nLab monoidal natural transformation

Idea

A monoidal natural transformation is a natural transformation between monoidal functors that respects the monoidal structure.

Definition

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

Then a monoidal natural transformation $f$ from $(F_1,\mu_1, \epsilon_1)$ to $(F_2,\mu_2, \epsilon_2)$ is a natural transformation $f(x) \;\colon\; F_1(x) \longrightarrow F_2(x)$ between the underlying functors that is compatible with the tensor product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:

$$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f_x \otimes_{\mathcal{D}} f_y}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f_{x \otimes_{\mathcal{C}} y} }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$$

and

$$\array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f_{1_{\mathcal{C}}}}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.$$

Related concepts

• symmetric monoidal natural transformation
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References

• Samuel Eilenberg, G. Max Kelly, p. 474 in: Closed Categories, in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

Exposition includes

• John Baez, Some definitions everyone should know, pdf.