Let \(L\) be the language of strings of length \(3\) over the alphabet \(\{0, 1\}\) containing at most one occurrence of \(1\). Each position is represented by a color: ⬤,⬤ and ⬤. The complex representing \(L\) is a triangle.

There is a function \(f:\{0,1\}^3\to\{0,1\}^3\) whose image is \(L\), and such that the value of \(f(x)\) at a position \(i\in\{0,1,2\}\) only depends on the values of \(x\) at the other two positions. One can imagine that each \(i\) has a partial view on the input \(x\), sufficient to evaluate \(f(x)\) at position \(i\). The inputs of \(f\) can be represented as a simplicial complex whose triangles are the possible inputs, and the vertices are the local views of the output positions on these inputs [1].

The function \(f\) can then be represented as a color-preserving simplicial map from the input complex to the complex representing \(L\). The animation shows how the input complex is mapped to the output complex.

[1] Maurice Herlihy and Nir Shavit. The topological structure of asynchronous computability. J. ACM, 46(6):858–923, November 1999.