# 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 $ϵ:L\circ R\to {1}_{D}$, the triangle identities are the following:

### As equations

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

and

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

are identities.

### As diagrams

$Rϵ.\eta R={1}_{R}\phantom{\rule{2em}{0ex}}\text{i.e.}\phantom{\rule{2em}{0ex}}\begin{array}{cccccc}& & & & {1}_{C}& \\ & & \multicolumn{5}{c}{ {⇓}^{\eta } }\\ D& \stackrel{R}{\to }& C& \stackrel{L}{\to }& D& \stackrel{R}{\to }& C\\ \multicolumn{4}{c}{ {⇓}^{ϵ} }\\ & & {1}_{D}& \end{array}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}D\stackrel{R}{\to }C$R\epsilon . \eta R = 1_R \qquad \text{i.e.} \qquad \array{\arrayopts{ \padding{0} } &&&&1_C& \\ &&\cellopts{\colspan{5}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="8.5em" height="2em" viewBox="0 0 85 20"> <defs> <marker id='svg195arrowhead' markerHeight='5' markerUnits='strokeWidth' markerWidth='8' orient='auto' refX='0' refY='5' viewBox='0 0 10 10'> <path d='M 0 0 L 10 5 L 0 10 z'/> </marker> </defs> <path marker-end='url(#svg195arrowhead)' stroke-width="1" stroke="#000" fill="none" d="M5 15q40-28 75 0"/> <foreignObject height='20' width='20' x='40' y='3' font-size='10'><math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><msup><mo>&#8659;</mo><mi>&#951;</mi></msup>[/itex]</foreignObject> </svg> \end{svg}\\ D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C \\ \cellopts{\colspan{4}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="8.5em" height="2em" viewBox="0 0 85 20"> <path marker-end='url(#svg195arrowhead)' stroke-width="1" stroke="#000" fill="none" d="M5 5q40 28 75 0"/> <foreignObject height='20' width='20' x='40' y='0' font-size='10'><math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><msup><mo>&#8659;</mo><mi>&#1013;</mi></msup>[/itex]</foreignObject> </svg> \end{svg} \\ &&1_D& } \quad = \quad D \stackrel{R}{\to} C

and

$ϵL.L\eta ={1}_{L}\phantom{\rule{2em}{0ex}}\text{i.e.}\phantom{\rule{2em}{0ex}}\begin{array}{cccc}& & {1}_{C}& \\ \multicolumn{5}{c}{ {⇓}^{\eta } }\\ C& \stackrel{L}{\to }& D& \stackrel{R}{\to }& C& \stackrel{L}{\to }& D\\ & & \multicolumn{4}{c}{ {⇓}^{ϵ} }\\ & & & & {1}_{D}& \end{array}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}C\stackrel{L}{\to }D$\epsilon L . L\eta = 1_L \qquad \text{i.e.} \qquad \array{\arrayopts{ \padding{0} } &&1_C& \\ \cellopts{\colspan{5}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="8.5em" height="2em" viewBox="0 0 85 20"> <defs> <marker id='svg195arrowhead' markerHeight='5' markerUnits='strokeWidth' markerWidth='8' orient='auto' refX='0' refY='5' viewBox='0 0 10 10'> <path d='M 0 0 L 10 5 L 0 10 z'/> </marker> </defs> <path marker-end='url(#svg195arrowhead)' stroke-width="1" stroke="#000" fill="none" d="M5 15q40-28 75 0"/> <foreignObject height='20' width='20' x='40' y='3' font-size='10'><math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><msup><mo>&#8659;</mo><mi>&#951;</mi></msup>[/itex]</foreignObject> </svg> \end{svg}\\ C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D \\ &&\cellopts{\colspan{4}}\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="8.5em" height="2em" viewBox="0 0 85 20"> <path marker-end='url(#svg195arrowhead)' stroke-width="1" stroke="#000" fill="none" d="M5 5q40 28 75 0"/> <foreignObject height='20' width='20' x='40' y='0' font-size='10'><math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><msup><mo>&#8659;</mo><mi>&#1013;</mi></msup>[/itex]</foreignObject> </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 April 15, 2013 17:31:34 by Tim Porter (95.147.236.104)