> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/00-prereqs/01-sets-maps-composition.md).

# 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"] } }

\`\`\`