Sets, maps, composition

Before giving a precise definition of category, we should become familiar with one example, the category of finite sets and maps.

An object in this category is a finite set or collection.

A map f in this category consists of 3 things: 1. set A, called the domain of the map 2. set B, called the codomain of the map 3. rule assigning to each element a ∈ A an element b ∈ B denoted and determined by f(a) = b

```dot {engine="circo"} digraph { A -> B [color=cornflowerblue, label="f "]; B -> C [color=cornflowerblue]; A -> C [color=crimson]; A -> A [style=dotted] }

```dot {engine="circo"}
digraph g {
  node [shape=plaintext];
  A1 -> B1;
  A2 -> B2;
  A3 -> B3;

  A1 -> A2 [label=f];
  A2 -> A3 [label=g];
  B2 -> B3 [label="g'"];
  B1 -> B3 [label="(g o f)'" tailport=s headport=s];

  { rank=same; A1 A2 A3 }
  { rank=same; B1 B2 B3 } 
}

```dot {engine="circo"} digraph G { subgraph clusterC { label="Category" subgraph clusterCInitial { label="Initial" style="rounded" "" [shape="plaintext"] } subgraph clusterCX { label="X" style="rounded" "x1" [shape="point"] "x2" [shape="point"] "x1" -> "x1" [label="1X"] "x2" -> "x2" [label="1X"] "x1" -> "x2" [label="1X"] "x2" -> "x1" [label="1X"] } subgraph clusterCTerminal { label="Terminal" style="rounded" "t" [shape="point"] "t" -> "t" [label="1Terminal"] } "" -> "x1" [label="initial"] "" -> "x2" [label="initial"] "" -> "t" [label="initial / terminal"] "" -> "" [label="1Initial"] "x1" -> "t" [label="terminal"] "x2" -> "t" [label="terminal"] } }

```