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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.).
Consider:
$\mathcal{C}, \mathcal{D}$ a pair of categories, or, generally, of objects in a given 2-category;
$L \colon \mathcal{C} \to \mathcal{D}$ and $R \colon \mathcal{D} \to \mathcal{C}$ a pair of functors between these, or generally 1-morphisms in the ambient 2-category;
$\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L$ and $\epsilon \colon L \circ R \Rightarrow id_{\mathcal{D}}$ 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:
$\,$
As equations, the triangle identities read
Here juxtaposition denotes the whiskering operation of 1-morphisms on 2-morphisms, as made more manifest in the diagrammatic unravelling of these expressions:
In terms of diagrams in the functor categories this means
and
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 β categories and toposes).
As string diagrams, the triangle identities appear as the action of βpulling zigzags straightβ (hence the name):
With labels left implicit, this notation becomes very economical:
Textbook accounts include
See the references at category theory for more.
Last revised on June 22, 2023 at 16:20:30. See the history of this page for a list of all contributions to it.