TikZ lab: whiskering and coherence - Imported Knowlpedia Content

Let $\eta:F\Rightarrow G$ be a natural transformation between functors $F,G:\mathcal C\to\mathcal D$ , and let $H:\mathcal D\to\mathcal E$ . Left whiskering produces

H\eta:HF\Rightarrow HG,\qquad (H\eta)_X=H(\eta_X).

Right whiskering by $K:\mathcal B\to\mathcal C$ produces

\eta K:F K\Rightarrow G K,\qquad (\eta K)_B=\eta_{K B}.

Both operations preserve vertical composition:

H(\theta\circ\eta)=H\theta\circ H\eta,\qquad (\theta\circ\eta)K=(\theta K)\circ(\eta K).

The associator in a monoidal category is governed by Mac Lane's pentagon. In formulas, the two composites

(((A\otimes B)\otimes C)\otimes D)\longrightarrow A\otimes(B\otimes(C\otimes D))

agree.