symmetric monoidal (∞,1)-category of spectra
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: every element in $k$ acts on the vector space by multiplication of vectors, and this action respects the addition of vectors.
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 commutative ring (or even a general rig) $R$. The analog of a vector space for fields replaced by rings is that of a module over the ring $R$.
This is the traditional and maybe most common notion of modules. But the basic notion is easily much more general.
The theory or monoids or rings and their modules, its “meaning” and usage, is naturally understood via the duality between algebra and geometry:
a ring $R$ is to be thought of as the ring of functions on some space,
an $R$-module is to be thought of as the space of sections of a vector bundle on that space.
A classical situation where this correspondence holds precisely is topology, where
the Gelfand duality theorem says that sending a compact topological space $X$ to its C-star algebra $C(X,\mathbb{C})$ of continuous functions with values in the complex numbers constitutes an equivalence of categories between compact topological spaces and the opposite category of commutative $C^\ast$-alegebras;
the Serre-Swan theorem says that sending a Hausdorff topological complex vector bundle $E \to X$ over a compact topological space to the $C(X,\mathbb{C})$-module of its continuous sections establishes an equivalence of categories between that of topological complex vector bundles over $X$ and that of finitely generated projective modules over $C(X,\mathbb{C})$.
In fact, as this example already shows, modules faithfully subsume vector bundles, but are in fact more general. In many contexts one regard modules as the canonical generalization of the notion of vector bundles, with better formal properties.
This identification of vector bundles with $R$-modules being the spaces of sections of a vector bundle on the space whose ring of functions is $R$ is can then taken as the very definition: notably in algebraic geometry Gelfand duality is taken to “hold by definition” in that an algebraic variety is essentially by defintion the formal dual of a given ring, and the Serre-Swan theore similarly becomes the statement that the space of section of a vector bundle over a variety is equivalently given by a module over that ring. (See also at quasicoherent module for more on this).
This duality between geometry and algebra allows to re-interpret many statement about modules in terms of vector bundles. For instance
the direct sum of modules corresponds to fiberwise direct sum of vector bundles;
the extension of scalars of a module along a ring homomorphism corresponds to pullback of vector bundles along the dual map of spaces;
etc.
Using this disctionary for instance the notion of descent of vector bundles can be expressed in terms of monadic descent, see at Sweedler coring for discussion of this point.
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.
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 $Set$: the modules for a free algebra monad (for certain kind of algebras) on Set, which are the composition of the free algebra functor and its right adjoint forgetful functor are exactly algebras of that type. Modules over a fixed monad (in $Cat$) 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.
The action $\rho$ of a monoid $A$ in a closed monoidal category $V$ may be equivalently encoded in terms of a $V$-enriched functor
from the delooping one-object $V$-enriched category $\mathbf{B}A$ corresponding to $A$ to $V$ itself.
This means that more generally it makes sense to replace $\mathbf{B}A$ by any $V$-enriched category $C$ – regarded as the horizontal categorification of a monoid, a “monoid-oid” – and think of a $V$-enriched functor $\rho : C \to V$ – a $V$-presheaf on $C$ – as a module over the category $C$.
From this perspective a $C$-$D$-bimodule is a $V$-enriched functor $C^{op}\times D \to V$, which is in this context known as a profunctor from $C$ to $D$. The notion of the bicategory $V Mod$ of $V$-enriched categories, $V$-profunctors between these and transformations between those is then a generalization of the category of monoids in $V$ and bimodules between them.
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
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 $CRing/R$. But a special such object: it is in fact canonically an abelian group object in $CRing/R$, 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 $cod : [I,C] \to C$ 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 $sAlg_k$ the (∞,1)-category of simplicial algebras over a ground field $k$ of characteristic 0, we have that the stabilization $Stab(sAlgk/A)$ of the over (∞,1)-category over $A$ is equivalent to the $(\infty,1)$-category $A Mod$ of $A$-modules.
We spell out the definition of module for
with the special classical cases of
and
Then we give more general definitions
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
and a unit element
satisfying the associativity law and the unit law.
A (left) module over $A$ in $(\mathcal{V}, \otimes, I)$ is
an object $N \in \mathcal{V}$
equipped with a morphism
in $\mathcal{V}$
such that this satisfies the axioms of an action, in that the following are commuting diagrams in $\mathcal{V}$:
and
A ring is (as discused there) equivalently a monoid object in the category Ab of abelian groups turned into a monoidal category by means of the tensor product of abelian groups $\otimes$. Accordingly a module over $R$ is a module in $(Ab,\otimes)$ accordin to def. 1.
