# nLab pseudonatural transformation

### Context

#### 2-Category theory

2-category theory

# Contents

## Idea

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

## Definition

###### Definition

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

$\begin{array}{ccc}U\left(r\right)& \stackrel{U\left(f\right)}{\to }& U\left(s\right)\\ \varphi \left(r\right)↓& \varphi \left(f\right)⇙& ↓\varphi \left(s\right)\\ V\left(r\right)& \underset{V\left(f\right)}{\to }& V\left(s\right)\end{array}$\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

$\begin{array}{ccc}& & U\left(s\right)\\ & {}^{U\left(f\right)}↗& {↓}^{\varphi \left(s\right)}& {↘}^{U\left(g\right)}\\ U\left(r\right)& {⇙}_{\varphi \left(f\right)}& V\left(s\right)& {⇙}_{\varphi \left(g\right)}& U\left(t\right)\\ {}^{\varphi \left(r\right)}↓& {}^{V\left(f\right)}↗& {⇓}^{V\left(f,g\right)}& {↘}^{V\left(g\right)}& {↓}^{\varphi \left(t\right)}\\ V\left(r\right)& & \underset{V\left(g\circ f\right)}{\to }& & V\left(s\right)\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccc}& & U\left(s\right)\\ & {}^{U\left(f\right)}↗& {⇓}^{U\left(f,g\right)}& {↘}^{U\left(g\right)}\\ U\left(r\right)& & \stackrel{U\left(g\circ f\right)}{\to }& & U\left(t\right)\\ {}^{\varphi \left(r\right)}↓& & {⇙}_{\varphi \left(g\circ f\right)}& & {↓}^{\varphi \left(t\right)}\\ V\left(r\right)& & \underset{V\left(g\circ f\right)}{\to }& & V\left(t\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 $U\left(f,g\right)$ and $V\left(f,g\right)$ denote the compositors of the 2-functors $U$ and $V$,

2. respect for units, (…)

3. naturality

for every 2-morphism

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

in $S$ an equality

$\begin{array}{ccc}& & \stackrel{U\left(f\right)}{\to }\\ & ↗& {⇓}^{U\left(F\right)}& ↘\\ U\left(r\right)& & \stackrel{U\left(g\right)}{\to }& & U\left(s\right)\\ {}^{\varphi \left(r\right)}↓& & {⇙}_{\varphi \left(g\right)}& & {↓}^{\varphi \left(s\right)}\\ V\left(r\right)& & \underset{V\left(g\right)}{\to }& & V\left(s\right)\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccccc}U\left(r\right)& & \stackrel{U\left(f\right)}{\to }& & U\left(s\right)\\ {}^{\varphi \left(r\right)}↓& & {⇙}_{\varphi \left(f\right)}& & {↓}^{\varphi \left(s\right)}\\ V\left(r\right)& & \stackrel{V\left(f\right)}{\to }& & V\left(s\right)\\ & ↘& {⇓}^{V\left(F\right)}& ↗\\ & & \underset{V\left(g\right)}{\to }\end{array}$\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 $\varphi \left(s\right)$ is an equivalence in the 2-category $C$. This is equivalent to $\varphi$ itself being an equivalence in the 2-category $\left[S,C\right]$ of 2-functors, pseudonatural transformations, and modifications.

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