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. π’ž,π’Ÿ\mathcal{C}, \mathcal{D} a pair of categories, or, generally, of objects in a given 2-category;

  2. L:π’žβ†’π’ŸL \colon \mathcal{C} \to \mathcal{D} and R:π’Ÿβ†’π’žR \colon \mathcal{D} \to \mathcal{C} a pair of functors between these, or generally 1-morphisms in the ambient 2-category;

  3. Ξ·:id π’žβ‡’R∘L\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L and Ο΅:L∘Rβ‡’id π’Ÿ\epsilon \colon L \circ R \Rightarrow id_{\mathcal{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

As equations, the triangle identities read

(ϡL)∘(Lη)=id L \big( \epsilon L \big) \circ \big( L \eta \big) \;=\; id_L
(Rϡ)∘(ηR)=id R \big( R \epsilon \big) \circ \big( \eta R \big) \;=\; id_R

Here juxtaposition denotes the whiskering operation of 1-morphisms on 2-morphisms, as made more manifest in the diagrammatic unravelling of these expressions:

As diagrams

In terms of diagrams in the functor categories this means

L⇒LηLRL⇒ϡLL=L⇒id LL L \overset{\;\;L\eta\;\;}{\Rightarrow} L R L \overset{\;\;\epsilon L\;\;}{\Rightarrow} L \;\; = \;\; L \overset{\;\;id_L\;\;}{\Rightarrow} L

and

R⇒ηRRLR⇒RϡR=R⇒id RR R \overset{\;\;\eta R\;\;}{\Rightarrow} R L R \overset{\;\;R\epsilon\;\;}{\Rightarrow} R \;\; = \;\; R \overset{\;\;id_R\;\;}{\Rightarrow} R

In terms of diagrams of 2-morphisms in the ambient 2-category, this looks as follows:

where on the right the identity 2-morphisms are left notationally implicit.

If we leave the identity 1-morphisms on the left notationally implicit, then we get the following suggestive form of the triangle identities:

(taken from geometry of physics – categories and toposes).

As string diagrams

As string diagrams, the triangle identities appear as the action of β€œpulling 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 December 30, 2020 at 07:47:43. See the history of this page for a list of all contributions to it.