capture's LaTeX alt text)

[exposition] The standard glue Tao leaves implicit: the continuous characters (1-dimensional representations) of the torus (ℝ⁺)ⁿ are exactly the power laws (λ₁,…,λₙ) ↦ λ₁^(d₁) ⋯ λₙ^(dₙ), one for each real exponent vector (d₁,…,dₙ) ∈ ℝⁿ — so the character group of the scaling torus is ℝⁿ, and the integer dimension vectors ℤⁿ of ordinary dimensional analysis form a sublattice of it. Multiplying quantities multiplies characters, i.e. adds weights; the free abelian group of dimensions is the character lattice of the scaling torus. Tao's own exponents are deliberately arbitrary reals (his convex-hull analysis of hybrid inequalities quantifies over (a,b,c) ∈ ℝ³), and his model also contains parameter dependences with no weight at all — the contrived L^{sin(M+T)} "does not have any specific dimension attached to it" (Tao, §1).

Two torsor remarks bracket the post, both genuinely present in the capture. Near the start: "it would be slightly more natural to use a parameter space which was a torsor of the structure group, rather than the structure group itself; we discuss this at the very end of the post." And at the very end, after the abstract model of one-dimensional ordered vector spaces V^M, V^L, V^T (the tensor-of-lines picture) has been set up:

"One way to avoid this (which was alluded to previously) is to interpret M, L, T not as scalars in ℝ, but rather as elements of the ℝ-torsors V^M, V^L, V^T respectively. With this modification to the parametric framework, the reference units M₀, L₀, T₀ can now be omitted." — Tao, closing section (formulas restored from LaTeX alt text)

That is: the "value" of a dimensional parameter is itself a torsorial choice, not a number; numbers only appear after an arbitrary reference unit turns the torsor into the group. Tao immediately names the price — recorded under Open problems & frontier below. The same V^M, V^L, V^T are, on tensor of lines, the primitive carriers: there the passive law (1) is derived from basis change on the lines ("is of course just (1)", in the dictionary theorem's proof), whereas here it is the postulated torus action.

Dimensioned rings and units as sections (Zapata-Carratala)

Zapata-Carratala starts from the observed algebra of working physicists — "Addition can only be performed between quantities specified by the same unit of measurement … addition is otherwise undefined", while "Multiplication can be performed between any two arbitrary physical quantities" (Zapata-Carratala, §2.1) — and rebuilds algebra so that this is structure, not folklore:

"From a mathematical point of view, the set of physical quantities is R × Zⁿ with addition only partially defined in the first argument and multiplication defined as a direct product in both arguments. The Zⁿ component corresponds to the types of physical quantities and the domains of partial addition are precisely the subsets of matching type." — Zapata-Carratala, §2.1 (pdftotext extraction, p. 5)

The machinery, in his own terms:

Dimensioned set:      a surjection  δ : A → D          D = "set of dimensions"
                      slices  A_d := δ⁻¹(d)            (a_d denotes an element with δ(a) = d)

Dimensional binar:    a ∗ b defined ONLY within a slice  (δ(a) = δ(b) = δ(a ∗ b))
                      — addition-like: a family of ordinary operations indexed by D
Dimensioned binar:    a ∗ b TOTAL, mapping slices to slices:  A_d ∗ A_e ⊆ A_{d·e}
                      — multiplication-like: D itself acquires an operation and
                        δ becomes a morphism  (A, ∗) → (D, ·)

Dimensioned ring (R_D, +_D, ·_D):
    (R_D, +_D)   a dimensional abelian group          (partial, slice-wise +)
    (R_D, ·_D)   a dimensioned (commutative) monoid   (total ·; D becomes a monoid)
    (a + b)·c = a·c + b·c   whenever defined          (distributivity)
    R₁ := the slice over the monoid identity          ("dimensionless slice" — an ordinary ring)

Unit (choice of units):  a section  u : D → R  of δ  with  u_{de} = u_d · u_e  and  u_d ≠ 0_d
                          — a multiplicative SPLITTING of the monoid surjection δ
Dimensioned field:        every non-zero element invertible   (forces (D, ·) to be a GROUP)

Two structural points deserve emphasis. First, the direction of explanation is unusual: the fact that dimensions themselves multiply (D is a monoid) is derived, not assumed — see Addition across dimensions for the distributivity argument. Second, units are global sections, and sections can fail to exist: taking the Möbius band as a dimensioned ring over the circle, "we find an explicit example of a dimensioned ring that does not admit units, since they would correspond to global non-vanishing sections of a non-trivialisable vector bundle" (Zapata-Carratala, §3). Unit existence is a triviality condition, exactly as for principal bundles — Baez's fiber-of-a-principal-G-bundle example ("what it actually is, is a G-torsor") is the same geometry.

