nLab
module

Idea

Basic idea

The basic idea is that a module $V$ is an object equipped with an action by a monoid $A$. This is closely related to the concept of a representation of a group.

A familiar example of a module is a vector space $V$ over a field $k$: this is a module over $k$ in the category Ab of abelian groups. But nothing in the definition of vector space really depends on the fact that $k$ here is a field: more generally it could be any ring $R$. The analog of a vector space for fields replaced by rings is that of a module over the ring $R$.

More general perspectives

The notion of monoid in a monoidal category generalizes directly to that of a monoid in a 2-category, where it is called a monad. Accordingly the notion of module generalizes to this more general case, where however it is called an algebra over a monad . For more on this see Modules for monoids in 2-categories: algebras over monads below.

Apart from this direct generalization, there are two distinct and separately important perspective on the notion of module from the nPOV:

• modules may usefully be thought of in the context of enriched category theory (and the enrichment may be over a 2-category);

• modules may usefully be thought of in terms of abelianization/stabilization of overcategories.

Modules for monoids in 2-categories: modules over monads

The notion of monoid generalizes straightforwardly from monoids in a monoidal category to monoids in a 2-category: for the 2-category Cat, and more generally for arbitrary 2-categories, these are called monads.

A module over a monad (see there for more details) is defined essentially exactly as that of module over a monoid. For historical reasons, a module over a monad in Cat is called an algebra over a monad, because the algebras in the sense of universal algebra can be obtained as algebras/modules over a finitary monad in $\mathrm{Set}$: the modules for a free algebra monad (for certain kind of algebras) on Set, which are the composition of the free alegbra functor and its right adjoint forgetful functor are exactly algebras of that type. Modules over a fixed monad (in $\mathrm{Cat}$) are the objects of the Eilenberg-Moore category of the monad; in arbitrary bicategory, this category generalized to Eilenberg-Moore objects which may or may not exist.

Enriched presheaves

The action $\rho$ of a monoid $A$ in a monoidal category $V$ may be equivalently encoded in terms of a $V$-enriched functor

$$\rho :BA\to V$$\rho : \mathbf{B}A \to V

from the delooping one-object $V$-enriched category $BA$ corresponding to $A$ to $V$ itself.

This means that more generally it makes sense to replace $BA$ by any $V$-enriched category $C$ – regarded as the horizontal categorification of a monoid, a “monoid-oid” – and think of $V$-enriched functors $\rho :C\to V$ – $V$-presheaves – as modules for $C$.

From this perspective a $C$-$D$-bimodule is a $V$-enriched functor $C^{\mathrm{op}}\times D\to V$, which is in this context known as a profunctor from $C$ to $D$. The notion of the bicategory $V\mathrm{Mod}$ of $V$-enriched categories, $V$-profunctors between these and transformations between those is then a generalization of the

Stabilized overcategories

A module $N$ over a (commutative, unital) ring $R$ may be encoded in another ring: the one that as an abelian group is the direct sum $R\oplus N$ and whose product is defined by the formulas

$$(r_1,n_1)\cdot (r_2,n_2):=(r_1{r}_2,r_1{n}_1+r_2{n}_2).$$(r_1, n_1) \cdot (r_2,n_2) := (r_1 r_2, r_1 n_1 + r_2 n_2) \,.

This is a square-0 extension of $R$. It is canonically equipped with a ring homomorphism $R\oplus N\to R$ which is the identity on $R$ and sends all elements of $N$ to 0. As such, $R\oplus N\to R$ is an object in the overcategory $\mathrm{CRing}/R$. But a special such object: it is in fact canonically an abelian group object in $\mathrm{CRing}/R$, where the group operation (over $R$!) is given by addition of elements in $N$.

From this perspective, it makes sense for general categories $C$ to think of the abelianization of their overcategories $C/A$ as categories of modules over the object $A$.

Taken all together, this makes the fiberwise abelianization of their codomain fibration $\mathrm{cod}:[I,C]\to C$ the category of all possible modules over all objects of $C$.

This general perspective has a nice vertical categorification to the context of (∞,1)-categories: abelianization becomes stabilization in this context, and the fiberwise stabilization of the codomain fibration of any (∞,1)-category $C$ is the tangent (∞,1)-category $T_C\to C$.

