# nLab triangle identities

category theory

## Applications

#### 2-Category theory

2-category theory

# Contents

## Idea

The triangle identities or zigzag identities are identities satisfied by the unit and counit of an adjunction.

## Statement

Given $C, D$ (categories, or otherwise objects of a $2$-category) with functors (or otherwise morphisms) $L: C \to D$ and $R : D \to C$ and natural isomorphisms (or otherwise $2$-morphisms) $\eta: 1_C \to R \circ L$ and $\epsilon: L \circ R \to 1_D$, the triangle identities are the following:

### As equations

$L \stackrel{L\eta}\to L R L\stackrel{\epsilon L}\to L$

and

$R\stackrel{\eta R}\to R L R \stackrel{R\epsilon}\to R$

are identities.

### As diagrams

$R\epsilon . \eta R = 1_R$ i.e.

$\array{\arrayopts{ \padding{0} } &&&&1_C& \\ &&\cellopts{\colspan{5}}\begin{svg} \end{svg}\\ D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C \\ \cellopts{\colspan{4}}\begin{svg} \end{svg} \\ &&1_D& } \quad = \quad D \stackrel{R}{\to} C$

and $\epsilon L . L\eta = 1_L$ i.e.

$\array{\arrayopts{ \padding{0} } &&1_C& \\ \cellopts{\colspan{5}}\begin{svg} \end{svg}\\ C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D \\ &&\cellopts{\colspan{4}}\begin{svg} \end{svg} \\ &&&&1_D& } \quad = \quad C \stackrel{L}{\to} D$

### As string diagrams

In string diagrams, the identities appear as the action of “pulling zigzags straight” (hence the name):

, .

With labels left implicit, this notation becomes very economical:

, .

Revised on September 22, 2014 02:22:46 by John Dougherty (68.101.162.59)