PlanetPhysics/Alternative Definition of Small Category

A (small) category ${\mathcal {C}}$ consists of a set of objects $C_{0}$ and a set of arrows $C_{1}$ together with the following structure:

• a source map $s:C_{1}\to C_{0}$ assigning an object $s(f)$ to each arrow $f\in C_{1}$ ,
• a target map: $t:C_{1}\to C_{0}$ assigning an object t(f) to each arrow $f\in C_{1}$ ,
• an identity map $1:C_{0}\to C_{1}$ assigning to each object $A$ an arrow $1_{A}$ with $s(1_{A})=t(1_{A})=A,$
• a composition map $\circ :C_{1}\times C_{1}\to C_{1}$ assigning to each pair of arrows $f,g$ , such that $s(g)=t(f)$, a third arrow $g\circ f$ with $s(g\circ f)=s(f)$ and $t(g\circ f)=t(g)$.
• The composition thus defined "$\circ$" is associative, that is, $h\circ (g\circ f)=(h\circ g)\circ f$ whenever these compositions make sense.
• the identity map satisfies $f\circ 1_{A}=f$ for any $f$ such that $s(f)=A$ and $1_{A}\circ g=g$, and any $g$ such that $t(g)=A$.