Index of the slides:

Ring objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A ring object: the tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Another ring object: a ring of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A homomorphism between ring objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Ring objects of line type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Lemma: the map β is known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ring objects (ℝ, 0, 1, +, ·, −) can be seen as a “ring object” in Set, that is, as five functions from powers of ℝ to ℝ, one for each operation:

[figure: 1; ℝ; 0; 1; ℝ; 1; ℝ; ℝ^{2}; +; ℝ; ℝ^{2}; ·; ∗; 0; ∗; 1; a + b; a, b; ab; a, b]

(we will never draw the subtraction − : ℝ → ℝ). These arrows must obey some equations — for example, (a + b)c = ac + bc, that becomes:

\begin{matrix}\begin{matrix}{a, b, c \atop a} & {a, b, c \atop b} \\ a + b \end{matrix} & {a, b, c \atop c} \\ (a + b)c \end{matrix} = \begin{matrix}\begin{matrix}{a, b, c \atop a} & {a, b, c \atop c} \\ ac \end{matrix} & \begin{aligned}{a, b, c \atop b} & & {a, b, c \atop c} \\ & bc \end{aligned} \\ ac + bc \end{matrix}

\begin{matrix}\begin{matrix}{id \atop π_{1}} & {id \atop π_{2}} \\ ⟨π_{1}, π_{2}⟩; + \end{matrix} & {id \atop π_{3}} \\ ⟨(⟨π_{1}, π_{2}⟩; +), π_{3}⟩; · \end{matrix} = \begin{matrix}{id \atop π_{1}} & {id \atop π_{3}} \\ ⟨π_{1}, π_{3}⟩; · \end{matrix} \begin{matrix}{id \atop π_{2}} & {id \atop π_{3}} \\ ⟨π_{2}, π_{3}⟩; · \end{matrix} ⟨(⟨π_{1}, π_{3}⟩; ·), (⟨π_{2}, π_{3}⟩; ·)⟩; +

As Set has finite products every (ℝ, 0, 1, +, ·, −)-polynomial in n variables can be represented as a morphism ℝ^{n} → ℝ; each of the ring axioms becomes the statement that two “(ℝ, 0, 1, +, ·, −)-polynomials” are equal.

A ring object: the tangent space The tangent space of ℝ, T ℝ, has the same points as ℝ^{2}, and a ring structure, with special definitions for ‘1’ and ‘·’. We will denote its points as (a, a_{x}), (b, b_{x}), … Here is its ring structure:

[figure: 1; T ℝ; 0; 1; T ℝ; 1; T ℝ; (T ℝ)^{2}; +; T ℝ; (T ℝ)^{2}; ·; ∗; (0, 0); ∗; (1, 0); (a + b, a_{x} + b_{x}); (a, a_{x}), (b, b_{x}); (ab, a_{x}b + b_{x}a); (a, a_{x}), (b, b_{x})]

Another ring object: a ring of functions For any set S the space of functions S → ℝ (a.k.a. “ℝ^{S}”) has a natural ring structure:

[figure: 1; (S → ℝ); 0; 1; (S → ℝ); 1; (S → ℝ); (S → ℝ)^{2}; +; (S → ℝ); (S → ℝ)^{2}; ·; ∗; (s ↦ 0); ∗; (s ↦ 1); (s ↦ a[s] + b[s]); (s ↦ a[s], (s ↦ b[s]); (s ↦ a[s]b[s]); (s ↦ a[s], (s ↦ b[s])]

If S ⊆ ℝ then some functions S → ℝ are “affine linear”, in the sense that they can be characterized by two reals — a “constant part” (‘a’) and a “slope” (‘a_{x}’).

Let’s write these functions as s ↦ a + a_{x}s.

Then the set of affine linear functions in S → ℝ is \underline{almost} closed by the ring operations — the only problem is the second-order term in the result of the multiplication (underlined below):

[figure: ∗; (s ↦ 0); ∗; (s ↦ 1); (s ↦ a + b + (a_{x} + b_{x})s); (…), (…); (s ↦ ab + (a_{x}b + ab_{x})s + \underline{a_{x}b_{x}s^{2}}); (…), (…)]

However, if S ⊆ { x ∈ ℝ | x^{2} = 0 } then the second-order term disappears, and the set of affine linear functions

AffLin(S → R) := { f : S → ℝ | f is affine linear } ⊆ (S → ℝ)

is a subring of S → ℝ, and, furthermore, there is a ring homeomorphism φ : T ℝ → (S → ℝ)...

