module object




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}.



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 } \,,



(geometric actions)

A group object-action


(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


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

  • Alexander Grothendieck, p. 106 (9 of 21) of: 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)

following the general principle of internalization formulated in:

  • Alexander Grothendieck, p. 370 (3 of 23) in: 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)

See also:

Last revised on June 18, 2021 at 18:31:48. See the history of this page for a list of all contributions to it.