Central theorem: a choice of units is a trivialization

The page's central theorem is Zapata-Carratala's Proposition 3.4 together with its concrete realisation, Theorem 4.1. In a dimensioned field, every non-zero element a_d induces slice-wise multiplication bijections a_d· : R_e → R_{de} (inverse: multiplication by 1/a_d), each an isomorphism of the slices' additive groups by distributivity. Then:

"These maps allow to prove a general result that confers a role to choices of unit on dimensioned fields similar to that of a trivialization of a fibre bundle." — Zapata-Carratala, §3 (pdftotext extraction, p. 13)

Proposition 3.4 (units trivialize). Let (R_D, +_D, ·) be a dimensioned field. A choice of units u : D → R induces an isomorphism of dimensioned fields R_D ≅ R₁ × D, where R₁ × D is the product dimensioned field (numeral, dimension-tag) with (r,d)·(s,e) = (rs, de) and slice-wise +.

Proof sketch (following the paper):

Φ_u : R₁ × D → R_D           Φ_u(r, d) := u_d · r          (slice-wise multiplication by u_d)

bijective:   inverse  Φ_u⁻¹(a_d) := u_{d⁻¹} · a_d          (D is a group; u_{d⁻¹} inverts u_d)
additive:    each Φ_u|_{R₁ × {d}} : R₁ → R_d  is an abelian-group isomorphism  (slice-wise mult.)
multiplicative (u is a monoid morphism):
    Φ_u((r₁,d)·(r₂,e)) = Φ_u((r₁r₂, de)) = u_{de}·r₁·r₂ = (u_d·r₁)·(u_e·r₂) = Φ_u(r₁,d)·Φ_u(r₂,e)   ∎

Theorem 4.1 (the power functor realises physics). For the category Line of 1-dimensional vector spaces over a field F (morphisms: invertible linear maps, which the paper calls factors — "a unit-free conversion factor"), tensor powers

Lⁿ := L ⊗ ⋯ ⊗ L  (n > 0)        L⁰ := F        L⁻ⁿ := (L*)⊗ⁿ        Lⁿ ⊗ Lᵐ = Lⁿ⁺ᵐ

power of L               :=  ⋃_{n ∈ ℤ} Lⁿ                          dimension set  ℤ
power ring of L₁, …, L_k :=  ⋃ L₁^{n₁} ⊗ ⋯ ⊗ L_k^{n_k}             dimension group ℤᵏ

assemble into a functor Line → DimRing; each power is a dimensioned field; a choice of non-zero u ∈ L^× induces a choice of units n ↦ uⁿ and hence a (non-canonical) isomorphism with F × ℤ; and for k base lines with chosen units uᵢ ∈ Lᵢ^×, the induced U : ℤᵏ → R, (n₁,…,n_k) ↦ u₁^{n₁} ⋯ u_k^{n_k} yields the power ring ≅ F × ℤᵏ (Zapata-Carratala, §4). The paper closes the loop explicitly: "We thus recover the explicit algebraic structure of physical quantities identified in Section 2.1 from the standard practice in dimensional analysis." Note the domain of the functor: Line is exactly the carrier category of the tensor-of-lines picture, so Theorem 4.1 is the survey's formal bridge between that page and this one — it assembles the lines and their tensor powers (that page's primitives) into a dimensioned field (this page's object), on which unit choices then act torsorially (next paragraph).

This is Baez's non-canonical-isomorphism theorem industrialised: F × ℤᵏ (numeral + exponent vector, the naive "reals with attached units") is what a quantity algebra looks like after an arbitrary choice; the invariant object is R_D, and the choice is a point of a torsor. Concretely [exposition]: in a dimensioned field, for any two non-zero a_d, b_d in the same slice there is a unique non-zero dimensionless r = b_d · (1/a_d) ∈ R₁^× with b_d = r · a_d — existence and uniqueness are exactly Baez's torsor axiom, so the non-zero part of every slice R_d^× is an R₁^×-torsor, and two unit systems u, u′ : D → R differ by a unique monoid morphism c : D → R₁^× (a character!). For D = ℤⁿ and R₁ = ℝ restricted to positive elements, the group of such characters is (ℝ⁺)ⁿ — the scaling torus reappears as precisely the group that acts freely and transitively on coherent unit systems. Zapata-Carratala never uses the word "torsor"; this paragraph is standard mathematics layered on his Propositions 3.4/4.1.

