nLab
pseudonatural transformation
Redirected from "pseudo-natural transformation".
Contents
Context
2-Category theory
2-category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Contents
Idea
A pseudonatural transformation is a lax natural transformation whose 2 2 -cell components are all invertible .
Definition
Definition
Given two 2-functors U , V : S → → C U, V: S \stackrel{\to}{\to} C between 2-categories , a pseudonatural transformation ϕ : U → V \phi: U \to V is a rule that assigns to each object s s of S S a morphism ϕ ( s ) : U ( s ) → V ( s ) \phi(s): U(s) \to V(s) of C C , and to each morphism f : r → s f: r \to s of S S an invertible 2-morphism ϕ ( f ) \phi(f) of C C :
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 law s are satisfied in C C (throughout we leave the associator s and unitor s in C C implicit):
respect for composition: for all composable morphisms r → f s → g t r \stackrel{f}{\to} s \stackrel{g}{\to} t in S S 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 ( g ∘ f ) V ( t ) = U ( s ) U ( f ) ↗ ⇓ U ( f , g ) ↘ U ( g ) U ( r ) → U ( g ∘ f ) U ( t ) ϕ ( r ) ↓ ⇙ ϕ ( g ∘ f ) ↓ ϕ ( t ) V ( r ) → V ( g ∘ f ) 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(t)
}
\;\;\;
=
\;\;\;
\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-functor s U U and V V ,
respect for units, (…)
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 S S 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 C C .
A pseudonatural transformation is called a pseudonatural equivalence if each component ϕ ( s ) \phi(s) is an equivalence in the 2-category C C . 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 .
References
A generalization to extranatural transformations can be found in
Alexander S. Corner, A universal characterisation of codescent objects , TAC 2019.
Discussion of the globular approach can be found in
Camell Kachour : Kamel Kachour, Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2008), volume 1, pages 1–68.
Camell Kachour : Steps toward the Weak ω-category of the Weak ω-categories in the globular setting, Published in : Categories and General Algebraic Structures with Applications (2015).
Last revised on June 11, 2022 at 20:54:24.
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