nLab horizontal morphism

Idea

A double category consists, in particular, of two categories $\mathcal{C}_h$, $\mathcal{C}_v$ with the same class of objects $Obj_{\mathcal{C}_h} = Obj_{\mathcal{C}_v}$. The 1-morphisms of $\mathcal{C}_h$ are called the horizontal morphisms, those of $\mathcal{C}_v$ are called the vertical morphisms of the double category.

A 2-morphism in the double category is of the form

$$\array{ a &\overset{ \phantom{AA}f \in \mathcal{C}_h\phantom{AA} }{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\Big\downarrow &\swArrow& \Big\downarrow{}^{ \mathrlap{k \in \mathcal{C}_v} } \\ c &\underset{ \phantom{AA}g \in \mathcal{C}_h\phantom{AA}}{\longrightarrow}& d }$$

Related concepts

• n-morphism