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\begin{document}
\begin{definition}
A (small) \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{C}$ consists of a set of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $C_0$ and a set of arrows
$C_1$ together with the following structure:
\begin{itemize}
\item a source map $s: C_1 \to C_0$ assigning an object $s(f)$ to each arrow $f \in C_1$ ,
\item a target map: $t: C_1 \to C_0$ assigning an object t(f) to each arrow $f \in C_1$ ,
\item an \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} map $1 : C_0 \to C_1$ assigning to each object $A$ an arrow $1_A$ with
$$s(1_A) = t(1_A) = A,$$
\item a \htmladdnormallink{composition map}{http://planetphysics.us/encyclopedia/Identity2.html} $\circ : C_1 \times C_1 \to C_1$ assigning to each pair of arrows
$f,g$ , such that $$s(g) = t(f)$$, a third arrow $g \circ f$ with $s(g \circ f) = s(f)$ and $t(g \circ f) = t(g)$.
\item The \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} thus defined ``$\circ$'' is associative, that is,
$$ h \circ (g \circ f) = (h \circ g) \circ f$$ whenever these compositions make sense.
\item the identity map satisfies $f \circ 1_A = f $ for any $f$ such that
$s(f) = A$ and $1_A \circ g = g$, and any $g$ such that $ t(g) = A$.
\end{itemize}
\end{definition}
\begin{thebibliography}{9}
P. A. Zito. 2008. [arXiv: math.CT].
\htmladdnormallink{2-$C^*$ -Categories with non-simple units.
}{http://arxiv.org/PS_cache/math/pdf/0509/0509266v1.pdf},(Preprint).
\end{thebibliography}
\end{document}
