$$\newcommand{\cat}[1]{\mathcal{#1}}$$
$$\newcommand{\action}{\bullet}$$
# Actions of monoidal categories

## Actegories

**Definition.** Let $\cat M$ be a monoidal category. An $\cat M$-**actegory** is a category $\cat C$ together with a bifunctor $\action : \cat M \times \cat C \to \cat C$ and a pair of natural isomorphisms
$$
\varepsilon : j \action x \to x\\
\delta : (m \odot n) \action x \to x
$$
called, respectively, *unitor* and *multiplicator*, satisfying... [bunch of coherence laws]

**Definition.** Let $\cat M$ be a monoidal category and let $\cat C$ and $\cat D$ be $\cat M$-actegories. An **equivariant functor** $\cat C \to \cat D$ is a functor $F : \cat C \to \cat D$ together with a natural isomorphism
$$
\ell : m \action F(x) \to F(m \action x)
$$
called the **lineator** satisfying... [bunch of coherence laws]

If $\ell$ is not invertible, we call the functor **lax equivariant**. If $\ell$ is not invertible and points the other way, we call the functor **oplax equivariant**.

**Definition.** Let $F,G : \cat C \to \cat D$ be equivariant functor between $\cat M$-actegories. An **equivariant natural transformation** is a natural transformation $\alpha : F \Rightarrow G$ such that [commutes with lineator, see para notes]

**Proposition.** Let $\cat M$ be a monoidal category. Then there is a 2-category $\cat M-\mathrm{Mod}$ whose objects are $\cat M$-actegories, whose 1-morphisms are equivariant functors, and whose 2-morphisms are equivariant natural transformations.

**Proposition.** Actegories are equivalenty presented by strong monoidal functors $\action : \cat M \to [\cat C, \cat C]$:
$$
MonCat_{st}(\cat M, [\cat C, \cat C]) \cong \cat M-\mathrm{Mod}(\cat C) %\cat M \to [-,=] \simeq \cat M \times - \to {=}
$$
where $\cat M-\mathrm{Mod}(\cat C)$ is the subcategory of $\cat M-\mathrm{Mod}$ of $\cat M$-actegories supported by the same category $\cat C$, i.e. the fiber of $U: \cat M-\mathrm{Mod} \to \cat{Cat}$ over $\cat C$.
**Proof.** One sends a strong monoidal functor $F : \cat M \to [\cat C, \cat C]$ to its uncurried version. A monoidal natural transformation between two such functors equips the identity of $\cat C$ with a scalator and thus becomes an equivariant functor between $\cat M$-actegories. A is an equivariant natural transformation.

**Proposition.** There is a forgetful 2-functor $U : \cat M-\mathrm{Mod} \to \mathrm{Cat}$.

**Proposition.** Let $\cat C$ be a category. Then $\cat M \times \cat C$ can be given the structure of an $\cat M$-actegory, by $m \bullet (n, x) = (m \odot n, x)$. This defines a 2-functor $\cat M \times - : \mathrm{Cat} \to \cat M-\mathrm{Mod}$.

**Proposition.** There is a 2-adjunction $\cat M \times - \dashv U$. That is, $\cat M \times \cat C$ is the free $\cat M$-actegory on the category $\cat C$.

## Actegories from monoidal functors

Fix monoidal categories $(\cat M,j,\odot)$ and $(\cat C,i,\otimes)$.
**Proposition.** Let $f: \cat M \to \cat C$ be a {-,lax,oplax} monoidal functor.
Then the formula $m \bullet c = f(m) \otimes c$ defines a {-,lax,oplax} action of $\cat M$ on $\cat C$.
**Proof.** [write things out].

We can recover the functor $f$ from the action as $f(m) = m \bullet i$, indicating that this construction may be fully faithful.

**Definition.** Call a functor $h: \cat C \to \cat C$ **left linear** if there is a natural isomorphism $h(c \otimes c') \cong h(c) \otimes c'$,
compatible with the monoidal structure. Note that this is the same as being a **strong** endofunctor, plus the assumption of invertibility.
Let $[\cat C, \cat C]_{lin}$ denote the category of left linear functors.
Call an action of $\cat M$ on $\cat C$ left linear if each functor $m \bullet -$ is left linear.

**Proposition.** Left linear {-,lax,oplax} actions of $\cat M$ on $\cat C$ are equivalent to {-,lax,oplax} monoidal functors $\cat M \to \cat C$.
**Proof.** By [??reference??], this amounts to the statement that {-,lax,oplax} monoidal functors $\cat M \to \cat C$ are equivalent to ones into $[\cat C, \cat C]_{lin}$ equipped with the composition monoidal structure. So we have to prove that $\cat C \cong [\cat C, \cat C]_{lin}$ as monoidal categories.
The isomorphism carries an object $c$ to $c \otimes -$, equipped with a linear structure by the associator of $\cat C$, and the inverse carries a functor $f$ to $f(I)$. Clearly $c \otimes I \cong c$, and $f(I) \otimes c \cong f(c)$ using the linearity of $f$, so these are mutual inverses.
And $c \otimes (c' \otimes -) \cong (c \otimes c') \otimes -$, so this equivalence is monoidal as well.

