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\begin{document}
The \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of category emerged in 1943-1945 from \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} in \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} and Homological Algebra by S. Eilenberg and S. Mac Lane \cite{Eilenberg-MacLane45}, as a generalization of the \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} concepts of \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{monoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}, etc., as well as an extension of \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} concepts and \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} employed in algebraic topology and homological algebra. Thus many properties of mathematical \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} can be unified by a presentation with \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/CommutativeSquareDiagram.html} of arrows that may represent \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html}, transformations, distributions, \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}, etc., and that-- in the case of concrete categories-- may also include \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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.
\subsection{Definitions}
To introduce the modern concept of category, according to S. MacLane
\cite{MacLane98} without using any set theory, one needs to introduce first the notions of {\em metagraph} and {\em metacategory}.
\begin{definition}
A concrete \emph{metagraph} $\mathcal{M}_G$ consists of objects, $A, B, C,$...
and arrows $f, g, h,$... between objects, and two operations as follows:
\begin{itemize}
\item a {\em \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} operation}, $dom$, which assigns to each arrow $f$ an object $A~ =~dom ~f$
\item a {\em \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 $A \stackrel{f}{\longrightarrow} B$
\end{itemize}
\end{definition}
\begin{definition}
A \emph{metacategory} $\mathbb{C}$ is a metagraph with two additional operations:
\begin{itemize}
\item {\em Identity}, $id$ or {\bf 1}, which assigns to each object $A$ a unique arrow $id_A$, or $1_A$;
\item {\em Composition}, $\circ$, which assigns to each pair of arrows $<g,f>$
with $dom~ g = cod~ f$ a unique arrow $g \circ f$ called their \emph{composite},
such that $g \circ f : dom f \to cod g,$
\end{itemize}
that are subject to two axioms:
\begin{itemize}
\item {\em 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 $g \circ 1_B = g ;$
\item {\em c2. Associativity}: for given objects and arrows in the
\htmladdnormallink{(categorical) sequence}{http://planetphysics.us/encyclopedia/CategoricalSequence.html}:
$$A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} D , $$ one always the equality
$$ h \circ(g \circ f) = (h \circ g) \circ f , $$
whenever the composition $\circ$ is defined.
\end{itemize}
\end{definition}
\begin{definition}
A \emph{category} $\mathcal{C}$ is an interpretation of a metacategory
within set theory. Thus, a {\em category} is a {\em graph} -- defined by a
set $Ob \mathcal{C}:=\mathbb{O}$, a set of arrows* (called also \emph{\htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}})
$Mor\mathcal{C}:= \mathbb{A}$, 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
\emph{identity}, and a {\em composition} $c = ``\circ''$, 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 {\em c1} and {\em c2} hold.
\end{definition}
*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:
$$Hom(B,C) := [f|f \in \mathcal{C}, dom f= B, cod f = C],$$
which is also denoted as $[B,C]_\mathcal{C}$, or simply $[B,C].$
\subsection{Alternative definitions}
There are several alternative definitions of a category.
Thus, as defined by W.F. Lawvere, a {\em category} is an interpretation of the
\htmladdnormallink{ETAC axioms}{http://planetphysics.us/encyclopedia/Formula.html} from his \htmladdnormallink{elementary theory of abstract categories}{http://planetphysics.us/encyclopedia/Formula.html} \cite{Lawvere66}.
For small categories-- whose $Ob \mathcal{C}$ is a set and also $Mor\mathcal{C}$
is a set-- one has a \htmladdnormallink{\em direct 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 {\em large}.
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\bibitem{EML1}
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\bibitem{Eilenberg-MacLane45}
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\bibitem{BBGG2k6}
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., {\em Axiomathes}, \textbf{16} Nos. 1--2: 65--122.
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\bibitem{Other}
See also a more extensive \htmladdnormallink{category theory bibliography}{http://planetphysics.us/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html}
\end{thebibliography}
\end{document}
