Category

A category defines the abstraction of composition. We primarily understand a category as a collection of some objects which satisfy a set of formal rules concerning Composition, Associativity, and Identity.

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Formal Definition

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Example

A common example of categories in action is when defining relations between Sets. Say we want establish a Bijection for the set of all shapes $S$ to a modified set of natural numbers $N$:

$$S=\{\text{triangle},\text{square},\text{pentagon},\dots \}$$ $$N=\{x|x\in \mathbb{N},x>2\}$$

We can define a relation $f$ that does just this

$$f_{edges}:S\to N$$

Where $f$ maps the number of edges of a given shape to its corresponding natural number.

Now say we have a set $B$ of the booleans:

$$B=\{true,false\}$$

And we define a new relation $g$ that maps the even numbers to $true$, and the odd numbers to $false$:

$$g_{even?}:N\to B$$

By the formal definition of a category we now have a Composition $g\circ f$ that maps shapes directly to a boolean:

$$g\circ f:$$