A homomorphism between ring objects “φ : T ℝ → (S → ℝ) is a ring homomorphism” means that for each of the five operations, 0, 1, +, ·, −, a certain square commutes...

[figure: 1; T ℝ; 0; 1; T ℝ; 1; T ℝ; T ℝ × T ℝ; +; T ℝ; T ℝ × T ℝ; ·; 1; (S → ℝ); 0; 1; (S → ℝ); 1; (S → ℝ); (S → ℝ) × (S → ℝ); +; (S → ℝ); (S → ℝ) × (S → ℝ); ·; 1; 1; id; T ℝ; (S → ℝ); φ; T ℝ × T ℝ; (S → ℝ) × (S → ℝ); φ×φ]

(We do not draw the ‘−’ arrows). The less trivial case is the square for ‘·’:

[figure: (ab, a_{x}b + ab_{x}); (a, a_{x}), (b, b_{x}); (ab, a_{x}b + ab_{x}); (s ↦ ab + (a_{x}b + ab_{x})s); (a, a_{x}), (b, b_{x}); (s ↦ a + a_{x}s), (s ↦ b + b_{x}s); (s ↦ ab + (a_{x}b + ab_{x})s + \underline{a_{x}b_{x}s^{2}}); (s ↦ a + a_{x}s), (s ↦ b + b_{x}s)]

As we are supposing that S ⊆ { x ∈ ℝ | x^{2} = 0 }, the term a_{x}b_{x}s^{2} is zero, and that square commutes.

In ℝ the set of square-zero elements, { x ∈ ℝ | x^{2} = 0 }, is too small for this to be interesting — but the same constructions work for any ring R.

Example: R := ℝ[X, Y]/⟨X^{2}, Y^{2}⟩ — the ring of polynomials on two variables, ‘X’ and ‘Y ’, with coefficients on ℝ, divided by an ideal to force X^{2} = 0 and Y^{2} = 0.

Notational convention: ϵ^{2} = 0 and δ^{2} = 0. Then, using ‘ϵ’ and ‘δ’ as variables, we can write just “ℝ[ϵ, δ]” instead of “ℝ[ϵ, δ]/⟨ϵ^{2}, δ^{2}⟩”.

Note that (ϵ + δ)^{2} = ϵ^{2} + 2ϵδ + δ^{2} = 2ϵδ ≠ 0 — so ϵ + δ is not a square-zero element in ℝ[ϵ, δ].

Ring objects of line type Fact (a.k.a. “Main Theorem”, proved in the next slides): When the arrow α below is invertible we can use the composite γ := (α^{−1}; π_{2}) to define, for any f : R → R, its derivative f′ : R → R, and these derivatives behave as expected:

\begin{aligned}(kf)′ & = & kf′ \\ (f + g)′ & = & f′ + g′, \\ (fg)′ & = & f′g + fg′, \\ (f ∘ g)′ & = & (f′ ∘ g) g′. \end{aligned}

[figure: R^{D}; R; β; R^{D}; R × R; α; R^{D}; R × R; α^{−1}; R^{D}; R; γ; R; R × R; π_{1}; R × R; R; π_{2}; (dx ↦ a + a_{x}dx); a; β; (dx ↦ a + a_{x}dx); a, a_{x}; α; (dx ↦ a + a_{x}dx); a, a_{x}; α^{−1}; (dx ↦ a + a_{x}dx); a_{x}; γ; a; a, a_{x}; π_{1}; a, a_{x}; a_{x}; π_{2}]

The hypotheses are just these: C is a category with finite limits, (R, 0, 1, +, ·, −) is a ring object in C, and D := { dx ∈ R | dx^{2} = 0 } (that is definable as an equalizer) is exponentiable.

(Stronger hypotheses, simpler to understand: C is cartesian closed and has pullbacks, (R, 0, 1, +, ·, −) is a ring object in C.)

Then if the (definable) map α : R × R → R^{D} is invertible, we have a notion of “derivative” for functions R → R, that behaves as expected.

A ring (R, 0, 1, +, ·, −) for which α : R × R → R^{D} is invertible is said to be “of line type”.

Lemma: the map β is known Lemma: even when α^{−1} does not exist β is known... More precisely: define β as “evaluate dx ↦ a + a_{x}dx at dx := 0”; then (α; β) = π_{1}. If α^{−1} exists then (α; β) = π_{1} iff β = (α^{−1}; π_{1}).