Lecture 2

Check out the exercises for Lecture 2 (not included in the video lecture) at the end of this post. The rest of the post is mostly based on the video lecture. 2.1. The Big Secret We ended last lecture saying that we should discuss examples of categories, as the next step. The two examples we had in mind had to do with the divisibility relation between numbers and composition of functions. Before discussing these examples, let us step back a bit and ask the question: what is the broad intuition behind the concept of a category -- what sort of phenomena of life/nature/science is it supposed to represent? Let us first answer this question for the concept of a directed graph. Directed graphs arise in life/nature/science as situations where a certain type of objects is identified and we would like to investigate a certain type of relation between these objects. For instance, the objects in question could be natural numbers (i.e., measures of whole quantities) and the relation we may be int

Lecture 1

Write-up of the Lecture (notes for reflection - feel free to interact via comments): 1.1. Directed Graphs The goal of this lecture is to define what a category is and to give examples of categories. To explain what a category is, first let us revisit the notion of a directed graph . The latter is a collection of objects, called vertices of the directed graph, and directed edges (represented as arrows) connecting them. Here is an example: This directed graph has four vertices and various arrows between them. Note that in our notion of a directed graph we allow there to be multiple arrows/edges connecting the same pair of vertices, and even arrows that loop in the same vertex. The contour around the diagram of points and arrows signifies that we want to think of the multitude of constituents of a directed graph (the vertices, the arrows, and the data of which arrow start and ends with which vertex) as a single entity. There can thus be many different directed graphs. Here is an