# nLab module object

Contents

### Context

#### Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

# Contents

## Idea

The notion of action object or module object is the internalization of the notion of action/module of monoids (such as groups or rings) on sets (such as group representations or modules), into any monoidal category $\mathcal{C}$ to yield a notion of actions of monoid objects (such as group objects or ring objects) on the objects of that category $\mathcal{C}$.

## Definition

###### Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$, then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an object $N \in \mathcal{C}$;

2. a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);

such that

1. (unitality) the following diagram commutes:

$\array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,$

where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

2. (action property) the following diagram commutes

$\array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$

## Generalisation

A module and the monoid it lies over do not necessarily belong to the same category, a fact suggested by the microcosm principle:

###### Definition

Given a monoidal category $(\mathcal{M}, \odot, 1)$ and an $\mathcal{M}$-module (also called $\mathcal{M}$-actegory) $\mathcal{C}$ (supported by the monoidal action $\bullet : \mathcal{M} \times \mathcal{C} \to \mathcal{C}$), and given $(A,\mu,e)$ a monoid in $(\mathcal{M}, \odot, 1)$, then a left module object in $\mathcal{C}$ over $(A,\mu,e)$ is

1. an object $N \in \mathcal{C}$;

2. a morphism $\rho \;\colon\; A \bullet N \longrightarrow N$ (called the action);

such that

1. (unitality) the following diagram commutes:

$\array{ 1 \bullet N &\overset{e \bullet id}{\longrightarrow}& A \bullet N \\ & {}_{\mathllap{\lambda}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,$

where $\lambda$ is the unitor of $\bullet$.

2. (action property) the following diagram commutes

$\array{ (A \odot A) \bullet N &\underoverset{\simeq}{\alpha_{A,A,N}}{\longrightarrow}& A \bullet (A \bullet N) &\overset{A \bullet \rho}{\longrightarrow}& A \bullet N \\ {}^{\mathllap{\mu \bullet N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \bullet N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$

where $\alpha$ is the actor of $\bullet$.

## Examples

###### Example

(geometric actions)

A group object-action

###### Example

(equivariant principal bundles)

A $G$-equivariant principal bundle is an internal action of a group object internal to a category of internal $G$-actions as in Example , such as G-sets/G-spaces/G-manifolds (an “equivariant group”) which satisfies, internally, principality and local triviality-condition.

###### Example

A module object in a symmetric monoidal category of spectra is a module spectrum over a ring spectrum.

Internal to just the stable homotopy category it is a homotopy module spectrum.

categorical algebra – contents

internalization and categorical algebra

universal algebra

categorical semantics

## References

The general definition of internal actions seems to have first been formulated in:

following the general principle of internalization formulated in:

Review: