# nLab Kock-Lawvere axiom

KockLawvere axiom

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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 $T X \to X$ 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

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

is an isomorphism.

### KL axiom for spectra of internal Weil algebras

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

For $A$ an algebra object in $T$, write $Spec_R(A) \coloneqq Hom_{R Alg(T)}(A,R) \subset R^A$ 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 morphism

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

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.

The Kock-Lawvere axiom was introduced in

• Anders Kock, A simple axiomatics for differentiation, Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 (JSTOR)

Textbook accounts are in

• 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)