nLab
monoidal natural transformation

Contents

Context

Monoidal categories

monoidal categories

With symmetry

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 (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) be two monoidal categories. Let (F i,μ i,ϵ i)(F_i, \mu_i, \epsilon_i) for i{1,2}i \in \{1,2\} be two lax monoidal functors 𝒞𝒟\mathcal{C}\longrightarrow \mathcal{D}.

Then a monoidal natural transformation ff from (F 1,μ 1,ϵ 1)(F_1,\mu_1, \epsilon_1) to (F 2,μ 2,ϵ 2)(F_2,\mu_2, \epsilon_2) is a natural transformation f(x):F 1(x)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,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f x 𝒟f y F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f x 𝒞y F 2(x 𝒞y) \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 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f 1 𝒞 F 2(1 𝒞). \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 May 3, 2019 at 20:52:45. See the history of this page for a list of all contributions to it.