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Lax natural transformation

Definition

Definition. A lax-natural transformation α:FG\alpha \colon F \Rightarrow G of 2-functors F,G:KLF, G \colon K \to L consists of a family of arrows α A:FAGA\alpha_A \colon F A \to G A in LL indexed over the objects AKA \in K together with, for each arrow ff of KK, a distinguished 2-cell

α f:Gfα Aα BFf\alpha_f \colon Gf \circ \alpha_A \Rightarrow \alpha_B \circ Ff

satisfying the two following coherence conditions.

  1. for any composable arrows ff and gg in KK, α f*α g=α gf\alpha_f \ast \alpha_g = \alpha_{gf} where *\ast means pasting the two squares.
  2. for any 2-cell θ:fg\theta \colon f \Rightarrow g in KK, FθF\theta and GθG\theta form the top and bottom of a commutative cylinder connecting the squares for α f\alpha_f and α g\alpha_g

If the α f\alpha_f are all invertible, α\alpha is called a pseudonatural transformation. If the α f\alpha_f are all identities, α\alpha is called a 2-natural transformation.

References