nLab whiskering

Whiskering

Context

2-Category theory

2-category theory

Whiskering

Idea

In a 2-category, the horizontal composition of a 2-morphism with 1-morphisms is sometimes called whiskering.

Whiskering from the left with an equivalence and from the right with an inverse equivalence is a conjugation action of equivalences on 2-morphisms.

Examples

An important use of whiskering is the usual definition of adjoint functors via the triangle identities: in Cat whiskering is the composition of a functor with a natural transformation to produce a natural transformation.

If we identify a functor or 1-morphism with its identity natural transformation or identity 2-morphism?, then whiskering is a special case of horizontal composition, and composition of 1-morphisms is a special case of whiskering.

In detail:

• If $F,G\colon C \to D$ and $H\colon D\to E$ are functors and $\eta\colon F \to G$ is a natural transformation whose coordinate at any object $A$ of $C$ is $\eta_A$, then whiskering $H$ and $\eta$ yields the natural transformation $H \circ \eta\colon (H \circ F) \to (H \circ G)$ whose coordinate at $A$ is $H(\eta_A)$.
• If $F\colon C \to D$ and $G,H\colon D \to E$ are functors and $\eta\colon G\to H$ is a natural transformation whose coordinate at $A$ is $\eta_A$, then whiskering $\eta$ and $F$ yields the natural transformation $\eta \circ F\colon (G \circ F) \to (H \circ F)$ whose coordinate at $A$ is $\eta_{F(A)}$.