dirty-haskell.org: Exercises in Category Theory — 1.1

It's not crazy — it's having fun with types.

Sets and functions do form a category $\ca{Set}$ .

The objects of $\ca{Set}$ are sets and its arrows are functions.

For any set $A$ we identify $\idarr{A}$ with the identity function on $A$ , $\id : A \to A$ .

$\circ$ is indeed associative.

Each poset is a category.

The objects of each poset are its elements and we construct an arrow $\arr{a}{\leq}{b}$ for every pair of objects $(a, b)$ iff $a \leq b$ .

Reflexivity ( $a \leq a$ ) and transitivity ( $a \leq b \land b \leq c \implies a \leq c$ ) provide a construction for $\idarr{a}$ and $\circ$ , respectively.

Each monoid is a category.

The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\arr{a}{\cdot b}{a \cdot b}$ .

The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for $\id$ and associativity of $\circ$ .