For instance for ${\mathrm{sAlg}}_k$ the (∞,1)-category of simplicial algebras over a ground field $k$ of characteristic 0, we have that the stabilization $\mathrm{Stab}(\mathrm{sAlgk}/A)$ of the over (∞,1)-category over $A$ is equivalent to the $(\mathrm{\infty }\mathrm{,1})$-category $A\mathrm{Mod}$ of $A$-modules.

Details

We spell out details of the definition of module for

• the classical case of modules over rings;

• the generalized case of modules as presheaves in enriched category theory;

• the generalized case of modules as objects in stabilized overcategories.

Ordinary concept

A (right) module over a monoid $A$ internal to a monoidal category $(V,\otimes ,I)$ is an object $N$ of $V$ equipped with a morphism

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

in $C$ which satisfies the usual axioms of an action.

Example: Modules over rings

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

$$\mathrm{Mod}\to \mathrm{CRing}$$Mod \to CRing

over CRing.

A ring is a monoid in Ab. Hence a module over a ring is first of all an object $N$ in Ab, hence an abelian group. Moreover, it is equipped with a morphism

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

in Ab. On the left we have the tensor product of abelian groups. So this is a morphism that sends

$$(r,n)\mapsto rn$$(r,n ) \mapsto r n

such that

$$(r,n_1+n_2)\mapsto r{n}_1+r{n}_2$$(r, n_1 + n_2) \mapsto r n_1 + r n_2

and

$$(r_1+r_2,n)\mapsto r_{1}n+r_{2}n.$$(r_1 + r_2, n) \mapsto r_1 n + r_2 n \,.

Moreover, this morphism respects the monoid structure on $R$, in that the diagram

$$\begin{array}{ccc}R\otimes R\otimes N& \stackrel{{\cdot }_R\otimes {\mathrm{Id}}_N}{\to }& R\otimes N\\ {}^{{\mathrm{Id}}_R\otimes \alpha }\downarrow & & {\downarrow }^{\mathrm{alpha}}\\ R\otimes N& \to & N\end{array}$$\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 formulas this means that

$$(r_1\cdot r_2)n=r_1(r_{2}n).$$(r_1 \cdot r_2) n = r_1 (r_2 n) \,.

Finally the unit axioms says that the identity element of $R$ acts as the identity on $N$.

Saying the same fully in terms of enriched category theory:

Write $BR$ for the Ab-enriched category with a single object and $R=BA(•,•)$. A module $N$ is an $\mathrm{Ab}$-functor

$$N:BR\to \mathrm{Ab}.$$N : \mathbf{B}R \to Ab \,.

The category $R\mathrm{Mod}$ has $R$-modules as objects and $R$-module homomorphisms as morphisms. More abstractly, this is the Ab-enriched functor category $[BR,\mathrm{Ab}]$.

Example: $G$-sets

Classically the notion of module is always regarded internal to Ab, so that a module is always an abelian group with extra structure. But noticing that such abelian ring modules are just enriched presheaves in Ab-enriched category theory, it makes sense to consider enriched presheaves in general $V$-enriched category theory as a natural generalization of the notion of module.

For that generalization the case of Set-enriched category theory plays a special basic role:

a group $G$ (with no extra structre, i.e. just a set with group structure) is a monoid in Set. A module over $G$ in the sense of Set-enriched functor (just an ordinary functor)

$$BG\to \mathrm{Set}$$\mathbf{B}G \to Set

is nothing but a $G$-set: a set equipped with a $G$-action:

$BG$ is the small category that is the delooping groupoid of $G$, which has a single object and ${\mathrm{Hom}}_{BG}(•,•)=G$. The functor $BG\to \mathrm{Set}$ takes the single object to some set $S$ and takes each morphism $(•\stackrel{g}{\to }•)$ to an automorphism $\rho (g):S\to S$ of that set, such that composition is respected. This is just a representation of $G$ on the set $S$.

Of course for this story to work, $G$ need not be a group, but could be any monoid.

In enriched category theory

Equivalently, regarding the monoid $A$ as a one-object $V$-enriched category $BA$, the module together with its action are given by a $V$-enriched functor

$$\rho :BA\to V.$$\rho : \mathbf{B}A \to V \,.