The scalable-monoid alternative (Jonsson)

Jonsson algebraicizes the same "scaling acts on magnitudes" intuition without any fibration apparatus — one sorted structure, one total multiplication, and a ring action:

Scalable monoid over a ring R:   a monoid (X, ∗, 1_X)  +  scaling action  ω : R × X → X  with
    1 · x = x        α · (β · x) = αβ · x        α · xy = (α·x)y = x(α·y)     (Def 2.1)
    — "an algebra without an additive group": keep (c) bilinearity, drop addition entirely

Commensurability:    x ∼ y  ⟺  α·x = β·y  for some α, β ∈ R
Orbitoid (dimension): an equivalence class of ∼   (orbits R·x may overlap; ∼ transitivises ≈)
Unit element for C:   u ∈ C  generating (every x ∈ C is λ·u)  and faithful (λ·u = λ′·u ⟹ λ = λ′)
Derived addition:     x + y := (ρ + σ)·u    where x = ρ·u, y = σ·u          (Def 2.35;
                      well defined independently of which unit element u is used — Lemma 2.34)

Proposition 2.36: a non-trivial orbitoid possessing a unit element is a free module of rank 1 over R — so a scalable monoid whose orbitoids all have unit elements "is the union of disjoint isomorphic free modules of rank 1 over a non-trivial commutative ring, a result that may be compared to definitions of systems of quantities in terms of unions of one-dimensional vector spaces by Quade and Raposo" (Jonsson, §2.6) — the bridge to tensor of lines. A quantity space is a commutative scalable monoid over a field with a finite basis of invertible elements (unique expansion x = µ · ∏ᵢ eᵢ^(kᵢ)); its monoid of dimensions Q/∼ is then a free abelian group of rank n (Props 3.19–3.20), each dimension is a 1-dimensional vector space (Prop 3.9), and every non-zero quantity is a unit quantity for its dimension (Prop 3.7). Jonsson also never says "torsor", but Definition 2.32's two conditions on a unit element — generating and faithful — are exactly transitivity and freeness of the scalar action on the orbitoid [exposition]. The character transformation law appears as his change-of-basis formula for measures (Prop 3.15):

basis E = {e₁, …, eₙ}   rescaled to   E′ = {λ₁·e₁, …, λₙ·eₙ}     (λᵢ ≠ 0):

    µ_{E′}(x) = ( λ₁^(−k₁) ⋯ λₙ^(−kₙ) ) · µ_E(x)        for  x = µ_E(x) · ∏ᵢ eᵢ^(kᵢ)

— numerically identical to Tao's passive law (1): the measure of a weight-k quantity transforms under the character −k of the rescaling (λ₁,…,λₙ).

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Structural anatomy

Primary structure — objects and morphisms

Each source takes a different structure as primary, and the differences are the content:

• Baez: the primary structure is a group action G × X → X subject to the free-and-transitive axiom. Objects: G-torsors; morphisms (implicit): G-equivariant maps. There is deliberately no algebra on X itself — that absence is the point.
• Tao (parametric): the primary structure is a family indexed by the structure group — a quantity is a function of the parameters (M,L,T) ∈ (ℝ⁺)³, and all operations are applied pointwise per parameter choice. The structure group is explicitly generalisable: GL₃(ℝ) for frame-dependent vectors, E(2) for points vs displacements, the Poincaré group ("the principle of special relativity can be interpreted as the assertion that all physical quantities transform cleanly with respect to this group action"), diffeomorphism and gauge groups. Dimensional analysis is the abelian, rank-n-torus rung of a ladder of representation theories.
• Zapata-Carratala: the primary structure is a surjection δ : A → D ("dimensioned set") and operations classified by how they interact with it — the category DimSet with commutative-square morphisms, then DimRing, DimMod, dimensioned algebras, up to dimensioned Poisson algebras. The group action is recovered (slice-wise multiplication by invertibles), not postulated.
• Jonsson: the primary structure is a variety of universal algebras — one carrier set, operations ∗, 1_X, and a unary ω_λ per ring element, with equations. Orbits, quotients, tensor products, and homomorphisms all come from general universal algebra; the fibration over dimensions is derived as the quotient X → X/∼.

