nLab
whiskering

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

For instance in Cat whiskering is the composition of a functor with a natural transformation to produce a natural transformation, If we identify a functor or morphism with its identity natural transformation or identity 2-morphism?, then whiskering is a special case of horizontal composition, and composition of morphisms is a special case of whiskering.

In detail:

  • If F,G:CDF,G\colon C \to D and H:DEH\colon D\to E are functors and η:FG\eta\colon F \to G is a natural transformation whose coordinate at any object AA of CC is η A\eta_A, then whiskering HH and η\eta yields the natural transformation Hη:(HF)(HG)H \circ \eta\colon (H \circ F) \to (H \circ G) whose coordinate at AA is H(η A)H(\eta_A).
  • If F:CDF\colon C \to D and G,H:DEG,H\colon D \to E are functors and η:GH\eta\colon G\to H is a natural transformation whose coordinate at AA is η A\eta_A, then whiskering η\eta and FF yields the natural transformation ηF:(GF)(HF)\eta \circ F\colon (G \circ F) \to (H \circ F) whose coordinate at AA is η F(A)\eta_{F(A)}.

References

Revised on February 14, 2013 11:39:48 by Urs Schreiber (89.204.130.41)