Correspondingly, a left module over $A$ is a functor

$$\rho :(BA{)}^{\mathrm{op}}\to V.$$\rho : (\mathbf{B}A)^{\mathrm{op}} \to V \,.

In this language the concept directly generalizes to the horizontal categorification of monoids $A$. Let $K$ be any $V$-enriched category, then $V$-functors $\rho :K\to V$ give right modules and functors $\rho :K^{\mathrm{op}}\to V$ give left modules over $K$. Accordingly, for $K$ and $L$ two $V$-enriched categories one says that $V$-functors

$$K^{\mathrm{op}}\otimes L\to V$$K^{op} \otimes L \to V

are $K$-$L$-bimodules, also known as profunctors or distributors from $K$ to $L$.

In terms of stabilized overcategories

The ordinary case of modules over rings is phrased in terms of stabilized overcategories by the following observation, wich apparently goes back to Quillen.

Proposition

Let $R\in \mathrm{CRing}$ be a commutative ring. Then there is a canonical equivalence between the category $R\mathrm{Mod}$ of $R$-modules and the category $\mathrm{Ab}(\mathrm{CRing}/R)$ of abelian group objects in the overcategory of $\mathrm{CRing}$ over $R$

$$R\mathrm{Mod}\simeq \mathrm{Ab}(\mathrm{CRing}/R).$$R Mod \simeq Ab(CRing/R) \,.

Proof

Unwind the structure encoded in an abelian group object $(p:K\to R)$ in the overcaregory $\mathrm{CRing}/R$.

The unit of the abelian group object in $\mathrm{CRing}/R$ is a diagram

$$\begin{array}{ccccc}R& & \to & & K\\ & {}_{\mathrm{Id}}\searrow & & {\swarrow }^p\\ & & R\end{array}.$$\array{ R &&\to&& K \\ & {}_{\mathllap{Id}}\searrow && \swarrow^{\mathrlap{p}} \\ && R } \,.

This diagram identifies $K$ with a ring of the form $R\oplus \ker (p)$ in which for $r\in R$ and $n\in \ker (p)$ we have $r\cdot n\in \ker (p)$.

The product of $R\oplus N\to R$ with itself in the overcategory is the fiber product over $R$ in the original category, hence is $R\oplus N\oplus N$.

The addition operation on the abelian group object is therefore a morphism

$$\begin{array}{ccccc}R\oplus N\oplus N& & \to & & R\oplus N\\ & \searrow & & \swarrow \\ & & R\end{array}.$$\array{ R \oplus N \oplus N &&\to&& R \oplus N \\ & \searrow && \swarrow \\ && R } \,.

With the above unit, the unit axiom on this operation says that this morphism is

$$R\oplus N\oplus N\stackrel{\mathrm{Id}\oplus (\mathrm{Id}+\mathrm{Id})}{\to }R\oplus N.$$R \oplus N \oplus N \stackrel{Id \oplus (Id + Id)}{\to} R \oplus N \,.

Since the ring product in the direct product ring $R\oplus N\oplus N$ between two elements in the two copies of $N$ vanishes, it therefore has to vanish between two elements in the same copy, too.

This says that $R\oplus N$ is a square-0 extension of $R$. Conversely, for every square-0-extension we obtain an abelian group object this way.

For instance the square-0-extension of a ring $R$ corresponding to the canonical $R$-module structure on $R$ itself is the ring of dual numbers for $R$.

Example: Modules over simplicial rings

Let ${\mathrm{sAlg}}_k$ (or $\mathrm{sAlg}$ for short) be the (∞,1)-category of commutative simplicial algebras over a base field $k$.

For $A\in {\mathrm{sAlg}}_k$ there is generally a functor

$$A\mathrm{Mod}\to \mathrm{Stab}({\mathrm{sAlg}}_k/A)$$A Mod \to Stab(sAlg_k/A)

from the stable (∞,1)-category of $A$-modules to the stabilization of the overcategory of $\mathrm{sAlg}$. But in general this functor is neither essentially surjective nor full. If however $k$ has characteristic 0, then this is an equivalence.