## Strong actegories

Fix a monoidal category $(M, j, \odot)$.

**Definition.** A *left strong* $\cat M$-actegory is an $\cat M$-actegory $\cat C$ equipped with a monoidal structure $(I, \otimes)$ and a natural morphism
$$
\kappa : m \action (x \otimes y) \to (m \action x) \otimes y
$$
called **left strength** satisfying [bunch of coherence laws]

**Lemma.** If $\cat C$ is an $\cat M$-actegory, so is $[\cat C, \cat C]$.
**Proof.** The action of $\cat M$ is defined pointwise, that is, $(m \action F)(x) := m \action F(x)$. The structure morphisms $\varepsilon$ and $\delta$ are likewise defined, e.g. $\varepsilon : j \action F \Rightarrow F$ is defined by whiskering: on the object $x$, is defined as $\varepsilon_{F(x)}$.

**Proposition.** The structure of a left strength is equivalent to a lax equivariant structure on the functor
$$
\cat C \to [\cat C, \cat C]\\
y \mapsto - \otimes y
$$

Dually, a **right strength** is a natural morphism
$$
\zeta : m \action (x \otimes y) \to x \otimes (m \action y)
$$
and corresponds to a lax equivariant structure on the functor $x \mapsto x \otimes -$.

A **bistrength** is defined as a natural morphism
$$
\beta : (m \odot n) \action (x \otimes y) \to (n \action x) \otimes (m \action y)
$$
satisfying [bunch of coherence laws]

**Definition.** Let $\cat C$ be an $\cat M$-actegory admitting both a left strength $\kappa$ and a right strength $\zeta$. Then we say **$\kappa$ and $\zeta$ are compatible** if and only if the following diagram commutes:
[....]
Evidently, if $\cat C$ admits compatible strengths it admits a canonical bistrength given by the morphisms both paths of the previous diagram compose to.

**Proposition.** If $\cat C$ comes equipped with a symmetry, every left strength gives rise to a right strength and viceversa, and the pairs so obtained are compatible.
**Proof.** Just conjugate the strength with a symmetry.

**Definition.** Let $\cat M$ be symmetric monoidal, let $\cat C$ be a symmetric monoidal category equipped with a bistrong $\cat M$-action. We say that $\cat C$ is **strongly symmetric** if the following commutes:
[$\beta \sigma = \sigma \beta$]

**Proposition.** Every bistrong action can be symmetrized.
**Proof.** Conjugate with $\sigma$.

**Observation.** Any monoidal category $\cat C$ is isomorphic, as monoidal category, to the category $[\cat C, \cat C]_{st}$ of its left strong endofunctors whose strength is invertible.

**Definition.** Let $\cat C$ and $\cat D$ be strong $\cat M$-actegories. A strong equivariant functor $\cat C \to \cat D$ is an equivariant functor $F : \cat C \to \cat D$ satisyfing... [bunch of coherence laws]

**Theorem.** There is an equivalence of categories between
* left strong $\cat M$-actegories, str

and

## Closed actegories

In a monoidal category, the internal hom object and the monoidal structure are related by the formula
$$
\cat M(x \odot y, z) \cong \cat M(x, [y,z])
$$
In a similar way, an action of $\cat M$ on $\cat C$ may give rise to an enrichment of $\cat C$ in $\cat M$ (i.e an $\cat M$-category with underlying ordinary category $\cat C$).

**Definition.** Let $\cat C$ be a $\cat M$-**actegory**. We call it **closed** if, for $x,y \in \cat C$, there exists an object $[x,y] \in \cat M$ and an equivalence
${\cat M}(m,[x,y]) \cong {\cat C}(m \action x, y)$.
Note that in this case, these objects assemble in a unique way into a left adjoint $[x,-] : {\cat C} \to {\cat M}$ to the functor $- \action x: {\cat M} \to {\cat C}$.
Let ${\cat M}-\mathrm{Mod}^{cl} \subseteq {\cat M}-\mathrm{Mod}$ be the full sub-2-category spanned by the closed actegories.

Recall also the following definition:
**Definition.** Let $\cat C$ be enriched in $\cat M$. Then we call it **tensored over $\cat M$** if, for each $m \in {\cat M}, x \in {\cat C}$, there exists
$m \cdot x$ and an isomorphism ${\cat M}(m, {\cat C}(x,y)) \cong {\cat C}_0(m \cdot x, y)$.
Let ${\cat M}-\mathrm{Cat}^t \subseteq {\cat M}-\mathrm{Cat}$ be the full sub-2-category spanned by the tensored enriched categories.

**Proposition.** Given a closed $\cat M$-actegory, $\cat C$, there is a $\cat M$-category with the same object as $\cat C$, with $\tilde{\cat C}(x,y) = [x,y]$ (using the notation from above). This construction gives an equivalence of 2-category ${\cat M}-\mathrm{Mod}^{cl} \cong {\cat M}-\mathrm{Cat}^t$, with inverse given by assembling the tensors into an action.