nLab
pseudonatural transformation

Idea

A pseudonatural transformation is a lax natural transformation whose $2$-cell components are all invertible.

Definition

Definition

Given two 2-functors $U,V:S\overrightarrow{\to }C$ between 2-categories, a pseudonatural transformation $\varphi :U\to V$ is a rule that assigns to each object $s$ of $S$ a morphism $\varphi (s):U(s)\to V(s)$ of $C$, and to each morphism $f:r\to s$ of $S$ an invertible 2-morphism $\varphi (f)$ of $C$:

$$\begin{array}{ccc}U(r)& \stackrel{U(f)}{\to }& U(s)\\ \varphi (r)\downarrow & \varphi (f)⇙& \downarrow \varphi (s)\\ V(r)& \underset{V(f)}{\to }& V(s)\end{array}$$\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 $C$ (throughout we leave the associators and unitors in $C$ implicit):

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

   $$\begin{array}{ccc}& & U(s)\\ & {}^{U(f)}\nearrow & {\downarrow }^{\varphi (s)}& {\searrow }^{U(g)}\\ U(r)& {⇙}_{\varphi (f)}& V(s)& {⇙}_{\varphi (g)}& U(t)\\ {}^{\varphi (r)}\downarrow & {}^{V(f)}\nearrow & {\Downarrow }^{V(f,g)}& {\searrow }^{V(g)}& {\downarrow }^{\varphi (t)}\\ V(r)& & \underset{V(g\circ f)}{\to }& & V(s)\end{array}=\begin{array}{ccc}& & U(s)\\ & {}^{U(f)}\nearrow & {\Downarrow }^{U(f,g)}& {\searrow }^{U(g)}\\ U(r)& & \stackrel{U(g\circ f)}{\to }& & U(t)\\ {}^{\varphi (r)}\downarrow & & {⇙}_{\varphi (g\circ f)}& & {\downarrow }^{\varphi (t)}\\ V(r)& & \underset{V(g\circ f)}{\to }& & V(t)\end{array},$$\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 $U(f,g)$ and $V(f,g)$ denote the compositors of the 2-functors $U$ and $V$,

2. respect for units, (…)

3. naturality

   for every 2-morphism

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

   in $S$ an equality

   $$\begin{array}{ccc}& & \stackrel{U(f)}{\to }\\ & \nearrow & {\Downarrow }^{U(F)}& \searrow \\ U(r)& & \stackrel{U(g)}{\to }& & U(s)\\ {}^{\varphi (r)}\downarrow & & {⇙}_{\varphi (g)}& & {\downarrow }^{\varphi (s)}\\ V(r)& & \underset{V(g)}{\to }& & V(s)\end{array}=\begin{array}{ccccc}U(r)& & \stackrel{U(f)}{\to }& & U(s)\\ {}^{\varphi (r)}\downarrow & & {⇙}_{\varphi (f)}& & {\downarrow }^{\varphi (s)}\\ V(r)& & \stackrel{V(f)}{\to }& & V(s)\\ & \searrow & {\Downarrow }^{V(F)}& \nearrow \\ & & \underset{V(g)}{\to }\end{array}$$\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 $C$.

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

Related concepts

• homotopy

• transfor

  • natural transformation

    • dinatural transformation, extranatural transformation

  • pseudonatural transformation

  • lax natural transformation