Related concepts

Vector bundles and sheaves of modules

A vector space is a vector bundle over the point. For every vector bundle $E\to X$ over a space $X$, its collection $\Gamma (E)$ of sections is a module over the monoid/ring of functions on $X$. When $X$ is a ringed space, $\Gamma (X)$ is usefully thought of as a sheaf of modules over the structure sheaf of $X$:

For describing vector bundles and their generalization it turns out that this perspective of encoding them in terms of their modules of sections is useful. For instance the category of vector bundles on a space typically fails to be an abelian category. But if instead of looking just as sheaves of modules on $X$ that arise as sections of vector bundles one generalizes to coherent sheaves of modules then one obtains an abelian category, something like the completion of $\mathrm{Vect}(X)$ to an abelian category. If one further demands that the category be closed under push-forward operations, such as to obtain a bifibration of generalized vctor bundles over spaces, one arrives at the notion of quasicoherent sheaves of modules over the structure sheaf.

But it turns out that the category of quasicoherent sheaves over a test space (see there for details) is equivalent simply to the category of all modules over the (functions on) this test space. This means that quasicoherent sheaves of modules have a nice description in terms of the general-abstract-nonsense characterization of modules discussed above:

For $C$ our (∞,1)-category of of test spaces (hence the opposite category $C^{\mathrm{op}}$ our (∞,1)-category of “functions rings” on test spaces), by the above the assignment of all modules over a test space is given by

$$\mathrm{Mod}:C^{\mathrm{op}}\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$$Mod : C^{op} \to (\infty,1)Cat

$$\mathrm{Mod}:U\mapsto \mathrm{Stab}(C/U).$$Mod : U \mapsto Stab( C/U ) \,.

Then for $X$ any space regarded as an ∞-stack on $C$, a “quasicorent $\mathrm{\infty }$-stack of modules” on $X$ is a morphism

$$X\to \mathrm{Mod}.$$X \to Mod \,.

By the above discussion, this procedure yieelds the expected notions of modules in particular for the choice $C=({\mathrm{sAlg}}_k{)}^{\mathrm{op}}$ of simplicial algebras over a ground ring of characteristic 0. The theory of quasicoherent sheaves of modules in this case is discussed in great detail at geometric ∞-function theory. Some more general remarks along these lines are at ∞-vector bundle.

References

On modules as enriched presheaves

See also the references at enriched category theory and at profunctor.

On modules as stabilized overcategories

The observation that the category of modules over a ring $R$ is equivalent to the category of abelian group objects in the overcategory $\mathrm{CRing}/R$ was used by Quillen:

• Quillen, D. G. Quillen, On the (co-)homology of commutative rings, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.

A more ‘classical’ references include Jon Beck’s thesis

• Jon M. Beck, Triples, algebras and cohomology, thesis.

The fully abstract higher categorical concept in terms of stabilized overcategories and the tangent (∞,1)-category appears in

• Jacob Lurie, Deformation Theory

Discussion

An earlier version of this entry led to the following discussion.

Eric: The wikipedia page distinguishes left $R$-modules as covariant functors and right $R$-modules as contravariant functors. Is that distinction important?

John: Yes, very — but I didn’t have the energy to get into that yet. For any ring $R$ there’s a ring $R^{\mathrm{op}}$ in which $xy$ is redefined to be $yx$. I defined a left $R$-module above; a right $R$-module is the same as a left $R^{\mathrm{op}}$-module. Eventually we’ll have to discuss all this stuff, which becomes vastly more important when we start talking about bimodules. If we want to show off, we’ll do it all not just for rings, which are monoids in Ab, but more generally for monoids in any symmetric monoidal category. For any monoid $M$ in a symmetric monoidal category we can define a new monoid $M^{\mathrm{op}}$, and we can define left and right $M$-modules, and a right $M$-module is the same as a left $M^{\mathrm{op}}$-module.

Sridhar: Given that left modules on rings are the covariant functors while right modules on rings are the contravariant functors, why does the above definition of a module on a monoid make the left modules the contravariant functors and the right modules the covariant functors? Is this actually the conflicting convention?

Toby: One problem is that this mixes with the conventions that one adopts for composition. What one person thinks is left multiplication, another will think is right multiplication. I would rather talk about left/right modules for monoids or rings, then talk about covariant/contravariant functors from categories or additive categories.