# 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:

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

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

• an identity map $1:C_{0}\to C_{1}$ assigning to each object $A$ an arrow $1_{A}$ with

 $s(1_{A})=t(1_{A})=A,$
• 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 $t(g\circ f)=t(g)$.

• The composition thus defined “$\circ$” is associative, that is,

 $h\circ(g\circ f)=(h\circ g)\circ f$

whenever these compositions make sense.

• 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 non-simple units. ,(Preprint).

Title small category- alternative SmallCategoryAlternative 2013-03-22 18:26:38 2013-03-22 18:26:38 bci1 (20947) bci1 (20947) 14 bci1 (20947) Definition msc 22A22 msc 18E05 msc 18-00 Category 2CCategory