The Deranged Mathematician: How to be Universal and Natural

I figured it was high time that I did a follow-up on my original post on category theory---this time, to discuss universal properties and natural transformations. Why is this of any interest? Simple: those two notions give a framework for how to think about coordinate-free definitions. If you are unfamiliar with the concept, I can give a very concrete example. In linear algebra, one defines the trace of a square matrix as the sum of its (main) diagonal entries. A priori, this seems entirely random and it is perhaps a great surprise that this turns out to be coordinate-independent---you will get exactly the same result if you choose a different basis in which to express your matrix. This can be gainfully exploited (to aid with calculating eigenvalues, for example), but one is still left with the uneasy question of why exactly this just happens to work out. Alternatively, it is possible to give a coordinate-free definition of the trace (and I do so in this post), which doesn't make use of any particular basis. It is then immediately obvious why the trace doesn't depend on a choice of coordinates, but there are other benefits as well: one of them is that it offers some insight into what you need to extend this definition to work beyond simply finite-dimensional spaces. (The key property turns out to be that you need the vector space to be naturally isomorphic to its dual. This occurs, for instance, for Hilbert spaces.) Read the full post (for free) on Substack: How to be Universal and Natural