I found this definition in http://arxiv.org/abs/hep-th/0304074 (pp.13) for 2-categories: It is a collection $\mathcal{C}_0$ of objects, $\mathcal{C}_1$ of morphisms, and $\mathcal{C}_2$ of 2-morphisms such that:
$(\mathcal{C}_{0},\mathcal{C}_{1},s^{(1)},t^{(1)},id^{(1)},\bullet) $ is a small category
$ (\mathcal{C}_{1},\mathcal{C}_{2},s^{(2)},t^{(2)},id^{(2)},\circ) $ is a small category
- $ t^{(1)}(s^{(2)}(f))=t^{(1)}(t^{(2)}(f)) $ and $ s^{(1)}(s^{(2)}(f))=s^{(1)}(t^{(2)}(f)) $ meaning that 2-morphisms are bi-gones
- A map $\bullet:\mathcal{C}_{2}\times_{\mathcal{C}_{0}}\mathcal{C}_{2}\rightarrow\mathcal{C}_{2} $ of horizonal composition between 2-morphisms (the vertical one is already defined by item 2.)
Further axioms which the author says are in fact the requirement for the maps $s^{(2)}$, $t^{(2)}$ and $id^{(2)}$ to induce functors from the category $ (\mathcal{C}_{0},\mathcal{C}_{2},s^{(1)}\circ s^{(2)},t^{(1)}\circ t^{(2)},id^{(1)}\circ id^{(2)},\bullet) $ to the category $(\mathcal{C}_{0},\mathcal{C}_{1},s^{(1)},t^{(1)},\bullet) $, these axioms are:
- $ s^{(2)}(f_{2} \bullet f_{1})=s^{(2)}(f_{2})\bullet s^{(2)}(f_{1}) $
- $ t^{(2)}(f_{2}\bullet f_{1})=t^{(2)}(f_{2})\bullet t^{(2)}(f_{1}) $
- $ id^{(2)}(id^{(1)}(s^{(1)}(s^{(2)}(f))))\bullet f=f=f\bullet id^{(2)}(id^{(1)}(t^{(1)}(t^{(2)}(f)))) $
- $ (f_{1}\bullet f_{2})\bullet f_{3}=f_{1}\bullet(f_{2}\bullet f_{3}) $
- $ id^{(2)}(g_{1})\bullet id^{(2)}(g_{2})=id^{(2)}(g_{1}\bullet g_{2}) $
- (f_1 \circ f'_1) \bullet (f_2 \circ f'_2) = (f_1 \bullet f_2) \circ (f'_1 \bullet f'_2)
As I see it, items 1., 2. and 5. are suffisiant to ensure that $s^{(2)}$, $t^{(2)}$ and $id^{(2)}$ induce the required functors, so my questions are:
- What do the remaining axioms establish?
- Why do we construct a category in this way, i.e: requiring $(\mathcal{C}_{0},\mathcal{C}_{1},s^{(1)},t^{(1)},id^{(1)},\bullet) $ and $ (\mathcal{C}_{1},\mathcal{C}_{2},s^{(2)},t^{(2)},id^{(2)},\circ) $ to be small categories and requiring certain functors from $ (\mathcal{C}_{0},\mathcal{C}_{2},s^{(1)}\circ s^{(2)},t^{(1)}\circ t^{(2)},id^{(1)}\circ id^{(2)},\bullet)$ to $(\mathcal{C}_{0},\mathcal{C}_{1},s^{(1)},t^{(1)},\bullet) $
- Is there another way to construct a 2-category?
Note that $s^{(i)}$, $t^{(i)}$ and $id^{(i)}$ are the source, target and identity maps for the given category.