nLab
triangle identities

Contents

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,DC, D be two categories, or, generally, two objects of a given 2-category;

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

  3. η:id CRL\eta: id_C \Rightarrow R \circ L and ϵ:LRid D\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:

  1. As equations

  2. As diagrams

  3. As string diagrams

\,

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. (Here, the composition of the 11- with the 22-morphisms is sometimes called whiskering.)

As diagrams

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

ϵL.Lη = id L 1 C [[!include adjunction > zigzageta]] C L D R C L D [[!include adjunction > zigzagepsilon]] 1 D = CLDAAAAAandAAAAARϵ.ηR = id R 1 C [[!include adjunction > zigzageta]] D R C L D R C [[!include adjunction > zigzagepsilon]] 1 D = DRC \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):

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').

References

Textbook accounts include

See the references at category theory for more.

Last revised on July 23, 2018 at 10:28:20. See the history of this page for a list of all contributions to it.