category

Category

Definition A category is a quadruple ${ \mathcal{C}=(\mathcal{O},\hom,\mathrm{id},\circ) }$ consisting of

a class ${ \mathcal{O} }$, whose members are called ${ \mathcal{C} }$-objects (or simply objects),
for each pair ${ (A,B) }$ of ${ \mathcal{C} }$-objects, a set ${ \hom(A,B) }$, whose members are called ${ \mathcal{C} }$-morphisms (or simply morphisms) from ${ A }$ to ${ B }$. The statement “${ f \in \mathrm{hom}(A,B) }$” is expresseed more graphically by using arrows; e.g., by statements such as “${ f: A \to B }$ is a morphism” or “${ A\xrightarrow[]{f}B }$ is a morhphism”
for each ${ \mathcal{C} }$-object ${ A }$, a morphism ${ A \xrightarrow{\mathrm{id}_{A}} A }$, called the ${ \mathcal{C} }$-identity on ${ A }$
a composition law associating with each morphisms ${ A \xrightarrow{f} B }$ and ${ B \xrightarrow{g} C }$ a ${ \mathcal{C} }$-morphism ${ A \xrightarrow[]{g\circ f} C}$, called the composite of ${ f }$ and ${ g }$, subject to the following conditions:
- ${ A \xrightarrow[]{f} B,\, B \xrightarrow[]{g}C,\, C\xrightarrow[]{h}D }$, the equation ${ h\circ (g \circ f) = (h \circ g ) \circ f }$ holds,
- ${ \mathcal{C} }$-identities act as identities with respect to composition; i.e., for ${ \mathcal{C} }$-morphisms ${ A \xrightarrow[]{f}B }$, we have ${ \mathrm{id}_{B} \circ f=f}$ and ${ f \circ \mathrm{id}_{A} = f}$,
- the sets ${ \mathrm{hom}(A,B) }$ are pairwise disjoint.

If ${ \mathcal{C}=(\mathcal{O},\hom,\mathrm{id},\circ)}$ is a category, then

The class ${ \mathcal{O} }$ is usually denoted by ${ \mathrm{Ob}(\mathcal{C}) }$.
The class of all ${ \mathcal{C} }$-morphisms (denoted by ${ \mathrm{Mor}(\mathcal{C})) }$ is defined to be the union of all the sets ${ \mathrm{hom}(A,B) }$ in ${ \mathcal{C} }$.
If ${ A \xrightarrow[]{f}B }$ is an ${ \mathcal{C} }$-morphism, we call ${ A }$ the domain of ${ f }$ (and denote it by ${ \mathrm{dom}(f) }$) and call ${ B }$ the codomain of ${ f }$ (and denote it by ${ \mathrm{cod}(f) }$).