small category alternative
Definition 0.1.
A (small) category^{} $\mathcal{C}$ consists of a set of objects ${C}_{0}$ and a set of arrows ${C}_{1}$ together with the following structure^{}:

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a source map $s:{C}_{1}\to {C}_{0}$ assigning an object $s(f)$ to each arrow $f\in {C}_{1}$ ,

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a target map: $t:{C}_{1}\to {C}_{0}$ assigning an object t(f) to each arrow $f\in {C}_{1}$ ,

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an identity map $1:{C}_{0}\to {C}_{1}$ assigning to each object $A$ an arrow ${1}_{A}$ with
$$s({1}_{A})=t({1}_{A})=A,$$ 
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a composition map $\circ :{C}_{1}\times {C}_{1}\to {C}_{1}$ 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 $t(g\circ f)=t(g)$.

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The composition thus defined “$\circ $” is associative, that is,
$$h\circ (g\circ f)=(h\circ g)\circ f$$ whenever these compositions make sense.

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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 $t(g)=A$.
References
P. A. Zito. 2008. [arXiv: math.CT]. http://arxiv.org/PS_cache/math/pdf/0509/0509266v1.pdf2${C}^{*}$ Categories with nonsimple units. ,(Preprint).
Title  small category alternative 

Canonical name  SmallCategoryAlternative 
Date of creation  20130322 18:26:38 
Last modified on  20130322 18:26:38 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  14 
Author  bci1 (20947) 
Entry type  Definition 
Classification  msc 22A22 
Classification  msc 18E05 
Classification  msc 1800 
Related topic  Category 
Related topic  2CCategory 