Sandwich natural transformation between two functors
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In the Kleisli adjunction we have:
$Gvarepsilon F = mu$ where $varepsilon$ is a natural transformation called the counit.
How exactly is $Gvarepsilon F$ defined? I understand $Gvarepsilon$ and $varepsilon F$ but not the sandwiched form.
Followup question: Counit of the Kleisli adjunction
category-theory adjoint-functors functors natural-transformations
asked Nov 24 at 9:14
Roland
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In the Kleisli adjunction we have:
$Gvarepsilon F = mu$ where $varepsilon$ is a natural transformation called the counit.
How exactly is $Gvarepsilon F$ defined? I understand $Gvarepsilon$ and $varepsilon F$ but not the sandwiched form.
Followup question: Counit of the Kleisli adjunction
category-theory adjoint-functors functors natural-transformations
asked Nov 24 at 9:14
Roland
19311
2
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50
add a comment |
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0
down vote
favorite
up vote
0
down vote
favorite
In the Kleisli adjunction we have:
$Gvarepsilon F = mu$ where $varepsilon$ is a natural transformation called the counit.
How exactly is $Gvarepsilon F$ defined? I understand $Gvarepsilon$ and $varepsilon F$ but not the sandwiched form.
Followup question: Counit of the Kleisli adjunction
category-theory adjoint-functors functors natural-transformations
asked Nov 24 at 9:14
Roland
19311
In the Kleisli adjunction we have:
$Gvarepsilon F = mu$ where $varepsilon$ is a natural transformation called the counit.
How exactly is $Gvarepsilon F$ defined? I understand $Gvarepsilon$ and $varepsilon F$ but not the sandwiched form.
Followup question: Counit of the Kleisli adjunction
category-theory adjoint-functors functors natural-transformations
category-theory adjoint-functors functors natural-transformations
asked Nov 24 at 9:14
Roland
19311
asked Nov 24 at 9:14
Roland
19311
edited Nov 24 at 9:52
asked Nov 24 at 9:14
Roland
19311
asked Nov 24 at 9:14
Roland
19311
asked Nov 24 at 9:14
Roland
19311
19311
2
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50
add a comment |
2
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50
2
2
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50
add a comment |
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2
If I call $varepsilon F$ $alpha$ and ask what is $Galpha$, do you understand it then? It's compositional. There is, of course, a coherence constraint you could verify. Namely that $Galpha = beta F$ where $beta = Gvarepsilon$. Incidentally, I personally prefer representing right whiskering by subscripting, i.e. $Gvarepsilon_F$.
– Derek Elkins
Nov 24 at 9:20
@DerekElkins I posted a followup math.stackexchange.com/questions/3011366/…
– Roland
Nov 24 at 9:50