nLab
differential forms in synthetic differential geometry

Context

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

tangent cohesion

differential cohesion

graded differential 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& 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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Content

Idea

In the context of synthetic differential geometry a differential form ω\omega of degree kk on a manifold XX is literally a function on the space of infinitesimal cubes or infinitesimal simplices in XX.

We give the definition as available 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.

Definition

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.

differential forms

An infinitesimal kk-simplex in a synthetic differential space XX is a collection of k+1k+1-points in XX that are pairwise infinitesimal neighbours.

The spaces X Δ diff kX^{\Delta^k_{diff}} of infinitesimal kk-simplices arrange to form the infinitesimal singular simplicial complex X Δ diff X^{\Delta^\bullet_{diff}}.

The functions on the space of infinitesimal kk-simplices form a generalized smooth algebra C (X Δ inf k)C^\infty(X^{\Delta^k_{inf}}).

A differential kk-form (often called simplicial kk-form or, less accurately, combinatorial kk-form to distinguish it from similar but cubical definitions) on XX 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 simplices.

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 Ω k(X)C (X Δ inf k)\Omega^k(X) \subset C^\infty(X^{\Delta^k_{inf}}) as follows:

for k1k \geq 1 there are the obvious images

s i *:C (X Δ inf k)C (X Δ inf k1) s_i^* : C^\infty(X^{\Delta^{k}_{inf}}) \to C^\infty(X^{\Delta^{k-1}_{inf}})

of the degeneracy maps. As one can see, these act by restricting a function on infinitesimal kk-simplices to the degenerate ones and regarding these then as a (k1)(k-1)-simplex.

Therefore we may characterize the subobject Ω k(X)C (X Δ inf k)\Omega^k(X) \hookrightarrow C^\infty(X^{\Delta^k_{inf}}) as the joint kernel of the degeneracy maps

Ω k(X)= i=0 k1ker(s i *). \Omega^k(X) = \cap_{i = 0}^{k-1} ker(s_i^*) \,.

coboundary operator

According to section 3.2 of Andres Kock’s book, the coboundary operator d:Ω k(X)Ω k+1(X)d : \Omega^k(X) \to \Omega^{k+1}(X) sends a differential kk-form ω\omega to the (k+1)(k+1)-form dωd \omega that on an infinitesimal (k+1)(k+1)-simplex (x 0,x 1,,x k+1)(x_0, x_1, \cdots, x_{k+1}) in XX evaluates to

dω(x 0,x 1,,x k+1):= i=0 k+1ω(x 1,,x i^,,x k+1), d\omega(x_0, x_1, \cdots, x_{k+1}) := \sum_{i=0}^{k+1} \omega(x_1, \cdots , \hat{x_i}, \cdots, x_{k+1}) \,,

where the hat indicates that the corresponding variable is omitted, as usual.

We recognize this as the alternating sum of the face maps i *\partial_i^* of the cosimplicial object C (X Δ inf )C^\infty(X^{\Delta_{inf}^\bullet}).

d:= i=0 k+1 i *:Ω k(X)Ω k+1(X). d := \sum_{i=0}^{k+1} \partial_i^* : \Omega^k(X) \to \Omega^{k+1}(X) \,.

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

Proposition

Let XX be a synthetic differential space and C (X Δ inf )C^\infty(X^{\Delta_{inf}^\bullet}) the cosimplicial object of generalized smooth algebras of functions on the spaces of infinitesimal kk-simplices in XX.

Then the deRham complex (Ω (X),d)(\Omega^\bullet(X), d) of differential forms on XX is the normalized Moore complex of the cosimplicial object C (X Δ inf )C^\infty(X^{\Delta_{inf}^\bullet}).

In other words, in as far as the Dold-Kan correspondence is an equivalence, we find that:

the object of differential forms on XX is the cosimplicial generalized smooth algebra C (X Δ inf k)C^\infty(X^{\Delta_{inf}^k}).

References

Revised on May 31, 2017 00:31:18 by Elves? (172.222.163.170)