nLab module object

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

• internal to Sets is a G-set: a set equipped with a group action by a discrete group – this is the plain notion of a group action;

• internal to TopologicalSpaces is a topological G-space: a topological group equipped with an action on a topological space by continuous functions;

• internal to SmoothManifolds is a G-manifold: a Lie group equipped with an action on a smooth manifold by smooth functions;

• internal to SimplicialSets is a simplicial group 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

(2-actions)
The notion of coherent action object in the 2-category Cat (of categories with functors and natural transformations) is a categorified notion of “action” (namely of monoidal categories), known as module categories (also: “actegories”), see also 2-module and n-module).

Example

(module spectrum)

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.

Related concepts

• module over a monoidal functor

• (infinity,1)-module, (infinity,n)-module

categorical algebra – contents

internalization and categorical algebra

• monoid object

• group object

• ring object

• algebra object (associative, Lie, …)

• module object/action object

• internal locale

• internal category

  • internal groupoid

  • internal site

  • internal diagram

universal algebra

• algebras over$\,$ algebraic theories,

• algebras over$\,$ monads,

• algebras over$\,$ operads.

categorical semantics

• internal logic, internal language

• relation between category theory and type theory

References

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

• Alexander Grothendieck, p. 106 (9 of 21) of: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English transl. pdf)

following the general principle of internalization formulated in:

• Alexander Grothendieck, p. 370 (3 of 23) in: FGA Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf, English transl. pdf)

Review:

• John Michael Boardman, Algebraic objects in categories, Chapter 7 of: Stable Operations in Generalized Cohomology (pdf) in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

See also:

• Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)