### Context

#### Synthetic differential geometry

#### Topos Theory

*Could not include topos theory - contents*

# Kock–Lawvere axiom

## Idea

The Kock–Lawvere axiom is the crucial axiom for the theory of synthetic differential geometry.

Imposed on a topos equipped with an internal algebra object $R$ over an internal ring object $k$, the Kock–Lawvere axiom says essentially that morphisms $D \to R$ from the infinitesimal interval $D \subset R$ into $R$ are necessarily *linear* maps, in that they always and uniquely extend to linear maps $R \to R$.

This linearity condition is what in synthetic differential geometry allows to identify the tangent bundle $T X \to X$ of a space $X$ with its fiberwise linearity by simply the internal hom object $X^D \to X$.

Put the other way round, the Kock–Lawvere axiom axiomatizes the familiar statement that “to first order every smooth map is linear”.

## Details

### KL axiom for the infinitesimal interval

The plain Kock–Lawevere axiom on a ring object $R$ in a topos $T$ is that for $D = \{x \in R| x^2 = 0\}$ the infinitesimal interval the canonical map

$R \times R \to R^D$

given by

$(x,d) \mapsto (\epsilon \mapsto x + d \epsilon)$

is an isomorphism.

### KL axiom for spectra of internal Weil algebras

We can consider the internal $R$-algebra object $R \oplus \epsilon R := (R \times R, \cdot, +)$ in $T$, whose underlying object is $R \times R$, with addition $(x,q)+(x',q'):=(x+x',q+q')$ and multiplication $(x, q ) \cdot (x', q') = (x x',x q ' + q x')$.

For $A$ an algebra object in $T$, write $Spec_R(A) := Hom_{R Alg(T)}(A,R) \subset R^A$ for the object of $R$-algebra homomorphisms from $A$ to $R$.

Then one checks that

$D = Spec(R \oplus \epsilon R)
\,.$

The element $q \in D \subset R$, $q^2 = 0$ corresponds to the algebra homomorphism $(a,d) \mapsto a + q d$.

Using this, we can rephrase the standard Kock–Lawvere axiom by saying that the canonical moprhism

$R \oplus \epsilon R \to R^{Spec_R(R \oplus \epsilon R)}$

is an isomorphism.

Notice that $(R \oplus \epsilon R)$ is a Weil algebra/Artin algebra:

an $R$-algebra that is finite dimensional and whose underlying ring is a local ring, i.e. of the form $W = R \oplus m$, where $m$ is a maximal nilpotent ideal finite dimensional over $R$.

Then the general version of the Kock–Lawvere axiom for all Weil algebras says that

For all Weil algebra objects $W$ in $T$ the canonical morphism

$W \to R^{Spec_R(W)}$

is an isomorphism.