Relation to the tensor-of-lines picture

Tensor of lines and this page formalize the same phenomena from opposite ends. The local sources support three precise bridges between them — no more than these are claimed here:

• The units of one line form a torsor. For Janyška–Modugno–Vitolo's positive spaces the statement is verbatim: "The scalar multiplication s : ℝ⁺ × U → U turns out to be a free and transitive action of the group (ℝ⁺, ·) on the set U" (JMV, Note 2.3 — verified against the local capture). For the signed slices of a dimensioned field it is this page's derivation from Proposition 3.4 above: R_d^× is an R₁^×-torsor. And Tao's closing move says it for his own lines: reinterpret M, L, T "as elements of the ℝ-torsors V^M, V^L, V^T".
• A chosen basis vector of the line = a point of the torsor. Tao refuses to "designate a preferred unit in these spaces (which would identify each of them with ℝ)"; JMV define a unit as "a scale k ∈ S, regarded as a semi-basis of the scale space S" (§3.1, verified locally); Baez picks a torsor point — "declare it to be the identity". These are one act described at three levels of structure [exposition], and Zapata-Carratala's Theorem 4.1 is where it becomes a theorem: a single non-zero u ∈ L^× induces the unit system n ↦ uⁿ and the trivialization ≅ F × ℤ.
• The scaling torus acts on the tensor powers. Two unit systems differ by a unique character c : D → R₁^× (derived above from Prop 3.4), and for D = ℤⁿ the positive characters form (ℝ⁺)ⁿ — the scaling torus acts freely and transitively on unit systems for the power ring of n lines. Parametrically the same torus action is Tao's law (1), which the dictionary theorem on tensor of lines derives from basis change on the lines.

The two pictures are not interchangeable, and what each takes as primary is the genuine difference:

• Carriers first vs action first. The tensor-of-lines picture takes the carriers as primary — lines closed under ⊗ and duals, with no group in the signature; a change of units is, abstractly, not an operation on anything (the gauge freedom appears only once a dictionary to numbers is set up). The torsor/weight picture takes the group and its action as primary — Tao's parametric structure group, Baez's G, Jonsson's scaling action — and recovers the carriers as weight spaces, slices, or orbitoids; Zapata-Carratala goes furthest, deriving even the action from the fibration δ and distributivity.
• Zero and sign. A line contains 0 — Tao is "careful to keep the origins 0 … of each of these vector spaces … distinct from each other" — but no torsor does: the scaling action fixes 0, so freeness fails on any carrier containing it [exposition]. Accordingly units must be non-zero by fiat (u_d ≠ 0_d, Zapata-Carratala §3) or the carrier zero-free from the start (JMV's positive spaces). A line is strictly more structure than its torsor of units: torsor, plus zero, plus slice-wise addition [exposition].
• Is the torsor packaging a third formalization? Tensor of lines deliberately leaves this question to this page, and the answer splits. Within Tao's post, no: he deflates his own torsor variant as a re-parametrisation whose price means "one may as well work entirely in the abstract setting instead". But the action-first developments this page assembles — dimensioned rings, scalable monoids — are free-standing axiomatizations with their own primitives and theorems (trivializations, derived addition), not re-descriptions of the lines; the survey keeps the two pages separate for that reason.

Quantity, unit, dimension, kind

Notion	Tao (parametric)	Zapata-Carratala	Jonsson	Baez
Quantity	family x_{M,L,T} transforming by a power law (weight vector)	element a_d of a slice of a dimensioned ring	element of a quantity space	element of a G-torsor (for origin-like quantities)
Dimension	the weight (a,b,c) — a character of the torus	element d of the dimension monoid D; equivalently the slice R_d	orbitoid [x] — a commensurability class; Q/∼ is free abelian	the group G itself (implicitly: which G acts)
Unit	a reference choice M₀,L₀,T₀; better, a point of a torsor V^M …	a multiplicative section u : D → R, u_d ≠ 0 (may not exist globally)	unit element: a generating, faithful u ∈ C; systems: dense+sparse+closed	a chosen base point of the torsor ("declare it to be the identity")
Kind	— (see below)	— (silent; D is arbitrary, so a finer monoid could split kinds, undeveloped)	same kind ⟺ commensurable — an explicit, radical identification	— (silent)

On kind, Jonsson is the only one of the four to commit, and he commits to the collapse: "we introduce a seemingly more radical idea: quantities are of the same kind if and only if they are commensurable" (Jonsson, §2.3). Under this definition torque and energy (both M L² T⁻² in the classical dimension group) are the same kind — a genuine finding about this formalization's resolution, recorded as a limit below. Tao is formally silent on the word "kind" but supplies the mechanism the others lack: enlarging the structure group splits same-dimension quantities by transformation law — positions vs displacements under the Euclidean group E(2) (laws (8)/(9)), vectors vs covectors under GL₃(ℝ), and onward up the ladder catalogued on tensor of lines. The E(2) case is the one that matters here: position + displacement = position while position + position is of neither kind — the torsor/group distinction (Baez's positions/velocities) re-derived inside the parametric model as a representation distinction.

