# nLab horizontal morphism

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## 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

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

Created on July 12, 2018 at 09:08:32. See the history of this page for a list of all contributions to it.