Category Theory: July 18

Covered in Class

1. Categories

   • Definition, using Set as an example; also Set with inclusion rather than functions

   • Other algebraic structures (Mon, Group, Ring, etc) as categories (concrete categories)

   • Simple finite examples 0, 1, 2, 3; 1+1

   • Extreme examples: monoid and preorder. Contrast with category Mon.

Suggested Exercises

All exercises are optional; pick any that look interesting and feel free to create your own! I'll be around in the lab on Tuesday from 10-11 and from 12 to about 1:30 if anyone wants to discuss solutions; also I encourage you to work together and discuss the material with each other! Feel free to post also to the #category-theory Slack channel.

See also the exercises in Gilbert Bernstein's Machines, Monoids, Categories handout.

1. Prove carefully that for any object in a category, its identity arrow must be unique.

2. Pick your favorite "set with structure" from week one (for example Groups with group homomorphisms, Rings with ring homomorphisms, or Vector Spaces with linear transformations) and show that it satisfies the properties of being a category.

3. Try representing a few favorite monoids or groups as a category with one object, as was shown in class. This is a good way to get more comfortable with this representation. Some examples to try: ℕ, ℤ, finite cyclic groups (the integers under addition mod n) and the dihedral group of symmetries of an equilateral triangle. Convince yourself that the category properties hold.

4. Let C be a category with a single object and three arrows 1 (the identity), f and g. We drew this in class and started the 3x3 multiplication table in which five of the entries are forced by the identity rule, leaving the other four entries blank. Try completing the mutiplication table in all possible ways that satisfy the category properties. What monoids do these correspond to?

5. (Pierce, 1.1.20, exercise 3): Verify that each of the categories 0, 1, 2, and 3 corresponds to a partial order. What would the category 4 look like? The category 5? The category N with an object corresponding to each natural number? [This answers the question in class of whether there is a natural way to generalize these categories.]