Category (mathematics)

A category consists of a collection of objects with morphisms between them. A morphism $f$ goes from one object, say $X$, to another, say $Y$, and is drawn as an arrow from $X$ to $Y$. Note that $X$ may equal $Y$ (in which case $f$ is referred to as an [-endomorphism]). The object $X$ is called the source or domain of $f$ and $Y$ is called the target or codomain of $f$, though note that $f$ itself need not be a Function and $X$ and $Y$ need not be sets. This is written as $f: X \rightarrow Y$.

These morphisms must satisfy three conditions:

1. [Composition_of_functions Composition]: For any two morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, there exists a morphism $X \rightarrow Z$, written as $g \circ f$ or simply $gf$.
2. Associativity: For any morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$ and $h:Z \rightarrow W$ composition is associative, i.e., $h(gf) = (hg)f$.
3. [identity_map Identity]: For any object $X$, there is a (unique) morphism, $1_X : X \rightarrow X$ which, when composed with another morphism, leaves it unchanged. I.e., given $f:W \rightarrow X$ and $g:X \rightarrow Y$ it holds that: $1_X f = f$ and $g 1_X = g$.