We unwind what this means in terms of abelian groups regarded as sets with extra structure:
A module $N$ over a ring $R$ is
an object $N \in$ Ab, hence an abelian group;
equipped with a morphism
in Ab; hence a function of the underlying sets that sends elements
and which is a bilinear function in that it satisfies
and
for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;
such that the diagram
commutes in Ab, which means that for all elements as before we have
such that the diagram
commutes, which means that on elements as above
The category of all modules over all commutative rings is Mod. It is a bifibration
over CRing.
This fibration may be characterized intrinsically, which gives yet another way of defining $R$-modules. This we turn to below.
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.
For $S \in$ Set and $G$ a discrete group, a $G$-action of $G$ on $S$ is a function
such that
the neutral element acts trivially
the action property holds: for all $g_1, g_2 \in G$ and $s \in S$ we have $\rho(g_1,\rho(g_2, s)) = \rho(g_1 \cdot g_2, s)$.
If a discrete group acts, as in def. 3, 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. 2.
For $G$ a discrete group, write $\mathbb{Z}[G] \in$ Ring for the ring
whose underlying abelian group is the free abelian group on the set underlying $G$;
whose multiplication is given on basis elements by the group operation in $G$.
For $G$ a finite group an element $r$r of $\mathbb{Z}[G]$ is for the form
with $r_g \in \mathbb{Z}$. Addition is given by addition of the coefficients $r_g$ and multiplication is given by the formula
For $A \in$ Ab an abelian group with underlying set $U(A)$, $G$-actions $\rho \colon G \times U(A) \to U(A)$ such that for each element $g \in G$ the function $\rho(g,-) \colon U(A) \to U(A)$ is an abelian group homomorphism are equivalently $\mathbb{Z}[G]$-module structures on $A$.
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 $U(A)$.
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.
Equivalently (in the case where $V$ is a closed monoidal category, where $V$ is regarded as enriched in itself), if we regard the monoid $A$ as a one-object $\mathcal{V}$-enriched category $\mathbf{B}A$, the module together with its action are given by a $V$-enriched functor
Correspondingly, a left module over $A$ is a functor
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
are $K$-$L$-bimodules, also known as profunctors or distributors from $K$ to $L$.
For $R$ a ring, write $\mathbf{B}R$ for the Ab-enriched category with a single object and hom-object $R = \mathbf{B}R(\bullet, \bullet)$.
Then a left $R$-module $N$ is equivalently an Ab-enriched functor
This makes manifest that the category $R$Mod is an Ab-enriched category, namely the Ab-enriched functor category
The right $R$-modules can be considered as $Ab$-functors $\mathbf{B}R^{op}\to Ab$. Then the usual tensor product (over $\mathbb{Z}$) $M\otimes N$ of a right and left $R$-modules can be considered as a functor $\mathbf{B}R^{op}\otimes \mathbf{B}R\to Ab$. The coend $\int^R M\otimes N$ computes then to $M\otimes_R N$.
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)
is nothing but a $G$-set: a set equipped with a $G$-action:
$\mathbf{B}G$ is the small category that is the delooping groupoid of $G$, which has a single object and $Hom_{\mathbf{B}G}(\bullet,\bullet) = G$. The functor $\mathbf{B}G \to Set$ takes the single object to some set $S$ and takes each morphism $(\bullet \stackrel{g}{\to} \bullet)$ 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.
There is a general definition of modules in terms of stabilized slice-categories of the category of monoids: tangent (infinity,1)-categories.
The ordinary case of modules over rings is phrased in terms of stabilized overcategories by the following observation, which goes back at least to Jon Beck’s 1967 thesis, and is found in the important paper of Daniel Quillen; both listed below.
Let $R \in CRing$ be a commutative ring. Then there is a canonical equivalence between the category $R Mod$ of $R$-modules and the category $Ab(CRing/R)$ of abelian group objects in the overcategory of $CRing$ over $R$
We first unwind what the structure of an abelian group object $(p: K \to R)$ in the overcaregory $CRing/R$ is explicitly
The unit of the abelian group object in $CRing/R$ is a diagram
This diagram identifies $K$ with a ring whose underlying abelian group is the direct sum $R \oplus ker(p)$ of some ring $R$ with the kernel of $p$ such that 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
With the above unit, the unit axiom on this operation together with the fact that the top morphism is a ring homomorphism says that this morphism is
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 $R$.
Let $G$ be a group. Taking together the above desriptions
one finds:
The category of $G$-modules is equivalent to the category of abelian group objects in the slice of Ring over the group ring
But there is also a more direct characterization along these lines, not involving the auxiliary construction of group rings.
