nLab monoidal action of a monoidal category

Idea

A monoidal actegory is a pseudoaction in the 2-category $\mathbf{MonCat}$ of monoidal categories, strong monoidal functors and monoidal natural transformations. Roughly, it is an actegory whose actee is monoidal and such that the action is compatible with this structure.

Definition

Since pseudoactions are actions of pseudomonoids, and since a pseudomonoid in $\mathbf{MonCat}$ is a braided monoidal category, monoidal actegories are always actions of braided monoidal categories.

The following comes from Definition 5.1.1 in (Capucci and Gavranović 2022):

Definition

Let $(\mathcal{M}, j, \otimes)$ be a braided monoidal category. A left monoidal $\mathcal{M}$-actegory is a monoidal category $(\mathcal{C}, i,\boxtimes)$ together with a strong monoidal functor, called action, $\odot:\mathcal{M} \times \mathcal{C} \to \mathcal{C}$, and monoidal natural transformations $\eta_x : j \odot x \to x$, $\mu_{m,n,x} : (m \otimes n) \odot x \to m \odot (n \odot x)$ satisfying the coherence laws of unitor and multiplicator of an actegory.

All the coherence laws of a monoidal actegory are spelled out in Appendix A of (Capucci and Gavranović 2022).

The fact $\odot$ is strong monoidal means it comes equipped with a structural morphism called mixed interchanger:

$$\iota_{m,x,n,y} : (m \otimes n) \odot (x \boxtimes y) \to (m \odot x) \boxtimes (n \odot y).$$

It would also come with $\upsilon : i \to j \odot i$, but the monoidality of $\eta$ makes it so that $\upsilon = \eta_i$, making the first redundant.

Braiding

Consider pseudoactions in $\mathbf{BrMonCat}$ and $\mathbf{SymMonCat}$. Now carriers of the action are braided functors, and the actee are braided monoidal categories.

The following appear as 5.4.1 and 5.4.3 in (Capucci and Gavranović 2022):

Definition

Let $(\mathcal{M}, j, \otimes, \sigma)$ be a symmetric monoidal category. A left braided monoidal $\mathcal{M}$-actegory is a braided monoidal category $(\mathcal{C}, i, \boxtimes, \beta)$ equipped with a braided monoidal functor $\odot: \mathcal{M} \times \mathcal{C} \to \mathcal{C}$ and two monoidal natural transformations $\eta$ and $\mu$ as above. Thus it satisfies the axioms of monoidal actegory and an additional compatibility axiom with the braided structure of $\mathcal{M}$ and $\mathcal{C}$:

If $\mathcal{C}$ is symmetric, then we call $(\mathcal{C}, \odot)$ a symmetric monoidal actegory.

Classifying objects

Actegories are famously equivalent to monoidal functors $\mathcal{M} \to [\mathcal{C},\mathcal{C}]$. We can thus say that $[\mathcal{C},\mathcal{C}]$ classifies actions on $\mathcal{C}$.

When the action is monoidal, braided or symmetric monoidal, the classifier is different:

• Monoidal actions are classified by $Z(\mathcal{C})$, i.e. the Drinfeld center of $\mathcal{C}$.
• Braided monoidal actions are classified by $\Sigma(\mathcal{C})$, i.e. the symmetric center? of $\mathcal{C}$, which is the full subcategory of $\mathcal{C}$ spanned by those objects $x:\mathcal{C}$ such that $\beta_{x,-}\beta_{-,x} = 1$.
• Symmetric monoidal actions are classified by $\mathcal{C}$ itself. In fact a symmetric monoidal action $\odot$ of $\mathcal{M}$ on $\mathcal{C}$ gives a symmetric monoidal functor $- \odot i : \mathcal{M} \to \mathcal{C}$, and viceversa $F(-) \boxtimes -$ is a symmetric monoidal action whenever $\mathcal{M} \to \mathcal{C}$ is a braided monoidal functor between symmetric monoidal categories.

This is all proven in Section 5.5 of Capucci & Gavranović 2022.

See also

• actegories, pseudoaction
• para construction ($\mathbf{Para}(\odot)$ is monoidal iff $\odot$ is)
• periodic table

References

• Matteo Capucci, Bruno Gavranović, Actegories for the working amthematician [arXiv:2203.16351]