Proposition [efr-6S13]

There is a limit sketch \mathcal {T}_\mathsf {Act}, \Theta _\mathsf {Act} whose models are tuples (M,C,\cdot ) of a pseudomonoid M = (M,\otimes ,I), an object C and an action \cdot of M on C. The strict natural transformations between models are strictly linear functors.

Proof

As above, there is nothing difficult about this. Let \mathcal {T}_\mathsf {Act} contain as a subcategory the sketch of pseudomonoids \mathcal {T}_\mathsf {PsMon}, along with three additional objects C, MC, MMC, projections \pi ^1_M, \pi ^1_C: MC \to M,C, \pi ^2_{M_1}, \pi ^2_{M_2}, \pi ^2_{C} : MMC \to M, M, C, and limit cones expressing these as the products M \times C and M \times M \times C respectively, a morphism \cdot : MC \to C, and isomorphism 2-cells \eta : 1_C \Rightarrow \cdot \langle I, 1_C \rangle and \mu : \cdot \langle \pi ^2_{M_1}, \cdot \langle \pi ^2_{M_2}, \pi ^2_C \rangle \rangle \to \cdot \langle \otimes \langle \pi ^2_{M_1}, \pi ^2_{M_2} \rangle , \pi ^2_C \rangle , subject to the coherence equations in Definition [efr-OFNV].

Context

Actegories [efr-NI76]

Definition Actegory [efr-OFNV]

Let \mathcal {M} be a monoidal category. A \mathcal {M}-actegory (also \mathcal {M}-module, \mathcal {M}-action) is a category \mathcal {C} equipped with a functor \cdot : \mathcal {M} \times \mathcal {C} \to \mathcal {C} called the action (written infix, M \cdot C), and natural isomorphisms \mu : (M \otimes M') \cdot C \to M \cdot (M' \cdot C), \eta : I \cdot C \to C satisfying the following coherence equations:

Actegories are a very natural concept, and as one would expect their history goes back a long way. The concept has been considered by Benabou all the way back in Reference [benabou-bicategories], and used many times since them. We will not delve too deeply into their theory here---see Reference [actegories-amthematician-capucci-gavranovic] for a thorough treatment.

Just as pseudo double categories, actegories really make sense in every 2-category, as a weakening of monoid actions.

Definition Pseudomonoid action [efr-IOLJ]

Let \mathbb {C} be a 2-category. A pseudomonoid action internal to \mathbb {C} consists of the following data: An internal pseudomonoid M, an object C, a morphism \cdot : M \times C \to C, natural isomorphisms \mu : \cdot (\otimes \times 1_C) \to \cdot (1_M \times \cdot ) and \eta : \cdot \langle e, 1_C \rangle \to 1_C, satisfying the coherence equations from Definition [efr-OFNV].

A strict homomorphism of actions is a pair F_m: M \to M', F_c: C \to C' so that F_m is a strictly monoidal functor and F_c preserves the action strictly, i.e F_m(M) \cdot F_c(C) = F_c(M \cdot C), F_c(\eta ) = \eta ', etc. Note that this makes sense even if the monoidal category or action is not itself strict.

Proposition [efr-6S13]

There is a limit sketch \mathcal {T}_\mathsf {Act}, \Theta _\mathsf {Act} whose models are tuples (M,C,\cdot ) of a pseudomonoid M = (M,\otimes ,I), an object C and an action \cdot of M on C. The strict natural transformations between models are strictly linear functors.

Proof

As above, there is nothing difficult about this. Let \mathcal {T}_\mathsf {Act} contain as a subcategory the sketch of pseudomonoids \mathcal {T}_\mathsf {PsMon}, along with three additional objects C, MC, MMC, projections \pi ^1_M, \pi ^1_C: MC \to M,C, \pi ^2_{M_1}, \pi ^2_{M_2}, \pi ^2_{C} : MMC \to M, M, C, and limit cones expressing these as the products M \times C and M \times M \times C respectively, a morphism \cdot : MC \to C, and isomorphism 2-cells \eta : 1_C \Rightarrow \cdot \langle I, 1_C \rangle and \mu : \cdot \langle \pi ^2_{M_1}, \cdot \langle \pi ^2_{M_2}, \pi ^2_C \rangle \rangle \to \cdot \langle \otimes \langle \pi ^2_{M_1}, \pi ^2_{M_2} \rangle , \pi ^2_C \rangle , subject to the coherence equations in Definition [efr-OFNV].

