§ Simplicial approxmation of maps (TODO)

§ What we want:

every map f:KLf: K \rightarrow L is homotopic to a simplicial map f:KLf_\triangle: K \rightarrow L: ie, a map that sends vertices to vertices, and sends other points through an extension of linearity. such that f(st(v))st(f(v))f(st(v)) \subseteq st(f_\triangle(v)) where ff_\triangle is the simplicial approximation of ff.
Recall that st(v)st(v) is the intersection of interiors of all simplices that contain the vertex vv. So on a graph, it's going to be a "star shaped" region around the vertex of all the edges around the vertex.

§ Why this can't happpen

Consider a triangle as a simplex of a circle. We want to represent rotations of the circle. I can rotate around a circle once, twice, thrice, ... As many times as I want. However, if all I have is a triangle, I can represent rotating once as the map 12,23,311 \mapsto 2, 2 \mapsto 3, 3 \mapsto 1 and rotating twice as maybe 13,32,211 \mapsto 3, 3 \mapsto 2, 2 \mapsto 1, but that's it. I've run out of room! So I need to subdivide the simplex to get "more points" to represent this map.

§ The correct statement