The diagram below depicts a partial arrow category of categories.

$\array{
{} & {} & A & \stackrel{f}{\to} & B & {} & {} \\
{} & {} & \mathllap{\scriptsize{\alpha_C}}\downarrow &
\mathllap{\scriptsize{\alpha}}\Downarrow &
\downarrow\mathrlap{\scriptsize{\alpha_D}} & {} & {} \\
{} & {} & C & \stackrel{g}{\to} & D & {} & {} \\
{} & \mathllap{\scriptsize{\beta_E}}\swarrow & \mathllap{\scriptsize{\beta}}\swArrow & \mathllap{\mathllap{\scriptsize{\gamma_G}}\searrow}{}\mathrlap{\swarrow\mathrlap{\scriptsize{\beta_F}}} & \seArrow\mathrlap{\scriptsize{\gamma}} & \searrow\mathrlap{\scriptsize{\gamma_H}} & {} \\
E\quad{} & \stackrel{h}{\to} & F & {} & G & \stackrel{i}{\to} & H
}$

Although it looks cluttered, the idea is simple. There are three morphisms here labeled $\alpha$, $\beta$, $\gamma$. These morphisms sweep transverse “arrow objects” across respective 2-cells.

The point of drawing the diagram like this is to highlight the branching of the 2-paths.

Note: The complete diagram would require the composite morphisms $\beta\circ\alpha$ and $\gamma\circ\alpha$.

Revised on February 21, 2010 at 17:34:51
by
Eric Forgy