triangle identities



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


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


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.

and ϵL.Lη=1 L \epsilon L . L\eta = 1_L i.e.

The RHS of the above diagrams have L and R interchanged. Furthermore, the LHS has C as target of L instead of 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 July 16, 2014 00:03:35 by Marco? (