nLab
Kock-Lawvere axiom

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Topos Theory

          topos theory

          Background

          Toposes

          Internal Logic

          Topos morphisms

          Extra stuff, structure, properties

          Cohomology and homotopy

          In higher category theory

          Theorems

          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={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.

          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.

          References

          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 November 4, 2017 at 19:51:44. See the history of this page for a list of all contributions to it.