nLab
module object

Contents

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 𝒞\mathcal{C} to yield a notion of actions of monoid objects (such as group objects or ring objects) on the objects of that category 𝒞\mathcal{C}.

Definition

Definition

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

  1. an object N𝒞N \in \mathcal{C};

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

such that

  1. (unitality) the following diagram commutes:

    1N eid AN ρ N, \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 𝒞\mathcal{C}.

  2. (action property) the following diagram commutes

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \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 (,,1)(\mathcal{M}, \odot, 1) and an \mathcal{M}-module (also called \mathcal{M}-actegory) 𝒞\mathcal{C} (supported by the monoidal action :×𝒞𝒞\bullet : \mathcal{M} \times \mathcal{C} \to \mathcal{C}), and given (A,μ,e)(A,\mu,e) a monoid in (,,1)(\mathcal{M}, \odot, 1), then a left module object in 𝒞\mathcal{C} over (A,μ,e)(A,\mu,e) is

  1. an object N𝒞N \in \mathcal{C};

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

such that

  1. (unitality) the following diagram commutes:

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

    where λ\lambda is the unitor of \bullet.

  2. (action property) the following diagram commutes

    (AA)N α A,A,N A(AN) Aρ AN μN ρ AN ρ N, \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 α\alpha is the actor of \bullet.

Examples

Example

(geometric actions)

A group object-action

Example

(equivariant principal bundles)

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


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:

See also:

Last revised on July 22, 2021 at 05:20:09. See the history of this page for a list of all contributions to it.