nLab
pseudonatural transformation

Contents

Idea

A pseudonatural transformation is a lax natural transformation whose 22-cell components are all invertible.

Definition

Definition

Given two 2-functors U,V:SCU, V: S \stackrel{\to}{\to} C between 2-categories, a pseudonatural transformation ϕ:UV\phi: U \to V is a rule that assigns to each object ss of SS a morphism ϕ(s):U(s)V(s)\phi(s): U(s) \to V(s) of CC, and to each morphism f:rsf: r \to s of SS an invertible 2-morphism ϕ(f)\phi(f) of CC:

U(r) U(f) U(s) ϕ(r) ϕ(f) ϕ(s) V(r) V(f) V(s) \array{ U(r) & \stackrel{U(f)}{\to} & U(s) \\ \phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) \\ V(r) & \underset{V(f)}{\to} & V(s) }

such that the following coherence laws are satisfied in CC (throughout we leave the associators and unitors in CC implicit):

  1. respect for composition: for all composable morphisms rfsgtr \stackrel{f}{\to} s \stackrel{g}{\to} t in SS we have an equality

    U(s) U(f) ϕ(s) U(g) U(r) ϕ(f) V(s) ϕ(g) U(t) ϕ(r) V(f) V(f,g) V(g) ϕ(t) V(r) V(gf) V(s)= U(s) U(f) U(f,g) U(g) U(r) U(gf) U(t) ϕ(r) ϕ(gf) ϕ(t) V(r) V(gf) V(t), \array{ && U(s) \\ & {}^{\mathllap{U(f)}}\nearrow &\downarrow^{\phi(s)}& \searrow^{\mathrlap{U(g)}} \\ U(r) &\swArrow_{\phi(f)}&V(s) &\swArrow_{\phi(g)}& U(t) \\ {}^{\mathllap{\phi(r)}}\downarrow &{}^{V(f)}\nearrow& \Downarrow^{V(f,g)} &\searrow^{V(g)}& \downarrow^{\mathrlap{\phi(t)}} \\ V(r) &&\underset{V( g\circ f)}{\to}&& V(s) } \;\;\; = \;\;\; \array{ && U(s) \\ & {}^{\mathllap{U(f)}}\nearrow &\Downarrow^{U(f,g)}& \searrow^{\mathrlap{U(g)}} \\ U(r) &&\stackrel{U(g \circ f)}{\to}&& U(t) \\ {}^{\mathllap{\phi(r)}}\downarrow && \swArrow_{\phi(g \circ f )} && \downarrow^{\mathrlap{\phi(t)}} \\ V(r) &&\underset{V(g \circ f)}{\to}&& V(t) } \,,

    of pasting 2-morphisms as indicated, where U(f,g)U(f,g) and V(f,g)V(f,g) denote the compositors of the 2-functors UU and VV,

  2. respect for units, (…)

  3. naturality

    for every 2-morphism

    f r F s g \array{ && \stackrel{f}{\to} \\ & \nearrow && \searrow \\ r &&\Downarrow^{F}&& s \\ & \searrow && \nearrow \\ && \underset{g}{\to} }

    in SS an equality

    U(f) U(F) U(r) U(g) U(s) ϕ(r) ϕ(g) ϕ(s) V(r) V(g) V(s)=U(r) U(f) U(s) ϕ(r) ϕ(f) ϕ(s) V(r) V(f) V(s) V(F) V(g) \array{ && \stackrel{U(f)}{\to} \\ & \nearrow &\Downarrow^{U(F)}& \searrow \\ U(r) &&\stackrel{U(g)}{\to}&& U(s) \\ {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(g)}&& \downarrow^{\mathrlap{\phi(s)}} \\ V(r) &&\underset{V(g)}{\to}&& V(s) } \;\;\; = \;\;\; \array{ U(r) &&\stackrel{U(f)}{\to}&& U(s) \\ {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(f)}&& \downarrow^{\mathrlap{\phi(s)}} \\ V(r) &&\stackrel{V(f)}{\to}&& V(s) \\ & \searrow &\Downarrow^{V(F)}& \nearrow \\ && \underset{V(g)}{\to} }

    in CC.

A pseudonatural transformation is called a pseudonatural equivalence if each component ϕ(s)\phi(s) is an equivalence in the 2-category CC. This is equivalent to ϕ\phi itself being an equivalence in the 2-category [S,C][S,C] of 2-functors, pseudonatural transformations, and modifications.

Revised on March 12, 2013 01:41:15 by Mike Shulman (108.247.159.128)