triangle identities (changes) in nLab

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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\mathcal{C}, \mathcal{D} a pair of categories, or, generally, of objects in a given 2-category;

2. L:𝒞→𝒟L \colon \mathcal{C} \to \mathcal{D} and R:𝒟→𝒞R \colon \mathcal{D} \to \mathcal{C} a pair of functors between these, or generally 1-morphisms in the ambient 2-category;

3. η:id 𝒞⇒R∘L\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L and ϵ:L∘R⇒id 𝒟\epsilon \colon L \circ R \Rightarrow id_{\mathcal{D}} two natural transformations or, generally 2-morphisms.

This data is called a 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

(ϵL)∘(Lη)=id L \big( \epsilon L \big) \circ \big( L \eta \big) \;=\; id_L

(Rϵ)∘(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⇒ϵLL=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⇒RϵR=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:

\begin{xymatrix} \mathcal{C} \ar[r]|-{ \;L\; } \ar@/^3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{C}}\; }_-{\ }=“s1” & \mathcal{D} \ar[r]|-{ \;R\; } \ar@/_3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{D}}\; }^-{\ }=“t2” & \mathcal{C} \ar[r]|-{ \;L\; } & \mathcal{D} & = & \mathcal{C} \ar[r]|-{ \;L\; } & \mathcal{D} % \ar@{=>}^\eta “s1”+(0,-2); “s1”+(0,-8) \ar@{=>}^\epsilon “t2”+(0,8); “t2”+(0,2) \end{xymatrix}

\begin{xymatrix} \mathcal{D} \ar[r]|-{ \;R\; } \ar@/_3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{D}}\; }^-{\ }=“s1” & \mathcal{C} \ar[r]|-{ \;L\; } \ar@/^3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{C}}\; }_-{\ }=“t2” & \mathcal{D} \ar[r]|-{ \;R\; } & \mathcal{C} & = & \mathcal{D} \ar[r]|-{ \;R\; } & \mathcal{C} % \ar@{=>}^\eta “t2”+(0,-2); “t2”+(0,-8) \ar@{=>}^\epsilon “s1”+(0,8); “s1”+(0,2) \end{xymatrix}

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 “pulling zigzags straight” (hence the name):

With labels left implicit, this notation becomes very economical:

, .

• adjunction

• unit of an adjunction

• adjunct

References

Textbook accounts include

• Francis Borceux, Theorem 3.1.5 and Diagram 3.3 in: Basic Category Theory, Vol. 1 of Handbook of Categorical Algebra, Cambridge University Press (1994)

See the references at category theory for more.