nLab module object

Contents

Context

Categorical algebra

Idea
Definition
Generalisation
Examples
Related concepts
References

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

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

an object $N \in \mathcal{C}$ ;
a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);

such that

(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 $\mathcal{C}$ .
(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

an object $N \in \mathcal{C}$ ;
a morphism $\rho \;\colon\; A \bullet N \longrightarrow N$ (called the action);

such that

(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 $\bullet$ .
(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

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:

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 translation: web version)

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 translation: web version)

Review:

Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Section 2.2 of: Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) [MR2007f:14001, ISBN:978-0-8218-4245-4, lecture notes]
Saunders MacLane, Sections VII.4 in: Categories for the Working Mathematician, Springer (1979, 2nd ed.) [doi:10.1007/978-1-4757-4721-8]
John Michael Boardman, Algebraic objects in categories, Chapter 7 of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford (1995) [doi:10.1016/B978-0-444-81779-2.X5000-7]
Magnus Forrester-Barker, Group Objects and Internal Categories [arXiv:math/0212065]