Lecture 3: Functors
See exercises below for clarification of some of the discussions during the video lecture, especially the discussions at the end of the lecture.
3.1. An Abstract Notion of a Dynamical System
We begin this lecture with an abstract notion of a dynamical system. After recalling this notion, we will try to express it using the language of categories. This process will lead us to a general notion of a mapping between categories: the notion of a functor.
A dynamical system is a mathematical structure that consists of a set "X" of states (of a real-life system being modeled), and for each time laps represented by a (non-negative) real number "t", a mapping of members of "X" to members of "X", i.e., a function from the set "X" to the set "X", which we will write as "f" with a subscript "t".