Categories Great and Small

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Prev: Types and Functions Next: Kleisli Categories

Exercises

1. Generate a free category from:
   1. A graph with one node and no edges
   2. A graph with one node and one (directed) edge (hint: this edge can be composed with itself)
   3. A graph with two nodes and a single arrow between them
   4. A graph with a single node and 26 arrows marked with the letters of the alphabet: a, b, c … z.
2. What kind of order is this?
   1. A set of sets with the inclusion relation: $A$ is included in $B$ if every element of $A$ is also an element of $B$.
   2. C++ types with the following subtyping relation: T1 is a subtype of T2 if a pointer to T1 can be passed to a function that expects a pointer to T2 without triggering a compilation error.
3. Considering that Bool is a set of two values True and False, show that it forms two (set-theoretical) monoids with respect to, respectively, operator && (AND) and || (OR).
4. Represent the Bool monoid with the AND operator as a category: List the morphisms and their rules of composition.
5. Represent addition modulo 3 as a monoid category.