synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 over an internal ring object , the Kock–Lawvere axiom says essentially that morphisms from the infinitesimal interval into are necessarily linear maps, in that they always and uniquely extend to linear maps .
This linearity condition is what in synthetic differential geometry allows to identify the tangent bundle of a space with its fiberwise linearity by simply the internal hom object .
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 in a topos is that for the infinitesimal interval the canonical map
given by
is an isomorphism of objects in .
We can consider the internal -algebra object in , whose underlying object is , with addition and multiplication .
For an algebra object in , write for the object of -algebra homomorphisms from to .
Then one checks that
The element , corresponds to the algebra homomorphism .
Using this, we can rephrase the standard Kock–Lawvere axiom by saying that the canonical morphism
is an isomorphism.
Notice that is a Weil algebra/Artin algebra: an -algebra that is finite dimensional and whose underlying ring is a local ring, i.e. of the form , where is a maximal nilpotent ideal finite dimensional over .
Then the general version of the Kock–Lawvere axiom for all Weil algebras says that
For all Weil algebra objects in the canonical morphism
is an isomorphism of objects in . Note that because the canonical morphism is also a homomorphism of -algebras, and inverses of algebra homomorphisms are algebra homomorphisms, this gives an isomorphism of internal -algebras.
The Kock-Lawvere axiom was introduced in
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.