I know that there exist differential forms on $\mathbb R^2$ which are closed but not exact. For example, the form $f = \frac{-y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2}{dy}$.
I now want an example of a discrete differential form, as defined on a possibly infinite simplex which shows the same behaviour.
I tried "triangulating" the above form in $\mathbb{R}^2$ in some reasonable way, but I was unable to do so.
Book on discrete differential geometry for the relevant definitions
On
Here, we have a simplicial complex with vertices $V$, edges $E$ and a single face $F$. the shaded area is a face, and the edges have the specified oriententation.
We have a form $f: E \rightarrow \mathbb R$ that is defined on the edges, and form $df = 0$, since it evaluates to $0$ on the face that exists in the complex.
However, the circulation around the other "non-existent-face" is $2$. Therefore, it is impossible to create a function $g: V \rightarrow \mathbb R$ such that $dg = f$, for if such a form existed, then $f$ evaluated around the top loop would have to be $0$.
It's clear how the failure of the space (not having a face) allows us to create a closed form which is not exact.
This is an elaboration of @ZxJx's answer with a picture.
Consider a triangle with vertices [1], [2], [3], and oriented edges e1 =[[2],[1]], e2 = [[3],[2]], and e3 = [[1],[3]]. The differential is d(e1) = [2] - [1], etc. Then d(e1 + e2 + e3) = [2] - [1] + [3] - [2] + [1] - [3] = 0. Thus e1 + e2 + e3 is closed. But it is not exact, because there are is no 2-simplex.