Example Discrete pseudomonoid [efr-I33G]

If \mathbb {C} = \mathcal {C} is an ordinary category with finite products (viewed as a discrete 2-category), a pseudomonoid is simply an internal monoid, and an action is just a monoid action in the ordinary sense.

Example Monoidal and symmetric monoidal actegories [efr-ARIX]

In Reference [actegories-amthematician-capucci-gavranovic], the authors study actegories with extra monoidal structure. Although they do not introduce pseudomonoid actions in a general 2-category, they study actegories internal to monoidal categories, which they call monoidal actegories. It is straightforward to check that pseudomonoid actions in \mathsf {MonCat} agree with their notion: such an action consists of a braided monoidal category \mathcal {M}, an ordinary monoidal category \mathcal {C}, a monoidal functor \mathcal {M} \times \mathcal {C} \to \mathcal {C}, and monoidal natural transformations \mu ,\eta satisfying the coherence equations for an actegory.

(Note that since the functor \otimes appears on one side of \mu , to speak of \mu being a monoidal natural transformation we must have a monoidal structure on \otimes ---this makes \mathcal {M} into a braided monoidal category).

For the case of symmetric monoidal actegories, since \mathsf {SymMonCat} is cocartesian, \mathsf {PsMon}(\mathsf {SymMonCat}) \simeq \mathsf {SymMonCat}. Given two symmetric monoidal categories \mathcal {M},\mathcal {C}, Reference [actegories-amthematician-capucci-gavranovic] show that an action is simply given by a strong symmetric monoidal functor f: \mathcal {M} \to \mathcal {C} (and in this case the action is M \cdot C = F(M) \otimes C). Unwinding this construction we see that \mathsf {Act}_s(\mathsf {SymMonCat}) is bi-equivalent to a category which has as objects strong (but not strict) symmetric monoidal functors C \to C', and morphisms squares

of symmetric monoidal functors, where the top map is strict, and which commute strictly.

Again, as expected, the notion of strict linear morphism is usually too strict. One can expect an equation like F_m(M) \cdot F_c(C) = F_c(M \cdot C) to hold only up to coherent isomorphism. Let us now make this clear:

Definition Pseudolinear map [efr-L5QG]

Let \mathbb {C} be a 2-category, and let (M,C,\cdot ), (N,D,\star ) be internal pseudomonoid actions. A pseudolinear map (or just linear) is a pair of maps F_m: M \to N, F_c: C \to D, equipped with a pseudomonoid pseudohomomorphism structure \phi , \tau on F_m and a natural isomorphism l: F_m(-) \cdot F_c(=) \to F_c(- \cdot =) satisfying the following coherence conditions (note that writing functors using element-notation like this is well-defined)

In Reference [actegories-amthematician-capucci-gavranovic], these are defined in two steps: first linear functors for the same \mathcal {M} (those with F_m = 1_\mathcal {M}) are considered. Then, given a strong monoidal \mathcal {M} \to \mathcal {N}, a functorial assignment of an \mathcal {M}-action to every \mathcal {N}-action is constructed, and the full category of actions is defined as the Grothendieck construction of this. There is nothing preventing this from working in the setting of a general 2-category, and it is straightforward to verify that our notion of linear morphism of actions agrees with theirs.

Related

Definition Actegory [efr-OFNV]

Let \mathcal {M} be a monoidal category. A \mathcal {M}-actegory (also \mathcal {M}-module, \mathcal {M}-action) is a category \mathcal {C} equipped with a functor \cdot : \mathcal {M} \times \mathcal {C} \to \mathcal {C} called the action (written infix, M \cdot C), and natural isomorphisms \mu : (M \otimes M') \cdot C \to M \cdot (M' \cdot C), \eta : I \cdot C \to C satisfying the following coherence equations: