nLab module over a monoid

Idea

In a monoidal category, there is a notion of modules over monoid objects which generalizes the classical notion of modules over rings. This is a special case of module over a monad where the monad is taken to be $A \otimes -$, with $A$ some monoid object.

Definition

Let $(\mathcal{V}, \otimes, I)$ be a monoidal category and $A$ a monoid object in $\mathcal{V}$, hence an object $A \in \mathcal{V}$ equipped with a multiplication morphism

$$\cdot : A \otimes A \to A$$

and a unit element

$$e : I \to A$$

satisfying the associativity law and the unit law.

Definition

A (left) module over $A$ in $(\mathcal{V}, \otimes, I)$ is

• an object $N \in \mathcal{V}$

• equipped with a morphism

  $$\lambda : A \otimes N \to N$$

  in $\mathcal{V}$

such that this satisfies the axioms of an action, in that the following are commuting diagrams in $\mathcal{V}$:

$$\array{ A \otimes A \otimes N &\stackrel{id_A \otimes \lambda}{\to}& A \otimes N \\ \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\lambda}} \\ A \otimes N &\stackrel{\lambda}{\to}& N }$$

and

$$\array{ I \otimes N &&\stackrel{e \otimes id_N}{\to}&& A \otimes N \\ & \searrow && \swarrow_{\mathrlap{\lambda}} \\ && N } \,.$$

Definition

Similarly a right module over $A$ in $(\mathcal{V}, \otimes, I)$ is

• an object $N \in \mathcal{V}$

• equipped with a morphism

  $$\rho : N \otimes A \to N$$

  in $\mathcal{V}$

such that this satisfies the axioms of a (right) action, in that the following are commuting diagrams in $\mathcal{V}$:

$$\array{ N \otimes A \otimes A &\stackrel{\rho \otimes id_A}{\to}& N \otimes A \\ \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\rho}} \\ N \otimes A &\stackrel{\rho}{\to}& N }$$

and

$$\array{ N \otimes I &&\stackrel{id_N \otimes e}{\to}&& N \otimes A \\ & \searrow && \swarrow_{\mathrlap{\rho}} \\ && N } \,.$$

Examples

Modules over monoids in abelian groups

Recall that a ring, in the classical sense, is a monoid object in the category Ab of abelian groups with monoidal structure given by the tensor product of abelian groups $\otimes$. Accordingly a module over $R$ is a module in $(Ab,\otimes)$ according to def. .

We unwind what this means in terms of abelian groups regarded as sets with extra structure:

Definition

A module $N$ over a ring $R$ is

1. an object $N \in$ Ab, hence an abelian group;

2. equipped with a morphism

   $$\alpha : R \otimes N \to N$$

   in Ab; hence a function of the underlying sets that sends elements

   $$(r,n) \mapsto r n \coloneqq \alpha(r,n)$$

   and which is a bilinear function in that it satisfies

   $$(r, n_1 + n_2) \mapsto r n_1 + r n_2$$

   and

   $$(r_1 + r_2, n) \mapsto r_1 n + r_2 n$$

   for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;

3. such that the diagram

   $$\array{ R \otimes R \otimes N &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N \\ {}^{\mathllap{Id_R \otimes \alpha}}\downarrow && \downarrow^{\mathrlap{\alpha}} \\ R \otimes N &\to& N }$$

   commutes in Ab, which means that for all elements as before we have

   $$(r_1 \cdot r_2) n = r_1 (r_2 n) \,.$$

4. such that the diagram

   $$\array{ 1 \otimes N &&\stackrel{1 \otimes id_N}{\to}&& R \otimes N \\ & \searrow && \swarrow_{\mathrlap{\alpha}} \\ && N }$$

   commutes, which means that on elements as above

   $$1 \cdot n = n \,.$$

Remark

The category of all modules over all commutative rings is Mod. It is a bifibration

$$Mod \to CRing$$

over CRing.

This fibration may be characterized intrinsically, which gives yet another way of defining $R$-modules. This we turn to below.

$G$-sets

Simpler than the traditionally default notion of a module in $(Ab,\otimes)$, as above is that of a module in Set, equipped with its cartesian monoidal structure. (These days one may want to think of this as a notion of modules over F1.)

A monoid object in $(Set,\times)$ is just a monoid, for instance a discrete group $G$. A $G$-module in $(Set,\times)$ is simpy an action, say a group action.

Definition

For $S \in$ Set and $G$ a discrete group, a $G$-action of $G$ on $S$ is a function

$$\lambda \colon G \times S \to S$$

such that

1. the neutral element acts trivially

   $$\array{ * \times S &&\stackrel{\simeq}{\to}&& S \\ & {}_{(e,id_S)}\searrow && \nearrow_{\mathrlap{\lambda}} \\ && G \times S }$$

2. the action property holds: for all $g_1, g_2 \in G$ and $s \in S$ we have $\lambda(g_1,\lambda(g_2, s)) = \lambda(g_1 \cdot g_2, s)$.

Abelian groups with $G$-action as modules over the group ring

If a discrete group acts, as in def. , on the set underlying an abelian group and acts by linear maps (abelian group homomorphisms), then this action is equivalently a module over the group ring $\mathbb{Z}[G]$ as in def. .

Definition

For $G$ a discrete group, write $\mathbb{Z}[G] \in$ Ring for the ring

1. whose underlying abelian group is the free abelian group on the set underlying $G$;

2. whose multiplication is given on basis elements by the group operation in $G$.

Remark

For $G$ a finite group an element $r$r of $\mathbb{Z}[G]$ is for the form

$$r = \sum_{g \in G} r_g g$$

with $r_g \in \mathbb{Z}$. Addition is given by addition of the coefficients $r_g$ and multiplication is given by the formula

$$\begin{aligned} r \cdot \tilde r & = \sum_{g \in G} \sum_{\tilde g \in G} (r_g \tilde r_{\tilde g}) g \cdot \tilde g \\ & = \sum_{q \in G} \left( \sum_{g \tilde g = q} r_g \tilde r_{\tilde g} \right) q \end{aligned} \,.$$

Proposition

For $A \in$ Ab an abelian group with underlying set $U(A)$, $G$-actions $\lambda \colon G \times U(A) \to U(A)$ such that for each element $g \in G$ the function $\lambda(g,-) \colon U(A) \to U(A)$ is an abelian group homomorphism are equivalently $\mathbb{Z}[G]$-module structures on $A$.

Proof

Since the underlying abelian group of $\mathbb{Z}[G]$ is a free by definition, a bilinear map $\mathbb{Z}[G] \times A \to A$ is equivalently for each basis element $g \in G$ a linear map $A \to A$. Similarly the module property is determined on basis elements, where it reduces manifestly to the action property of $G$ on $U(A)$.

Remark

This reformulation of linear $G$-actions in terms of modules allows to treat $G$-actions in terms of homological algebra. See at Ext – Relation to group cohomology.

more examples

• In a symmetric monoidal category of chain complexes equipped with the tensor product of chain complexes, then a monoid is a dg-algebra, and a module is a dg-module. See there for more.

Related concepts

• model structure on modules in a monoidal model category

References

The basic properties of categories of modules over monoid objects in symmetric monoidal categories are spelled out in sections 1.2 and 1.3 of

• Florian Marty, Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009, web

A summary is in section 4.1 of

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry, arXiv:1410.1716.

See also MO/180673, and the references at modules over a monad.

For the classical case of the symmetric monoidal category Ab, a standard textbook is

• F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York, (1992).