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 can be taken then as the very definition: notably in algebraic geometry Gelfand duality is taken to “hold by definition” in that an algebraic variety is essentially by definition the formal dual of a given ring, and the Serre-Swan theorem similarly becomes the statement that the space of sections 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 us 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.
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
Then we give more general definitions
See module over a 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. 1. 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