Contents

Idea

Thetriangle identities orzigzag identities are identities characterized by theunit and counit of anadjunction, such as apair ofadjoint functors. These identitiesdefine, equivalently, the nature of adjunction (this prop.).

Statement

Consider:

1. $𝒞,𝒟$ apair ofcategories, or, generally, ofobjects in a given2-category;

2. $L:𝒞\to 𝒟$ and$R:𝒟\to 𝒞$ a pair offunctors between these, or generally1-morphisms in the ambient2-category;

3. $\eta :{\mathrm{id}}_{𝒞}\Rightarrow R\circ L$ and$ϵ:L\circ R\Rightarrow {\mathrm{id}}_{𝒟}$ twonatural transformations or, generally2-morphisms.

This data is called apair ofadjoint functors (generally: anadjunction) if thetriangle 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

Asequations, the triangle identities read

Here juxtaposition denotes thewhiskering operation of1-morphisms on2-morphisms, as made more manifest in the diagrammatic unravelling of these expressions:

As diagrams

In terms ofdiagrams in thefunctor categories this means

and

In terms of diagrams of2-morphisms in the ambient2-category, this looks as follows:

where on the right theidentity2-morphisms are left notationally implicit.

If we leave theidentity1-morphisms on the left notationally implicit, then we get the following suggestive form of the triangle identities:

(taken fromgeometry of physics – categories and toposes).

As string diagrams

Asstring diagrams, the triangle identities appear as the action of “pulling zigzags straight” (hence the name):

With labels left implicit, this notation becomes very economical:

Related concepts

• 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 ofHandbook of Categorical Algebra, Cambridge University Press (1994)

See the references atcategory theory for more.

Last revised on June 22, 2023 at 16:20:30. See thehistory of this page for a list of all contributions to it.