The category of $G$-modules is equivalent to the category of abelian group objects in the slice category of groups over $G$
The proof is analogous to that of prop. 2. One checks that a group homomorphism $\hat G \to G$ with the structure of an abelian group object over $G$ is a central extension of $G$ by some abelian group $A$ which more over is a split extension (the is the neutral element of the abelian group object) and hence is a semidirect product group $\hat G \simeq G \ltimes A$. By the discussion there these are equivalently given by actions of $G$ on $A$ by group automorphisms. This is precisely what it means for $A$ to carry a $G$-module structure.
This construction generalizes to ∞-groups. See at ∞-action the section ∞-action – G-modules.
Let $sAlg_k$ (or $sAlg$ for short) be the (∞,1)-category of commutative simplicial algebras over a base field $k$.
For $A \in sAlg_k$ there is generally a functor
from the stable (∞,1)-category of $A$-modules to the stabilization of the overcategory of $sAlg$. But in general this functor is neither essentially surjective nor full. If however $k$ has characteristic 0, then this is an equivalence.
There is a notion of algebra over an operad. The corresponding notion of modules is described at module over an algebra over an operad.
Let $R$ be a commutative ring.
The ring $R$ is naturally a module over itself, by regarding its multiplication map $R \otimes R \to R$ as a module action $R \otimes N \to N$ with $N \coloneqq R$.
More generally, for $n \in \mathbb{N}$ the $n$-fold direct sum of the abelian group underlying $R$ is naturally a module over $R$
The module action is componentwise:
Even more generally, for $I \in$ Set any set, the direct sum $\oplus_{i \in I} R$ is an $R$-module.
This is the free module (over $R$) on the set $S$.
The set $I$ serves as the basis of a free module: a general element $v \in \oplus_i R$ is a formal linear combination of elements of $I$ with coefficients in $R$.
For special cases of the ring $R$, the notion of $R$-module is equivalent to other notions:
For $R = \mathbb{Z}$ the integers, an $R$-module is equivalently just an abelian group.
A $\mathbb{Z}$-module, hence an abelian group, is not a free module if it has a non-trivial torsion subgroup.
For $R = k$ a field, an $R$-module is equivalently a vector space over $k$.
Every finitely generated free $k$-module is a free module, hence everey finite dimensional vector space has a basis. For infinite dimensions this is true if the axiom of choice holds.
For $f : S \to R$ a homomorphism of rings, restriction of scalars produces $R$-modules $f_* N$ from $S$-modules $N$ and extension of scalars produces $S$-modules $f_! N$ from $R$-modules $N$.
For $N$ a module and $\{n_i\}_{i \in I}$ a set of elements, the linear span
(hence the completion of this set under addition in $N$ and multiplication by $R$) is a submodule of $N$.
Consider example 8 for the case that the module is $N = R$, the ring itself, as in example 1. Then a submodule is equivalently (called) an ideal of $R$.
Let $X$ be a topological space and let
be the ring of continuous functions on $X$ with values in the complex numbers.
Given a complex vector bundle $E \to X$ on $X$, write $\Gamma(E)$ for its set of continuous sections. Since for each point $x \in X$ the fiber $E_x$ of $E$ over $x$ is a $\mathbb{C}$-module (by example 6), $\Gamma(X)$ is a $C(X,\mathbb{C})$-module.
By the Serre-Swan theorem if $X$ is Hausdorff and compact, then $\Gamma(X)$ is a projective $C(X,\mathbb{C})$-module and indeed there is an equivalence between projective $C(X,\mathbb{C})$-modules and complex vector bundles over $X$.
More on this below in Vector bundle and 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 $Vect(X)$ 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 $C^{op}$ our (∞,1)-category of “functions rings” on test spaces), by the above the assignment of all modules over a test space is given by
Then for $X$ any space regarded as an ∞-stack on $C$, a “quasicorent $\infty$-stack of modules” on $X$ is a morphism
module, (∞,1)-module, 2-module
finitely generated module, presentable module, finitely presented module?
projective module, injective module, free module, flat module
A standard textbook is
Lectures notes on sheaves of modules / modules over a ringed space are in
Lecture notes with an eye towards physics are in
See also the references at enriched category theory and at profunctor.
The observation that the category of modules over a ring $R$ is equivalent to the category of abelian group objects in the overcategory $CRing/R$ was used by Quillen:
More ‘classical’ references for this include Jon Beck’s thesis
The fully abstract higher categorical concept in terms of stabilized overcategories and the tangent (∞,1)-category appears in
(∞,1)-modules over A-∞ algebras are discussed in section 4.2 of