Dimensional homogeneity as equivariance

Tao states the identification directly — the abstract model admits "only … those operations which are 'dimensionally consistent' or invariant (or more precisely, equivariant) with respect to the action of the underlying structure group" (Tao, introduction). A law relating quantities is dimensionally homogeneous iff it is stated between objects of a single weight and is preserved by every rescaling; a hybrid statement fails to transform "according to a single group action". Equivariance buys the transfer principle: "to verify a dimensionally consistent statement between dimensionful quantities, it suffices to do so for a single choice of the dimension parameters M,L,T" (stated with its proof sketch on tensor of lines) — Tao's worked example being E = mc²: verify it in Planck-style units where c = 1 (where it degenerates to the dimensionally inconsistent E = m), and equivariance propagates it to all unit systems; but having spent the consistency one cannot transfer back (Tao, §1). In Jonsson the same principle appears arithmetically: equal quantities have equal exponent vectors (Lemma 3.6), and "this is the essence of the principle of dimensional homogeneity formulated by Fourier" (Jonsson, §3.2). In Zapata-Carratala homogeneity is enforced syntactically — an expression is well-formed only if all its +-joined terms carry the same dimension subscript, and his notation "reflects the dimensioned structure explicitly so that we can keep track of the consistency of expressions" (§2.2); laws are morphism-level statements in DimRing, automatically slice-respecting. None of the four sources states the Buckingham π theorem in this language; the missing bridge (equivariant functions of quantities are functions of the weight-zero invariants) is exactly what that sibling page's sources supply — recorded here as a silence.

Change of units and its invariants

• Tao: a change of units is a point of the structure group acting passively: the numeral of a weight-d quantity is multiplied by the character −d of the rescaling (equation (1); "the presence of the negative signs in (1) … is describing the effect of a passive change of units rather than an active change of the object"). Invariant: weight-zero quantities — and relations between quantities that are equivariant.
• Zapata-Carratala: a change of units is a change of section u ↦ u′, equivalently a change of trivialization R_D ≅ R₁ × D; between two base lines it is a Line-morphism, a factor ("a unit-free conversion factor"). Invariant: the dimensioned ring R_D itself, its dimensionless slice R₁, and everything functorial in it. All algebraic operations "are compatible with conversion factors that allow change between units of the same kind" (§2.1) — compatibility is built into the definitions.
• Jonsson: a change of units is a change of basis E ↦ E′ = {λᵢ·eᵢ}; measures transform by ∏ᵢ λᵢ^(−kᵢ) (Prop 3.15). Invariant: the measures of the dimensionless orbitoid — "For every x ∈ [1_Q], µ_E(x) does not depend on E" (Prop 3.16), which he flags as the hinge of the π theorem (Remark 3.17).
• Baez: a change of units is the group element itself — the ratio of two torsor points. Invariant: ratios/differences, i.e. the G-valued pairings. His temperature coda adds the twist that both origin and scale can be torsorial at once: "As soon as we pick units of temperature, temperatures are elements of an R-torsor. When absolute zero was discovered, this R-torsor was revealed to be R itself" — and before picking units, temperatures live "on a line whose symmetries include not just translations but also dilations", needing "a more sophisticated concept than that of 'torsor' allowing both translations and dilations whenever you start with a ring."

Addition across dimensions

The four sources give four distinct, precise answers to why quantities multiply freely across dimensions while addition does not — this page reports them without reconciling them (that is comparison's job).

Zapata-Carratala: partial by definition, and distributivity is why multiplication is total. Addition is a dimensional binar — defined only within slices, "otherwise undefined", mirroring practice. The non-obvious theorem-shaped observation is the converse direction: given slice-wise addition, asking for any distributive law forces the multiplication to be a dimensioned binar, i.e. forces dimensions themselves to compose:

"Therefore, if we are to demand distributivity as generally as possible, the multiplicative operation must map transitively between dimension slices, in other words, the dimension of a_d · c_f only depends on d and f." — Zapata-Carratala, §3 (pdftotext extraction, p. 10; subscripts inlined)

So in dimensioned algebra the monoid structure on D — the very existence of a "multiplication of dimensions" — is explained by the partiality of addition plus distributivity, rather than both being posited independently.

Tao: defined, but hybrid — and provably useless rather than ill-formed. In the parametric model everything is a family of reals, so any two quantities can be added pointwise. The sum simply fails to lie in any single weight space: if x and y have different exponent vectors, "the sum x+y or difference x−y, while still defined as a dimensionful quantity, no longrer [sic] has any single dimension" (Tao, §1) — [exposition] in representation-theoretic terms the sum lives in the direct sum of two weight spaces and transforms by no single character. Tao then quantifies exactly how much law-like content hybrids retain — his convex-hull criterion for hybrid inequalities, developed with its AM–GM witness on tensor of lines — whence "one cannot bound a positive quantity of hybrid dimension by a quantity with a single dimension", which "helps explain why we almost never see such hybrid dimensional quantities appear in a physical problem". Addition across dimensions is thus neither forbidden nor meaningless in this model: it is equivariance-breaking, and equivariance is where the physics lives.

