nLab
triangle identities

Contents

Idea

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

Statement

Given C,DC, D (categories, or otherwise objects of a 22-category) with functors (or otherwise morphisms) L:CDL: C \to D and R:DCR : D \to C and natural isomorphisms (or otherwise 22-morphisms) η:1 CRL\eta: 1_C \to R \circ L and ϵ:LR1 D\epsilon: L \circ R \to 1_D, the triangle identities are the following:

As equations

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

and

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

are identities.

As diagrams

Rϵ.ηR=1 R R\epsilon . \eta R = 1_R i.e.

1 C η D R C L D R C ϵ 1 D =DRC \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.Lη=1 L \epsilon L . L\eta = 1_L i.e.

1 C η C L D R C L D ϵ 1 D =CLD \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):

String diagram of first zigzag identity (for 'Adjunction'), .

With labels left implicit, this notation becomes very economical:

Minimal string diagram of first zigzag identity (for 'Adjunction'), Minimal string diagram of second zigzag identity (for 'Adjunction').

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