nLab differential forms in synthetic differential geometry

Idea

In the context of synthetic differential geometry a differential form $\omega$ of degree $k$ on a manifold $X$ is literally a function on the space of infinitesimal $k$-cubes or infinitesimal $k$-simplices in $X$.

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

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

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

A differential $k$-form (often called simplicial $k$-form or, less accurately, combinatorial $k$-form to distinguish it from similar but cubical definitions) on $X$ 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

• Moerdijk-Reyes, Models for Smooth Infinitesimal Analysis

for a realization of the cubical version in models based on sheaves on generalized smooth algebras.

We may characterize the object $\Omega^k(X) \subset C^\infty(X^{\Delta^k_{inf}})$ as follows:

for $k \geq 1$ there are the obvious images

$$s_i^* : C^\infty(X^{\Delta^{k}_{inf}}) \to C^\infty(X^{\Delta^{k-1}_{inf}})$$

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

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

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

coboundary operator

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

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

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

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

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

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

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

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

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

References

• Anders Kock, Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010) [pdf, doi:10.1017/CBO9780511691690]

• Ieke Moerdijk, Gonzalo E. Reyes: Models for Smooth Infinitesimal Analysis, Springer (1991) [doi:10.1007/978-1-4757-4143-8]

• J. L Castiglioni, G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence, J. Pure Applied Algebra 191 (2004) 119-142 [arXiv:math/0306289, doi:10.1016/j.jpaa.2003.11.009]