Crash Monad Tutorial

Examples of Functors

Identity functor - for a category X an identity X → X keeps objects snd arrows intact. Still it is a functor.
Setf _↪ Set - a functor that includes Setf into Set, that is, maps each finite set to itself, and the same with functions. Note that this is not an identity functor.
Set → Top - similar to the previous example, this functor makes Set a part of Top. Each set is mapped to a discrete topological space.
For any set A we can define the following functor:
(- xA): Set → Set - it will map any set X to a cartesian product, X × A.
For any set A we can define a functor
P_A: Set → Set; it maps any set X to X^A, a set of functions from A to X.
Set _↪ Part embeds sets to sets with partial functions: it maps sets and functions to themselves.
An opposite to 6. is a functor +Null: Part → Set -this functor adds an "extension" Null to each set: X ↦ (X+Null), so that a partial function X → Y maps to a function (X+Null) → (Y+Null).
(Exercise. Define such an extension for partial functions.)