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Kock-Lawvere axiom

Context

Synthetic differential geometry

differential geometry

synthetic differential geometry

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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 RR over an internal ring object kk, the Kock–Lawvere axiom says essentially that morphisms DRD \to R from the infinitesimal interval DRD \subset R into RR are necessarily linear maps, in that they always and uniquely extend to linear maps RRR \to R.

This linearity condition is what in synthetic differential geometry allows to identify the tangent bundle TXXT X \to X of a space XX with its fiberwise linearity by simply the internal hom object X DXX^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 RR in a topos TT is that for D={xRx 2=0}D = \{x \in R| x^2 = 0\} the infinitesimal interval the canonical map

R×RR D R \times R \to R^D

given by

(x,d)(ϵx+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 RR-algebra object RϵR:=(R×R,,+)R \oplus \epsilon R := (R \times R, \cdot, +) in TT, whose underlying object is R×RR \times R, with addition (x,q)+(x,q):=(x+x,q+q)(x,q)+(x',q'):=(x+x',q+q') and multiplication (x,q)(x,q)=(xx,xq+qx)(x, q ) \cdot (x', q') = (x x',x q ' + q x').

For AA an algebra object in TT, write Spec R(A):=Hom RAlg(T)(A,R)R ASpec_R(A) := Hom_{R Alg(T)}(A,R) \subset R^A for the object of RR-algebra homomorphisms from AA to RR.

Then one checks that

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

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

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

RϵRR Spec R(RϵR) R \oplus \epsilon R \to R^{Spec_R(R \oplus \epsilon R)}

is an isomorphism.

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

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

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

For all Weil algebra objects WW in TT the canonical morphism

WR Spec R(W) W \to R^{Spec_R(W)}

is an isomorphism.

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