nLab monoidal action of a monoidal category



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory



A monoidal actegory is a pseudoaction in the 2-category MonCat\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 MonCat\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 (,j,)(\mathcal{M}, j, \otimes) be a braided monoidal category. A left monoidal \mathcal{M}-actegory is a monoidal category (𝒞,i,)(\mathcal{C}, i,\boxtimes) together with a strong monoidal functor, called action, :×𝒞𝒞\odot:\mathcal{M} \times \mathcal{C} \to \mathcal{C}, and monoidal natural transformations η x:jxx\eta_x : j \odot x \to x, μ m,n,x:(mn)xm(nx)\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:

ι m,x,n,y:(mn)(xy)(mx)(ny). \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 υ:iji\upsilon : i \to j \odot i, but the monoidality of η\eta makes it so that υ=η i\upsilon = \eta_i, making the first redundant.


Consider pseudoactions in BrMonCat\mathbf{BrMonCat} and SymMonCat\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 (,j,,σ)(\mathcal{M}, j, \otimes, \sigma) be a symmetric monoidal category. A left braided monoidal \mathcal{M}-actegory is a braided monoidal category (𝒞,i,,β)(\mathcal{C}, i, \boxtimes, \beta) equipped with a braided monoidal functor :×𝒞𝒞\odot: \mathcal{M} \times \mathcal{C} \to \mathcal{C} and two monoidal natural transformations η\eta and μ\mu as above. Thus it satisfies the axioms of monoidal actegory and an additional compatibility axiom with the braided structure of \mathcal{M} and 𝒞\mathcal{C}:

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

Classifying objects

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

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

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


Last revised on November 27, 2023 at 13:35:30. See the history of this page for a list of all contributions to it.