nLab
horizontal morphism

Contents

Contents

Idea

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

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

a AAf𝒞 hAA b h𝒞 v k𝒞 v c AAg𝒞 hAA d \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 }

Created on July 12, 2018 at 05:08:32. See the history of this page for a list of all contributions to it.