nLab
monoidal natural transformation
Redirected from "monoidal transformations".
Contents
Context
Monoidal categories
monoidal categories
With braiding
With duals for objects
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Category theory
Contents
Idea
A monoidal natural transformation is a natural transformation between monoidal functors that respects the monoidal structure.
Definition
Let ( π , β π , 1 π ) (\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and ( π , β π , 1 π ) (\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) be two monoidal categories . Let ( F i , ΞΌ i , Ο΅ i ) (F_i, \mu_i, \epsilon_i) for i β { 1 , 2 } i \in \{1,2\} be two lax monoidal functors π βΆ π \mathcal{C}\longrightarrow \mathcal{D} .
Then a monoidal natural transformation f f from ( F 1 , ΞΌ 1 , Ο΅ 1 ) (F_1,\mu_1, \epsilon_1) to ( F 2 , ΞΌ 2 , Ο΅ 2 ) (F_2,\mu_2, \epsilon_2) is a natural transformation f ( x ) : F 1 ( x ) βΆ F 2 ( x ) f(x) \;\colon\; F_1(x) \longrightarrow F_2(x) between the underlying functors that is compatible with the tensor product and the unit in that the following diagrams commute for all objects x , y β π x,y \in \mathcal{C} :
F 1 ( x ) β π F 1 ( y ) βΆ f x β π f y F 2 ( x ) β π F 2 ( y ) ( ΞΌ 1 ) x , y β β ( ΞΌ 2 ) x , y F 1 ( x β π y ) βΆ f x β π y F 2 ( x β π y )
\array{
F_1(x) \otimes_{\mathcal{D}} F_1(y)
&\overset{f_x \otimes_{\mathcal{D}} f_y}{\longrightarrow}&
F_2(x) \otimes_{\mathcal{D}} F_2(y)
\\
{}^{\mathllap{(\mu_1)_{x,y}}}\downarrow
&&
\downarrow^{\mathrlap{(\mu_2)_{x,y}}}
\\
F_1(x\otimes_{\mathcal{C}} y)
&\underset{f_{x \otimes_{\mathcal{C}} y} }{\longrightarrow}&
F_2(x \otimes_{\mathcal{C}} y)
}
and
1 π Ο΅ 1 β β Ο΅ 2 F 1 ( 1 π ) βΆ f 1 π F 2 ( 1 π ) .
\array{
&& 1_{\mathcal{D}}
\\
& {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}}
\\
F_1(1_{\mathcal{C}})
&&\underset{f_{1_{\mathcal{C}}}}{\longrightarrow}&&
F_2(1_{\mathcal{C}})
}
\,.
References
Exposition includes
Last revised on September 2, 2023 at 12:14:49.
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