Baez: for torsors, addition does not exist at all. The torsor answer applies to the affine layer (energies, positions, dates): "there's no good reason you'd want to add the day of your dentist appointment and Christmas day". Two torsor points subtract to a group element; they do not add. This is an answer about origin-relative quantities of a single dimension, complementary to the exponent-mismatch answers above — a silence worth recording: Baez's essay never discusses adding across dimensions (his G is one group at a time).

Jonsson: addition is not primitive, so there is nothing to forbid. A scalable monoid has no addition in its signature. Sums are constructed (Def 2.35) inside one orbitoid from a unit element, and Lemma 2.34 shows the construction is independent of which unit element is chosen. Across orbitoids the construction has no input data — x + y for incommensurable x, y is not "rejected"; it was never defined. Of the four, this is the most radical position: the partiality of addition is an artefact of addition being derived structure.

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Expressive power & limits

What the picture handles that naive "reals with attached units" cannot:

• Non-canonicity as a theorem. F × ℤᵏ — a numeral and an exponent vector — is exactly the naive representation, and the trivialization theorems say it is correct after a choice of units and canonical never. Everything unit-choice-independent factors through R_D; everything else is bookkeeping about a torsor point. The naive model cannot even state this distinction.
• Affine quantities natively. Temperature zero, voltage ground, energy origin, epochs and dates (ℤ-torsors!), quantum phase (U(1)) — Baez's examples are precisely the quantities the exponent-vector formalisms of Kennedy and Hart leave out. The torsor layer is the formal home for them: point − point = delta, point + delta = point, point + point undefined. Baez's temperature story even captures ontology updates — discovering absolute zero "revealed" the torsor to be the group.
• Kinds via bigger structure groups. Tao's E(2) example distinguishes position from displacement — two same-dimension quantities — by their transformation law, and his Poincaré/diffeomorphism/gauge ladder shows the mechanism scales all the way to "the principle of general relativity" (the ladder is catalogued case by case on tensor of lines). None of the exponent-lattice formalizations has a story of comparable reach.
• Fractional and irrational exponents without strain. A weight is any point of the character group; Tao's analysis quantifies over real exponent vectors, and a reader in the comment thread raises irrational powers from CFT correlation functions — the parametric framework absorbs them by construction, while the constructed lines stop at ℚ (the abstract-side constructions and the open irrational edge are on tensor of lines). In Zapata-Carratala the dimension monoid D is arbitrary, so ℚⁿ or ℝⁿ dimension groups are no harder than ℤⁿ (his §8 asks why physics only ever seems to need ℤ or ℚ — see below).
• Angles, honestly. Weight zero does not mean "is a number". Jonsson's Remark 3.17: "A 'dimensionless quantity' does not correspond to a unique number, but to a number that depends on the choice of a quantity unit for [1_Q]. For example, plane angles can be measured in both radians and degrees" — and in a coherent unit system the unit for [1_Q] is pinned to 1_Q, "for a plane angle 1_Q corresponds to the radian". The dimensionless slice is a 1-dimensional space with a torsor of unit choices like any other slice; coherence, not dimensionlessness, is what makes the radian special.

What it cannot express, or expresses only by silence:

• Logarithmic quantities (dB, pH, magnitudes). None of the four sources treats them — a fourfold silence, recorded as a finding. The picture is suggestive but undeveloped: [exposition] applying log to a positive weight-d quantity turns the multiplicative transformation into an additive one (log x ↦ log x̃ − a·log M − …), i.e. converts an ℝ⁺-torsor into an additive ℝ-torsor whose base-point choice is the reference level (the "re 20 µPa" of dB SPL) — but no local source states this, and Tao's model classifies log of a dimensionful quantity among the hybrid, no-single-dimension objects.
• Same-dimension different-kind distinctions, at the torus rung. With structure group (ℝ⁺)ⁿ and dimension lattice ℤⁿ, torque and energy have equal weight and are indistinguishable; Jonsson's commensurability definition of kind makes the collapse explicit and deliberate. The picture's own remedy (enlarge the group / refine D) is demonstrated by Tao only for the geometric cases; neither he nor Zapata-Carratala develops a torque-vs-energy refinement. Contrast the kind hierarchies that mp-units builds syntactically.
• Hybrid objects break the typing. Sums across weights exist in Tao's model but "do not correspond to any particular object in the abstract setting" short of "taking formal sums of spaces of different dimensionalities" (Tao, closing section — clause (iii) of the dictionary theorem as stated on tensor of lines) — the abstract/torsor side must be extended (direct sums, graded objects) to even name what the parametric side produces freely.
• No canonical exponent lattice. The torus picture works for any ℤⁿ, ℚⁿ, ℝⁿ — and therefore explains none of them. Why physical dimension groups are free abelian of small finite rank is an input from metrology (SI/VIM — concepts), not an output of the representation theory; Zapata-Carratala names this gap explicitly (§8, quoted below).

