Geometry & Topology 2024, 28: Composing Maps

Truth be told, I'm not sure which topic I'd like to tackle next, but since there's nothing stoppin me from revision, let's just check out more topology. Specifically, I'm checking out Geometry and Topology by Miles Reid and Balazs Szendroi. The first chapter is basics happening in metric spaces, which we don't need to think about, too much, so I'm starting with chapter 2, which I'm also relatively familiar with, but I think building maps is something that can always be expanded on, since it's very useful for cheating all sorts of problems.

We remind ourselves first of ways to think around maps. Their basic operation is the composition, which presupposes the correct relations between sets for the composition to make sense. Take maps f: X → Y, g: Y → Z, then their composition g ⚬ f : X → Z with (g ⚬ f)(x) = g(f(x)). This is also where one derives the rules for matrix multiplication from. Compositions of basic coordinate translations for any space should get each point in the space mapped onto every other point of the map. An affine linear map Tᵢ: ℝⁿ → ℝⁿ, T(x) = Aᵢx + bᵢ, composes as follows

A valid composition of multiple maps is always well defined. It can be bracketed however one wants or is computationally convenient. Composition is associative and commutative. Geometrically, a motion in an n-dimensional Euclidean space is a composite of at most n + 1 reflections.