nLab
triangle identities

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

$$Rϵ.\eta R=1_R\text{i.e.}\begin{array}{cccccc}& & & & 1_C& \\ & & \multicolumn{5}{c}{ $ {\Downarrow }^{\eta }$ }\\ D& \stackrel{R}{\to }& C& \stackrel{L}{\to }& D& \stackrel{R}{\to }& C\\ \multicolumn{4}{c}{ $ {\Downarrow }^{ϵ}$ }\\ & & 1_D& \end{array}=D\stackrel{R}{\to }C$$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></math></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></math></foreignObject> </svg> \end{svg} \\ &&1_D& } \quad = \quad D \stackrel{R}{\to} C

and

$$ϵL.L\eta =1_L\text{i.e.}\begin{array}{cccc}& & 1_C& \\ \multicolumn{5}{c}{ $ {\Downarrow }^{\eta }$ }\\ C& \stackrel{L}{\to }& D& \stackrel{R}{\to }& C& \stackrel{L}{\to }& D\\ & & \multicolumn{4}{c}{ $ {\Downarrow }^{ϵ}$ }\\ & & & & 1_D& \end{array}=C\stackrel{L}{\to }D$$\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></math></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></math></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:

, .

Related concepts

• adjunction

• zig-zag law / triangle identity

• unit of an adjunction

• adjunct