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Mechanization

No system in this survey implements "torsor" as a first-class abstraction, but the picture's two halves are both mechanized, separately:

• Equivariance as the semantics of unit polymorphism. Kennedy's POPL'97 theorem for the system behind F# units of measure is literally the statement that well-typed programs are equivariant under the scaling torus: "the behaviour of programs is invariant under changes to the units used. We prove this 'dimensional invariance' result" — with changes-of-units modelled as scaling environments; §5's natural first-cut notion assigns a scale factor to each unit variable and extends it "to a homomorphism from the Abelian group of (equivalence classes of) unit expressions into the Abelian group of scale factors", and the paper's more general relational definition is then shown to subsume exactly these homomorphisms ($REPOS/papers/units-of-measure/kennedy-1997-relational-parametricity-units-popl.pdf, abstract and §5). That is the character/weight picture in type-theoretic clothing — developed fully in Kennedy's dimension types and type-system mechanisms.
• Torsor discipline as "point" types. The affine layer ships in several libraries as a quantity/point split with exactly Baez's operation table (point − point = delta, point + delta = point, point + point rejected): mp-units has quantity_point.h and point_origin_concepts.h in src/core/include/mp-units/framework/; Au has au/quantity_point.hh; Pint implements offset units (degC) via OffsetConverter and logarithmic units via LogarithmicConverter in pint/facets/nonmultiplicative/definitions.py; Unitful.jl defines °C/°F through an @affineunit macro and an AffineUnits type (src/pkgdefaults.jl, src/user.jl, src/types.jl). All paths verified in the pinned repos of the survey's grounding map (grounding/_sources.md). An independent rediscovery of the same discipline — a reader's C++ embedding of 1-D vector spaces, duals, tensor products, and 1-dimensional affine spaces under strong typing — is recorded in the Tao comment thread and quoted on tensor of lines.
• Proof assistants. The Lean formalization mechanizes the dimension group (CommGroup (dimension B E) in LeanDimensionalAnalysis/Basic.lean) — the character lattice — but not the torus action or a torsor layer; neither dimensioned rings nor scalable monoids have a known mechanization (a silence of the sources and, as far as this survey's artifacts show, of the ecosystem).

Decision procedure. The torsor/weight picture adds no new decision problem beyond the free-abelian-group layer: equality of dimensions is equality of weights in ℤⁿ (decidable, linear algebra over ℤ), and checking a law's homogeneity is checking that all +-joined terms have equal weight — the same arithmetic every checker in this survey runs, from GNAT's sem_dim to F#'s UnifyMeasures (see free abelian group for the unification story and its complexity). What the torsor layer adds is bookkeeping, not search: origin tags on affine quantities and base-point management in conversions, enforced syntactically by the point types above. Tao's transfer principle is also the semantic licence behind a cheap dynamic strategy: verify an equivariant relation at one unit assignment (e.g. all SI base units set to 1) and conclude it for all — the strategy runtime unit systems like Pint implicitly rely on when they normalise to base units before comparing.

