Definition 1.2.1 (Category).label A category $\mathfrak{C}$ is a collection of objects $\ob{\frak{C}}$, such that for any $X,Y,Z\in\text{Ob}(\frak{C})$ there exists collections of morphisms $\mor{}{X }{Y },\mor{}{Y }{Z }$ and a composition law:

satisfying:

1. (CAT1)

   For any $X,Y,Z,W\in \ob{\frak{C}}$, $\mor{}{X }{Y }$ and $\mor{}{Z }{W }$ are disjoint or equal

2. (CAT2)

   For any $X,Y\in\ob{\frak{C}}$, there exists $\text{Id}_{X}\in \mor{}{X}{X},\text{Id}_{Y}\in \mor{}{Y }{Y }$ such that for each $f\in \mor{}{X }{Y }$, $f\circ \text{Id}_{X}=f=\text{Id}_{Y}\circ f$

3. (CAT3)

   For any $X,Y,Z,W\in \ob{\frak{C}}$, $f\in \mor{}{X }{Y },g\in \mor{}{Y }{Z }$ and $h\in \mor{}{Z }{W }$, $(h\circ g)\circ f=h\circ (g\circ f)$