nLab
lax F-natural transformation

Lax \mathcal{F}-natural transformations

Definition

Let KK and LL be F-categories (strict for simplicity), with 2-categories K τ,L τK_\tau,L_\tau of tight morphisms and K λ,L λK_\lambda,L_\lambda of loose morphisms. Let F,G:KLF,G:K\to L be \mathcal{F}-functors (also strict for simplicity).

A pseudo/lax \mathcal{F}-natural transformation α:FG\alpha : F \to G consists of:

  • A lax natural transformation α λ:F λG λ\alpha_\lambda : F_\lambda \to G_\lambda,
  • The 1-morphism components of α λ\alpha_\lambda are tight, and
  • The 2-morphism naturality constraint of α λ\alpha_\lambda at any tight arrow of KK is an isomorphism.

In particular, α λ\alpha_\lambda restricts to a pseudo natural transformation α τ:F τG τ\alpha_\tau : F_\tau \to G_\tau.

Similarly, an \mathcal{F}-natural transformation can be strict/lax, pseudo/oplax, and so on.

Examples

  • If KK and LL are chordate (all morphisms are tight), then a pseudo/lax transformation is simply a pseudo transformation.

  • If LL is chordate but KK is inchordate (only identities are tight), then a pseudo/lax transformation is simply a lax transformation.

  • If LL is inchordate, there are no nonidentity pseudo/lax transformations.

Applications

  • Lax \mathcal{F}-natural transformations appear in the notion of lax F-adjunction.

Created on March 4, 2018 at 00:15:47. See the history of this page for a list of all contributions to it.