# PlanetPhysics/Category

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### Definitions

To introduce the modern concept of category, according to S. MacLane without using any set theory, one needs to introduce first the notions of metagraph and metacategory .
A concrete metagraph ${\mathcal {M}}_{G}$ consists of objects, $A,B,C,$... and arrows $f,g,h,$... between objects, and two operations as follows:
* a \htmladdnormallink{domain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, $dom$, which assigns to each arrow $f$ an object $A~=~dom~f$
* a \htmladdnormallink{codomain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, $cod$, which assigns to each arrow $f$ an object $B~=~cod~f,$ represented as $f:A\to B$ or $A{\stackrel {f}{\longrightarrow }}B$
A metacategory $\mathbb {C}$ is a metagraph with two additional operations:
* Identity , $id$ or {\mathbf 1}, which assigns to each object $A$ a unique arrow $id_{A}$, or $1_{A}$;
* Composition , $\circ$, which assigns to each pair of arrows $<g,f>$ with $dom~g=cod~f$ a unique arrow $g\circ f$ called their composite , such that $g\circ f:domf\to codg,$
that are subject to two axioms:
* c1. (Unit law) : for all arrows $f:A\to B$ and $g:B\to C$ the composition with the identity arrow $1_{B}$ results in $1_{B}\circ f=f$ and $g\circ 1_{B}=g;$
* c2. Associativity : for given objects and arrows in the (categorical) sequence: $A{\stackrel {f}{\longrightarrow }}B{\stackrel {g}{\longrightarrow }}C{\stackrel {h}{\longrightarrow }}D,$ one always the equality $h\circ (g\circ f)=(h\circ g)\circ f,$ whenever the composition $\circ$ is defined.
A category ${\mathcal {C}}$ is an interpretation of a metacategory within set theory. Thus, a category is a graph -- defined by a set $Ob{\mathcal {C}}:=\mathbb {O}$, a set of arrows* (called also morphisms) $Mor{\mathcal {C}}:=\mathbb {A}$, and two functions:
$dom:Mor{\mathcal {C}}\to Ob{\mathcal {C}}$ and
$cod:Mor{\mathcal {C}}\to Ob{\mathcal {C}},~$ -- that also has two additional functions: $id:Ob{\mathcal {C}}\to Mor{\mathcal {C}}$ defined by the assignments $\mathbb {A} \times _{\mathbb {O} }\mathbb {A} \longrightarrow \mathbb {A}$ called identity , and a composition Failed to parse (syntax error): {\displaystyle c = "\circ"} , that is, $c\longmapsto id_{c}$, defined by the assignments $(g,f)\longmapsto g\circ f$, such that: $dom(id_{A})=A=cod(id_{A}),dom(g\circ f)=domf,cod(g\circ f)=codg,$ for all objects $A\in Ob{\mathcal {C}}$ and all composable pairs of arrows (morphisms) $(g,f)\in \mathbb {A} \times _{\mathbb {O} }\mathbb {A} ,$ and also such that the unit law and associativity axioms c1 and c2 hold.
* Note that the set of all morphisms $Mor~{\mathcal {C}}$ of a category ${\mathcal {C}}$ is sometimes denoted as ${\mathcal {M}}$, or in French publications as $Fl~{\mathcal {C}}$.
For convenience one also defines a $Hom$ (or $hom$) set as: $Hom(B,C):=[f|f\in {\mathcal {C}},domf=B,codf=C],$ which is also denoted as $[B,C]_{\mathcal {C}}$, or simply $[B,C].$

### Alternative definitions

There are several alternative definitions of a category. Thus, as defined by W.F. Lawvere, a category is an interpretation of the ETAC axioms from his elementary theory of abstract categories. For small categories-- whose $Ob{\mathcal {C}}$ is a set and also $Mor{\mathcal {C}}$ is a set-- one has a \htmladdnormallinkdirect definition. {http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html}
If, on the other hand, $Hom_{\mathcal {C}}(X,Y)$ is a class rather than a set then the category ${\mathcal {C}}$ is called large .