symmetric monoidal (∞,1)-category of spectra
A familiar example of a module is a vector space over a field : this is a module over in the category Ab of abelian groups: every element in 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 here is a field: more generally it could be any commutative ring (or even a general rig) . The analog of a vector space for fields replaced by rings is that of a module over the ring .
This is the traditional and maybe most common notion of modules. But the basic notion is easily much more general.
A classical situation where this correspondence holds precisely is topology, where
the Gelfand duality theorem says that sending a compact topological space to its C-star algebra of continuous functions with values in the complex numbers constitutes an equivalence of categories between compact topological spaces and the opposite category of commutative -alegebras;
the Serre-Swan theorem says that sending a Hausdorff topological complex vector bundle over a compact topological space to the -module of its continuous sections establishes an equivalence of categories between that of topological complex vector bundles over and that of finitely generated projective modules over .
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 -modules being the spaces of sections of a vector bundle on the space whose ring of functions is 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 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:
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 : 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 ) 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.
This means that more generally it makes sense to replace by any -enriched category – regarded as the horizontal categorification of a monoid, a “monoid-oid” – and think of a -enriched functor – a -presheaf on – as a module over the category .
From this perspective a --bimodule is a -enriched functor , which is in this context known as a profunctor from to . The notion of the bicategory of -enriched categories, -profunctors between these and transformations between those is then a generalization of the category of monoids in and bimodules between them.
A module over a (commutative, unital) ring may be encoded in another ring: the one that as an abelian group is the direct sum and whose product is defined by the formulas
This is a square-0 extension of . It is canonically equipped with a ring homomorphism which is the identity on and sends all elements of to 0. As such, is an object in the overcategory . But a special such object: it is in fact canonically an abelian group object in , where the group operation (over !) is given by addition of elements in .
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 is the tangent (∞,1)-category .
For instance for the (∞,1)-category of simplicial algebras over a ground field of characteristic 0, we have that the stabilization of the over (∞,1)-category over is equivalent to the -category of -modules.
We spell out the definition of module for
with the special classical cases of
Then we give more general definitions
and a unit element
A (left) module over in is
equipped with a morphism
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 . Accordingly a module over is a module in accordin to def. 1.
A module over a ring is
equipped with a morphism
and which is a bilinear function in that it satisfies
for all and ;
such that the diagram
such that the diagram
commutes, which means that on elements as above
This fibration may be characterized intrinsically, which gives yet another way of defining -modules. This we turn to below.
Simpler than the traditionally default notion of a module in , 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.)
the neutral element acts trivially
the action property holds: for all and we have .
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 as in def. 2.
For a finite group an element r of is for the form
with . Addition is given by addition of the coefficients and multiplication is given by the formula
Since the underlying abelian group of is a free by definition, a bilinear map is equivalently for each basis element a linear map . Similarly the module property is determined on basis elements, where it reduces manifestly to the action property of on .
Equivalently (in the case where is a closed monoidal category, where is regarded as enriched in itself), if we regard the monoid as a one-object -enriched category , the module together with its action are given by a -enriched functor
Correspondingly, a left module over is a functor
In this language the concept directly generalizes to the horizontal categorification of monoids . Let be any -enriched category, then -functors give right modules and functors give left modules over . Accordingly, for and two -enriched categories one says that -functors
The right -modules can be considered as -functors . Then the usual tensor product (over ) of a right and left -modules can be considered as a functor . The coend computes then to .
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 -enriched category theory as a natural generalization of the notion of module.
is nothing but a -set: a set equipped with a -action:
is the small category that is the delooping groupoid of , which has a single object and . The functor takes the single object to some set and takes each morphism to an automorphism of that set, such that composition is respected. This is just a representation of on the set .
Of course for this story to work, 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.
We first unwind what the structure of an abelian group object in the overcaregory is explicitly
The unit of the abelian group object in is a diagram
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 between two elements in the two copies of vanishes, it therefore has to vanish between two elements in the same copy, too.
This says that is a square-0 extension of . Conversely, for every square-0-extension we obtain an abelian group object this way.
For instance the square-0-extension of a ring corresponding to the canonical -module structure on itself is the ring of dual numbers for .
Let be a group. Taking together the above desriptions
But there is also a more direct characterization along these lines, not involving the auxiliary construction of group rings.
The proof is analogous to that of prop. 2. One checks that a group homomorphism with the structure of an abelian group object over is a central extension of by some abelian group which more over is a split extension (the is the neutral element of the abelian group object) and hence is a semidirect product group . By the discussion there these are equivalently given by actions of on by group automorphisms. This is precisely what it means for to carry a -module structure.
For there is generally a functor
from the stable (∞,1)-category of -modules to the stabilization of the overcategory of . But in general this functor is neither essentially surjective nor full. If however has characteristic 0, then this is an equivalence.
Let be a commutative ring.
The ring is naturally a module over itself, by regarding its multiplication map as a module action with .
The module action is componentwise:
This is the free module (over ) on the set .
For special cases of the ring , the notion of -module is equivalent to other notions:
A -module, hence an abelian group, is not a free module if it has a non-trivial torsion subgroup.
For a module and a set of elements, the linear span
(hence the completion of this set under addition in and multiplication by ) is a submodule of .
Let be a topological space and let
More on this below in Vector bundle and modules.
A vector space is a vector bundle over the point. For every vector bundle over a space , its collection of sections is a module over the monoid/ring of functions on . When is a ringed space, is usefully thought of as a sheaf of modules over the structure sheaf of :
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 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 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:
A standard textbook is
Lecture notes with an eye towards physics are in
The observation that the category of modules over a ring is equivalent to the category of abelian group objects in the overcategory was used by Quillen:
More ‘classical’ references for this include Jon Beck’s thesis