Category Theory For Beginners: Graphs And Dynamical Systems

In this video I discuss how we can make categories of structured sets (the category of graphs, and the category of dynamical systems, and the category of functions), by considering functors. In particular, I show how the category of functors from the category with two parallel arrows, to the category Set corresponds to the category of graphs. I illustrate how graph homomorphisms correspond to the natural transformations between such functors. I also describe the category of dynamical systems, and the category of functions using this idea. I also show that the category of dynamical systems is isomorphic to a subcategory of the category of functions, which in turn, is isomorphic to a subcategory of the category of graphs. Here I use the term "dynamical system", in the same loose sense as Lawvere does in his great book "Conceptual Mathematics", to just mean a function from a set to itself. Note that the kind of graphs we discuss are also called `quivers'. https://en.wikipedia.org/wiki/Quiver_(mathematics) This unlisted video describes how to find subobject classifiers in the kinds of functor categories I describe in this video, and may help people wishing to connect this material with my video on topos theory. https://youtu.be/ahjkqMcEcKk