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Open problems & frontier

• Why ℤⁿ (or ℚⁿ)? The dimensioned formalism deliberately allows any dimension monoid, which converts a modelling convention into an open question. Zapata-Carratala, §8: "ordinary physical quantities are always formulated with the particular groups Z or Q as dimension sets … no further justification has been found to single out Z or Q as canonical choices for dimension sets." And the deliberately provocative follow-up: "could it be scientifically justifiable or theoretically useful to consider physical theories whose observable quantities have dimensions displaying some exotic algebraic or topological properties?" (Zapata-Carratala, §8).
• Dimensioned vs graded. Zapata-Carratala's §8 notes that Vysoký's graded-geometry definitions "match our definitions of dimensioned set, dimensioned map and dimensioned abelian group respectively by replacing all our generic dimension sets with Z" — yet the graded tensor product "differs significantly from our general definition of dimensioned tensor product". Whether dimensioned algebra and ℤ-graded algebra are two presentations of one theory or genuinely diverge is posed as "a fruitful line of further research".
• The parametric/abstract trade-off. Tao's closing torsor move (parameters as torsor elements, no reference units) has a cost he states exactly: "one loses some of the power of the parametric model, namely the power to perform numerical operations even if they are dimensionally inconsistent, and so one may as well work entirely in the abstract setting instead" (Tao, closing section; the two settings and the dictionary between them are tensor of lines's subject). Where a mechanized system should sit on this axis — fully abstract (only equivariant operations typecheck) vs parametric with escape hatches (any operation, homogeneity checked separately) — is exactly the design spectrum the comparison capstone maps across libraries.
• Which structure group? Tao's ladder — torus, GL₃, E(2), Poincaré, diffeomorphisms, gauge groups — frames classical dimensional analysis as the smallest of a family of "representation theories of the dimensionful universe". No units library climbs past the torus rung (vector/frame-aware unit checking à la Tao's laws (6)–(9) is unmechanized in every system surveyed); Hart's dimensioned matrices is the nearest theoretical rung. How much of the ladder is worth mechanizing is open.
• Beyond torsors for affine-and-scaled quantities. Baez's final remark observes that pre-unit temperature needs a line with translations and dilations, "because R is not just a group, but a ring. So, there's a more sophisticated concept than that of 'torsor' allowing both translations and dilations whenever you start with a ring" — he names no such concept, and none of this survey's sources supplies one; the library quantity / quantity_point pairs are its untheorised engineering shadow.

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Sources

• J. Baez, "Torsors Made Easy", web essay, December 27, 2009 — torsor definition and axioms, the ratio operation, energy/voltage/phase and dates/notes/antiderivative examples, the non-canonical-isomorphism slogan, principal-bundle fibres, the temperature coda. (Quotes transcribed from the local capture $REPOS/papers/units-of-measure/baez-torsors-made-easy-web.html — the full essay.)
• T. Tao, "A mathematical formalisation of dimensional analysis", What's New, December 29, 2012 — used here for the structure group (ℝ⁺)³, transformation law (1), the weight-space passage, equivariance and the transfer principle, hybrid quantities (in brief), non-toral structure groups (in brief), and the closing torsor modification; the abstract model, the dictionary theorem, and the convex-hull criterion are grounded on tensor of lines. (Quotes verified against the local capture $REPOS/papers/units-of-measure/tao-2012-formalisation-dimensional-analysis-blog.html; formulas restored from the WordPress LaTeX alt attributes.)
• C. Zapata-Carratala, "Dimensioned Algebra: the mathematics of physical quantities", arXiv:2108.08703, 2021 — dimensioned sets/binars/rings/fields (§§2–3), units as sections and the Möbius counterexample, Proposition 3.4 (trivialization), the power functor and Theorem 4.1 (§4), dimensioned modules/algebras/Poisson structures (§§5–7), and the §8 research programme. (Quotes verified against a pdftotext -layout extraction of $REPOS/papers/units-of-measure/zapata-carratala-2021-dimensioned-algebra-arxiv.pdf.)
• D. Jonsson, "Magnitudes, scalable monoids and quantity spaces", arXiv:2108.02106v6, 2021/2023 — Definition 2.1 (scalable monoid), commensurability and orbitoids (§2.3), unit elements and derived addition (§2.6), rank-1 free-module structure (Prop 2.36), quantity spaces and bases (§3.1), measures and the change-of-basis law (Props 3.15–3.16), the free-abelian dimension group (§3.5). (Quotes verified against a pdftotext -layout extraction of $REPOS/papers/units-of-measure/jonsson-2021-magnitudes-scalable-monoids-quantity-spaces-arxiv.pdf.)
• J. Janyška, M. Modugno & R. Vitolo, "Semi-vector spaces and units of measurement", arXiv:0710.1313v1, 2007 — a primary source of tensor of lines, cited on this page only for Note 2.3 (the scalar multiplication of a positive space is "a free and transitive action of the group (ℝ⁺, ·)") and the §3.1 unit-as-semi-basis definition. (Both statements verified against a pdftotext -layout extraction of $REPOS/papers/units-of-measure/janyska-modugno-vitolo-2007-semi-vector-spaces-units-arxiv.pdf; the extraction renders ℝ⁺ as IR+ and sets semi–basis with an en-dash — normalized here.)
• A. Kennedy, "Relational parametricity and units of measure", POPL '97 — dimensional invariance as the semantics of unit-polymorphic typing; cited here from the local author copy $REPOS/papers/units-of-measure/kennedy-1997-relational-parametricity-units-popl.pdf and developed in Kennedy's dimension types.
• Related deep-dives: theory index · units-of-measure umbrella · concepts glossary · tensor of lines · free abelian group · Buckingham π · Kennedy's dimension types · Hart's multidimensional analysis · type-system mechanisms · comparison · system pages: F# · mp-units · Au · Pint · Unitful.jl · GNAT · Lean.