The infinitesimal singular simplicial complex of a space $X$ in a smooth topos $(\mathcal{T},R)$ is the infinitesimal analogue of the singular simplicial complex $X^{\Delta_R^\bullet}$ (see interval object) that in degree $k$ is the space of $k$-simplices $\Delta^k_R \to X$ in $X$: the infinitesimal singular simplicial complex has in degree $k$ the infinitesimal $k$-simplices in $X$.
There are several ways to make the notion of “infinitesimal $k$-simplex in $X$” precise. Here we describe a notion promoted by Anders Kock, where an “infinitesimal $k$-simplex” in $X$ for $X$ a suitably locally linear space , is a $(k+1)$-tuple $(x_0,\cdots, x_k) \in X^{\times^{k+1}}$ of points in $X$ that are pairwise infinitesimal neighbours in $X$.
One central application of the singular simplicial complex is in the definition of differential forms in synthetic differential geometry.
The basic definition applies to spaces of the form $R^n$ and is generalized from there to spaces that “locally look like” $R^n$ in one way or other.
Let here and in the following $(\mathcal{T},R)$ be a smooth topos.
Write, as usual
for the infinitesimal space of first order infinitesimal neighbours of the origin of $R^n$, with its canonical inclusion into $R^n$.
Two elements $x , y \in R^n$ are called first order infinitesimal neighbours, denoted $x \sim_1 y$, if their difference is in the image of this inclusion.
Write
This naturally forms a simplicial object $X^{\Delta_{inf}^{bullet}} : \Delta^{op} \to \mathcal{T}$. This is the infinitesimal simplicial singular complex of $R^n$.
A more detailed discussion of this is in the entry infinitesimal object in the section Spaces of infinitesimal simplices.
warning this section is as such not drawn from the literature, it seems
Recall that a smooth locus in $\mathcal{T}$ is an object $\ell A$ that is the joint limit over some
here $J \subset Hom_{\mathcal{T}}(R^n,R)$ is an ideal.
Declare that two generalized elements $x,y \in \ell A$ are infinitesimal neighbours if their image under the injection
is a pair of infinitesimal neighbour in $R^n$. Then let
be the sub-simplicial object of infinitesimal neighbours in $R^n$ that are points in $\ell A$.
(linearity of space of infinitesimal neighbours)
If $p, q \in \ell A$ are infinitesimal neighbours in the smooth locus $\ell A \subset R^n$, then for all $t \in R$ also the element $p + t(q-p)$ formed by linear combination in $R^n$ is in $\ell A$ and hence is an infinitesimal neighbour of $p$ there.
Because by the Kock-Lawvere axiom valid in the smooth topos $(\mathcal{T},R)$ we have for all $f : R^n \to R$
Therefore also
Consider the circle $S^1$ regarded as the smooth locus $S^1 = \{(x, y) \in R^2 | x^2 + y^2 = 1\}$.
For $a \in S^1 \subset R^2$ an infinitesimal neighbour $(a + \epsilon)$ in $R^2$ is again a point on the circle, and hence an infinitesimal neighbour of $a$ in $S^1$, if
which, due to $a_x^2 + a_y^2 = 1$ is equivalent to
This is solved by $\epsilon$ of the form
for some fixed $\delta \in D$.
use that each manifold is locally isomorphic to an $R^n$ and that the neighbourhood relation only needs an infinitesimal neighbourhood. Proceed locally as above and then patch. See references below.
The lined topos $(\mathcal{T}, R)$ also comes canonically for every object $X \in \mathcal{T}$ with the finite singular simplicial complex $\Pi(X) : [n] \mapsto X^{R^n}$ induced from regarding
as an interval object (see there for details).
(inclusion of infinitesimal into finite simplices)
For $\ell A =: X \hookrightarrow R^n$ a smooth locus define for all $n \in \mathbb{N}$ a morphism
by defining it on generalized elements as
The morphisms $\iota_n$ constitute a morphism of simplicial objects
in that they respects the face and degenracy maps on each side.
Straightforward checking:
For instance
The inner face maps $d_i$ on $X^{\Delta_{inf}^{k}}$ omit the $i$th point in the $(k+1)$-tuple of points, while on $X^{\Delta^{k}}$ they act by pullback along $(t_1, \cdots, t_k) \mapsto (t_1, \cdots, t_{i-1}, t_{i}, t_i, t_{i+1}, cdots, t_k)$. That means that in the sum above $t_i$ appears twice to yield
which indeed corresponds to omission of the $i$th point $x^i$.
The collection of first order infinitesimal neighbours of a space $X$ arranges itself into the infinitesimal path ∞-groupoid? $\Pi^{inf}(X)$. Various concepts derive from this one:
of differential forms may be understood in terms of functions on $\Pi(x)^{inf}$. This is described at
A deRham space is the colimit over a $\Pi^{inf}(X)$.
In the language of synthetic differential geometry the infinitesimal singular complex for “formal manifolds” (internally defined manifolds with an infinitesimal thickening to all orderes) is described (with the simplicial structure not made explicit) in
section I.18 of
and in section 2.8 of
Discussion of this that does make the simplicial structure explicit and relates it to the Dold-Kan correspondence is in
Herman Stel, ∞-Stacks and their Function Algebras – with applications to ∞-Lie theory.
Herman Stel, Cosimplicial C-infinity rings and the de Rham complex of Euclidean space (arXiv:1310.7407)
The details of what $X^{\Delta^k_{inf}}$ is like concretely on representables in the smooth topos $PSh(k-Alg^{op})$ of algebraic geometry, i.e. on affine schemes is worked out in detail in
The formulas given there should more or less directly carry over to smooth toposes with smooth loci by replacing ordinary rings with smooth algebras.
to be discussed
As the title suggests, the infinitesimal singular simplicial complex is tightly related to differential forms in synthetic differential geometry: the deRham complex is the normalized Moore cochain complex of the cosimplicial algebra $C^\infty(X^{\Delta^\bullet_{inf}})$ of functions on the spaces of infinitesimal simplices.
There is also
Dubuc, Kock, On 1-form classifiers , Communications in Algebra 12 (1984)
Dubuc, $C^\infty$-schemes, Amer. J. of Math. 103 (1981)
Kumpera, Spencer, Lie Equations , Annals of Math. Studies 73 (1973)
There is also a version of the infinitesimal singular simplicial context in the context of nonstandard analysis. See