With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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):
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:
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.
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):
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.
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:
This is all proven in Section 5.5 of Capucci & Gavranović 2022.
Last revised on November 27, 2023 at 13:35:30. See the history of this page for a list of all contributions to it.