# PlanetPhysics/Category

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The concept of category emerged in 1943-1945 from work in algebraic topology and Homological Algebra by S. Eilenberg and S. Mac Lane [1], as a generalization of the algebraic concepts of semigroup, monoid, group, groupoid, etc., as well as an extension of topological concepts and diagrams employed in algebraic topology and homological algebra. Thus many properties of mathematical systems can be unified by a presentation with diagrams of arrows that may represent functions, transformations, distributions, operators, etc., and that-- in the case of concrete categories-- may also include objects such as class elements, sets, topological spaces, etc. ; the usefulness of such diagrams comes from the composition of the arrows and the (fundamental) axioms that define any category which allow mathematical constructions to be represented by universal properties of diagrams.


### Definitions

To introduce the modern concept of category, according to S. MacLane [2] without using any set theory, one needs to introduce first the notions of metagraph and metacategory .

A concrete metagraph ${\displaystyle {\mathcal {M}}_{G}}$ consists of objects, ${\displaystyle A,B,C,}$... and arrows ${\displaystyle f,g,h,}$... between objects, and two operations as follows:

• a \htmladdnormallink{domain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, ${\displaystyle dom}$, which assigns to each arrow ${\displaystyle f}$ an object ${\displaystyle A~=~dom~f}$
• a \htmladdnormallink{codomain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, ${\displaystyle cod}$, which assigns to each arrow ${\displaystyle f}$ an object ${\displaystyle B~=~cod~f,}$ represented as ${\displaystyle f:A\to B}$ or ${\displaystyle A{\stackrel {f}{\longrightarrow }}B}$

A metacategory ${\displaystyle \mathbb {C} }$ is a metagraph with two additional operations:

• Identity , ${\displaystyle id}$ or {\mathbf 1}, which assigns to each object ${\displaystyle A}$ a unique arrow ${\displaystyle id_{A}}$, or ${\displaystyle 1_{A}}$;
• Composition , ${\displaystyle \circ }$, which assigns to each pair of arrows ${\displaystyle }$ with ${\displaystyle dom~g=cod~f}$ a unique arrow ${\displaystyle g\circ f}$ called their composite , such that ${\displaystyle g\circ f:domf\to codg,}$

that are subject to two axioms:

• c1. (Unit law) : for all arrows ${\displaystyle f:A\to B}$ and ${\displaystyle g:B\to C}$ the composition with the identity arrow ${\displaystyle 1_{B}}$ results in ${\displaystyle 1_{B}\circ f=f}$ and ${\displaystyle g\circ 1_{B}=g;}$
• c2. Associativity : for given objects and arrows in the (categorical) sequence: ${\displaystyle A{\stackrel {f}{\longrightarrow }}B{\stackrel {g}{\longrightarrow }}C{\stackrel {h}{\longrightarrow }}D,}$ one always the equality ${\displaystyle h\circ (g\circ f)=(h\circ g)\circ f,}$ whenever the composition ${\displaystyle \circ }$ is defined.

A category ${\displaystyle {\mathcal {C}}}$ is an interpretation of a metacategory within set theory. Thus, a category is a graph -- defined by a set ${\displaystyle Ob{\mathcal {C}}:=\mathbb {O} }$, a set of arrows* (called also morphisms) ${\displaystyle Mor{\mathcal {C}}:=\mathbb {A} }$, and two functions:

${\displaystyle dom:Mor{\mathcal {C}}\to Ob{\mathcal {C}}}$ and

${\displaystyle cod:Mor{\mathcal {C}}\to Ob{\mathcal {C}},~}$ -- that also has two additional functions: ${\displaystyle id:Ob{\mathcal {C}}\to Mor{\mathcal {C}}}$ defined by the assignments ${\displaystyle \mathbb {A} \times _{\mathbb {O} }\mathbb {A} \longrightarrow \mathbb {A} }$ called identity , and a composition $\displaystyle c = "\circ"$ , that is, ${\displaystyle c\longmapsto id_{c}}$, defined by the assignments ${\displaystyle (g,f)\longmapsto g\circ f}$, such that: ${\displaystyle dom(id_{A})=A=cod(id_{A}),dom(g\circ f)=domf,cod(g\circ f)=codg,}$ for all objects ${\displaystyle A\in Ob{\mathcal {C}}}$ and all composable pairs of arrows (morphisms) ${\displaystyle (g,f)\in \mathbb {A} \times _{\mathbb {O} }\mathbb {A} ,}$ and also such that the unit law and associativity axioms c1 and c2 hold.

• Note that the set of all morphisms ${\displaystyle Mor~{\mathcal {C}}}$ of a category ${\displaystyle {\mathcal {C}}}$ is sometimes denoted as ${\displaystyle {\mathcal {M}}}$, or in French publications as ${\displaystyle Fl~{\mathcal {C}}}$.

For convenience one also defines a ${\displaystyle Hom}$ (or ${\displaystyle hom}$) set as: ${\displaystyle Hom(B,C):=[f|f\in {\mathcal {C}},domf=B,codf=C],}$ which is also denoted as ${\displaystyle [B,C]_{\mathcal {C}}}$, or simply ${\displaystyle [B,C].}$

### 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 [3]. For small categories-- whose ${\displaystyle Ob{\mathcal {C}}}$ is a set and also ${\displaystyle Mor{\mathcal {C}}}$ is a set-- one has a \htmladdnormallinkdirect definition. {http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html}

If, on the other hand, ${\displaystyle Hom_{\mathcal {C}}(X,Y)}$ is a class rather than a set then the category ${\displaystyle {\mathcal {C}}}$ is called large .

## References

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