**synthetic differential geometry
* smooth topos
* infinitesimal space
* amazing right adjoint
* Kock-Lawvere axiom
* integration axiom
* microlinear space
* synthetic differential supergeometry
* super smooth topos
* . infinitesimal cohesion
* de Rham space
* formally smooth morphism
, formally unramified morphism
formally étale morphism
* jet bundle
## Models ##
* Models for Smooth Infinitesimal Analysis
* Fermat theory
* smooth algebra
* smooth locus
* smooth manifold
, formal smooth manifold
, derived smooth manifold
* smooth space
, diffeological space
, Frölicher space
* smooth natural numbers
* Cahiers topos
* smooth ∞-groupoid
* synthetic differential ∞-groupoid
* tangent bundle
* vector field
, tangent Lie algebroid
, chain rule
* differential forms
* differential equation
, variational calculus
* Euler-Lagrange equation
, de Donder-Weyl formalism
, variational bicomplex
, phase space
* connection on a bundle
, connection on an ∞-bundle
* Riemannian manifold
, Killing vector field
* Hadamard lemma
* Borel's theorem
* Boman's theorem
* Whitney extension theorem
* Steenrod-Wockel approximation theorem
* Poincare lemma
* Stokes theorem
* de Rham theorem
* Chern-Weil theory
* Lie theory
, ∞-Lie theory
* Chern-Weil theory
, ∞-Chern-Weil theory
* gauge theory
* ∞-Chern-Simons theory
* Klein geometry
, Klein 2-geometry
, higher Klein geometry
* Euclidean geometry
, Cartan geometry
, higher Cartan geometry
* Riemannian geometry
In the context of synthetic differential geometry a differential form of degree on a manifold is literally a function on the space of infinitesimal cubes or infinitesimal simplices in .
We give the definition as availalable in the literature and then interpret this in a more unified way in terms of the Chevalley-Eilenberg algebra of the infinitesimal singular simplicial complex.
missing here are details on what axioms the space we are working on has to satisfy for the following to make sense. See the case distinction at infinitesimal singular simplicial complex.
An infinitesimal -simplex in a synthetic differential space is a collection of -points in that are pairwise infinitesimal neighbours.
The spaces of infinitesimal -simplices arrange to form the infinitesimal singular simplicial complex .
The functions on the space of infinitesimal -simplices form a generalized smooth algebra .
A differential -form (oftent called simplicial -form or, les accurately, combinatorial -form to distinguish it from similar but cubical definitions) on is an element in this function algebra that has the property that it vanishes on degenerate infinitesimal simplices.
See definition 3.1.1 in
- Anders Kock, Synthetic geometry of manifolds (pdf)
for this simplicial definition. A detailed account of this is in the entry infinitesimal object in the section Spaces of infinitesimal simplicies.
This is a very simple-looking statement. The reason is the topos-theoretic language at work in the background, which takes care that we may talk about infinitesimal objects as if they were just plain ordinary sets. For a very detailed account of how the above statement is implemented concretely in terms of concrete models for synthetic differential spaces see section 1 of
- Breen, Messing, Combinatorial differential forms (arXiv)
There are also cubical variants of the above definition
- Anders Kock, Cubical version of combinatorial differential forms (pdf for fee)
See also section 4.1 of
for a realization of the cubical version in models based on sheaves on generalized smooth algebras.
We may characterize the object as follows:
for there are the obvious images
of the degeneracy maps. As one can see, these act by restricting a function on infinitesimal -simplices to the degenerate ones and regarding these then as a -simplex.
Therefore we may characterize the subobject as the joint kernel of the degeneracy maps
According to section 3.2 of Andres Kock’s book, the coboundary operator sends a differential -form to the -form that on an infinitesimal -simplex in evaluates to
where the hat indicates that the corresponding variable is omitted, as usual.
We recognize this as the alternating sum of the face maps of the cosimplicial object .
These constructions remind one and should be compared with the Dold-Kan correspondence. In particular with its dual (cosimplicial) version as recalled in section 4 of CastiglioniCortinas
In total this should show the following
Let be a synthetic differential space and the cosimplicial object of generalized smooth algebras of functions on the spaces of infinitesimal -simplices in .
Then the deRham complex of differential forms on is the normalized Moore complex of the cosimplicial object .
In other words, in as far as the Dold-Kan correspondence is an equivalence, we find that:
the object of differential forms on is the cosimplicial generalized smooth algebra .