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 $f:X\to Y$.

These morphisms must satisfy three conditions:

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