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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 $D=\{x\in R\mid x^2=0\}$ the infinitesimal interval the canonical map

$$R\times R\to R^D$$R \times R \to R^D

given by

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

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

Then one checks that

$$D=\mathrm{Spec}(R\oplus ϵR).$$D = Spec(R \oplus \epsilon R) \,.

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

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

$$R\oplus ϵR\to R^{{\mathrm{Spec}}_R(R\oplus ϵR)}$$R \oplus \epsilon R \to R^{Spec_R(R \oplus \epsilon R)}

is an isomorphism.

Notice that $(R\oplus ϵR)$ 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

$$W\to R^{{\mathrm{Spec}}_R(W)}$$W \to R^{Spec_R(W)}

is an isomorphism.