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”.
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
given by
is an isomorphism.
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
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
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
is an isomorphism.
Anders Kock, section I.12 of Synthetic differential geometry, Cambridge University Press, London Math. Society Lecture Notes Series No. 333 (1981, 2006) (pdf)
Anders Kock, section 1.3 of Synthetic geometry of manifolds, Cambridge Tracts in Mathematics, 180 (2010) (pdf)