Eric Forgy Arrow Category of Categories

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