# nLab Kock-Lawvere axiom

### Context

#### Synthetic differential geometry

differential geometry

synthetic differential geometry

topos theory

# 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 $TX\to X$ 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 $D=\left\{x\in R\mid {x}^{2}=0\right\}$ the infinitesimal interval the canonical map

$R×R\to {R}^{D}$R \times R \to R^D

given by

$\left(x,d\right)↦\left(ϵ↦x+dϵ\right)$(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 $R\oplus ϵR:=\left(R×R,\cdot ,+\right)$ in $T$, whose underlying object is $R×R$, with addition $\left(x,q\right)+\left(x\prime ,q\prime \right):=\left(x+x\prime ,q+q\prime \right)$ and multiplication $\left(x,q\right)\cdot \left(x\prime ,q\prime \right)=\left(xx\prime ,xq\prime +qx\prime \right)$.

For $A$ an algebra object in $T$, write ${\mathrm{Spec}}_{R}\left(A\right):={\mathrm{Hom}}_{R\mathrm{Alg}\left(T\right)}\left(A,R\right)\subset {R}^{A}$ for the object of $R$-algebra homomorphisms from $A$ to $R$.

Then one checks that

$D=\mathrm{Spec}\left(R\oplus ϵR\right)\phantom{\rule{thinmathspace}{0ex}}.$D = Spec(R \oplus \epsilon R) \,.

The element $q\in D\subset R$, ${q}^{2}=0$ corresponds to the algebra homomorphism $\left(a,d\right)↦a+qd$.

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

$R\oplus ϵR\to {R}^{{\mathrm{Spec}}_{R}\left(R\oplus ϵR\right)}$R \oplus \epsilon R \to R^{Spec_R(R \oplus \epsilon R)}

is an isomorphism.

Notice that $\left(R\oplus ϵR\right)$ 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

$W\to {R}^{{\mathrm{Spec}}_{R}\left(W\right)}$W \to R^{Spec_R(W)}

is an isomorphism.

Revised on October 3, 2012 23:17:41 by Urs Schreiber (82.169.65.155)