nLab triangle identities

Contents

category theory

Applications

2-Category theory

2-category theory

Contents

Idea

The triangle identities or zigzag identities are identities characterized by the unit and counit of an adjunction, such as a pair of adjoint functors. These identities define, equivalently, the nature of adjunction (this prop.).

Statement

Consider:

1. $C, D$ be two categories, or, generally, two objects of a given 2-category;

2. $L: C \to D$ and $R : D \to C$ two functors between these, or generally 1-morphisms in the ambient 2-category;

3. $\eta: id_C \Rightarrow R \circ L$ and $\epsilon: L \circ R \Rightarrow id_D$ two natural transformations or, generally 2-morphisms.

This data is called an pair of adjoint functors (generally: an adjunction) if the triangle identities are satisfied, which may be expressed in any of the following equivalent ways:

$\,$

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. (Here, the composition of the $1$- with the $2$-morphisms is sometimes called whiskering.)

As diagrams

As diagrams in the ambient 2-category, the triangle identities look as follows

$\array{ \epsilon L . L\eta &=& id_L \\ \array{\arrayopts{ \padding{0} } &&1_C& \\ \cellopts{\colspan{5}}\begin{svg} [[!include adjunction > zigzageta]] \end{svg}\\ C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D \\ &&\cellopts{\colspan{4}}\begin{svg} [[!include adjunction > zigzagepsilon]] \end{svg} \\ &&&&1_D& } & = & C \stackrel{L}{\to} D } \phantom{AAAAA} \text{and} \phantom{AAAAA} \array{ R\epsilon . \eta R &=& id_R \\ \array{\arrayopts{ \padding{0} } \\ &&&&1_C& \\ &&\cellopts{\colspan{5}}\begin{svg} [[!include adjunction > zigzageta]] \end{svg}\\ D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C \\ \cellopts{\colspan{4}}\begin{svg} [[!include adjunction > zigzagepsilon]] \end{svg} \\ &&1_D& } &=& D \stackrel{R}{\to} C }$

or, equivalently, like so: As string diagrams

As string diagrams, the triangle identities appear as the action of “pulling zigzags straight” (hence the name):  With labels left implicit, this notation becomes very economical: , .

Textbook accounts include

See the references at category theory for more.