nLab Kock-Lawvere axiom

KockLawvere axiom


Synthetic differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


Kock–Lawvere axiom


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”.


KL axiom for the infinitesimal interval

The plain Kock–Lawevere axiom on a ring object RR in a topos TT is that for D={xR|x 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+ϵd) (x,d) \mapsto (\epsilon \mapsto x + \epsilon d)

is an isomorphism of objects in TT.

KL axiom for spectra of internal Weil algebras

We can consider the internal RR-algebra object RϵR(R×R,,+)R \oplus \epsilon R \coloneqq (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') \coloneqq (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) \coloneqq 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 morphism

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 of objects in TT. Note that because the canonical morphism is also a homomorphism of RR-algebras, and inverses of algebra homomorphisms are algebra homomorphisms, this gives an isomorphism of internal RR-algebras.


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)

Last revised on April 4, 2023 at 21:37:43. See the history of this page for a list of all contributions to it.