Higher algebra

Higher linear algebra



Basic idea

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

A familiar example of a module is a vector space VV over a field kk: this is a module over kk in the category Ab of abelian groups: every element in kk 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 kk here is a field: more generally it could be any commutative ring (or even a general rig) RR. The analog of a vector space for fields replaced by rings is that of a module over the ring RR.

This is the traditional and maybe most common notion of modules. But the basic notion is easily much more general.

Motivation for and role of modules: generalized vector bundles

The theory of monoids or rings and their modules, its “meaning” and usage, is naturally understood via the duality between algebra and geometry:

  1. a ring RR is to be thought of as the ring of functions on some space,

  2. an RR-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

  1. the Gelfand duality theorem says that sending a compact topological space XX to its C-star algebra C(X,)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 *C^\ast-alegebras;

  2. the Serre-Swan theorem says that sending a Hausdorff topological complex vector bundle EXE \to X over a compact topological space to the C(X,)C(X,\mathbb{C})-module of its continuous sections establishes an equivalence of categories between that of topological complex vector bundles over XX and that of finitely generated projective modules over C(X,)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 RR-modules being the spaces of sections of a vector bundle on the space whose ring of functions is RR 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 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 dictionary 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.

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 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 SetSet: 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 CatCat) 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

See module over an enriched category.

Stabilized overcategories

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

(r 1,n 1)(r 2,n 2):=(r 1r 2,r 1n 1+r 2n 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 RR. It is canonically equipped with a ring homomorphism RNRR \oplus N \to R which is the identity on RR and sends all elements of NN to 0. As such, RNRR \oplus N \to R is an object in the overcategory CRing/RCRing/R. But a special such object: it is in fact canonically an abelian group object in CRing/RCRing/R, where the group operation (over RR!) is given by addition of elements in NN.

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

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

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 CC is the tangent (∞,1)-category T CCT_C \to C.

For instance for sAlg ksAlg_k the (∞,1)-category of simplicial algebras over a ground field kk of characteristic 0, we have that the stabilization Stab(sAlgk/A)Stab(sAlgk/A) of the over (∞,1)-category over AA is equivalent to the (,1)(\infty,1)-category AModA Mod of AA-modules.


We spell out the definition of module for

Then we give more general definitions

Modules over a monoid in a monoidal category

See module over a monoid.

Presheaves in enriched category theory

See module over an enriched category.

In terms of stabilized overcategories

There is a general definition of modules in terms of stabilized slice-categories of the category of monoids: tangent (infinity,1)-categories.

Modules over a ring

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 RCRingR \in CRing be a commutative ring. Then there is a canonical equivalence between the category RModR Mod of RR-modules and the category Ab(CRing/R)Ab(CRing/R) of abelian group objects in the overcategory of CRingCRing over RR

RModAb(CRing/R). R Mod \simeq Ab(CRing/R) \,.

We first unwind what the structure of an abelian group object (p:KR)(p: K \to R) in the overcaregory CRing/RCRing/R is explicitly

The unit of the abelian group object in CRing/RCRing/R is a diagram

R K Id p R. \array{ R &&\to&& K \\ & {}_{\mathllap{Id}}\searrow && \swarrow_{\mathrlap{p}} \\ && R } \,.

This diagram identifies KK with a ring whose underlying abelian group is the direct sum Rker(p)R \oplus ker(p) of some ring RR with the kernel of pp such that for rRr \in R and nker(p)n \in ker(p) we have rnker(p)r\cdot n \in ker(p).

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

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

RNN RN R. \array{ R \oplus N \oplus N &&\to&& R \oplus N \\ & \searrow && \swarrow \\ && R } \,.

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

RNNId(Id+Id)RN. R \oplus N \oplus N \stackrel{Id \oplus (Id + Id)}{\to} R \oplus N \,.

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

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

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

Modules over a group

Let GG be a group. Taking together the above desriptions

  1. of Modules over a group as modules over the group ring

  2. of Modules over a ring as stabilized overcategories

one finds:


The category of GG-modules is equivalent to the category of abelian group objects in the slice of Ring over the group ring

GModAb(Ring /[G]). G Mod \simeq Ab(Ring_{/\mathbb{Z}[G]}) \,.

But there is also a more direct characterization along these lines, not involving the auxiliary construction of group rings.


The category of GG-modules is equivalent to the category of abelian group objects in the slice category of groups over GG

GModAb(Grp /G). G Mod \simeq Ab(Grp_{/G}) \,.

The proof is analogous to that of prop. 1. One checks that a group homomorphism G^G\hat G \to G with the structure of an abelian group object over GG is a central extension of GG by some abelian group AA which more over is a split extension (the is the neutral element of the abelian group object) and hence is a semidirect product group G^GA\hat G \simeq G \ltimes A. By the discussion there these are equivalently given by actions of GG on AA by group automorphisms. This is precisely what it means for AA to carry a GG-module structure.

This construction generalizes to ∞-groups. See at ∞-action the section ∞-action – G-modules.

Modules over a simplicial ring

Let sAlg ksAlg_k (or sAlgsAlg for short) be the (∞,1)-category of commutative simplicial algebras over a base field kk.

For AsAlg kA \in sAlg_k there is generally a functor

AModStab(sAlg k/A) A Mod \to Stab(sAlg_k/A)

from the stable (∞,1)-category of AA-modules to the stabilization of the overcategory of sAlgsAlg. But in general this functor is neither essentially surjective nor full. If however kk has characteristic 0, then this is an equivalence.

Modules over an algebra over an operad

There is a notion of algebra over an operad. The corresponding notion of modules is described at module over an algebra over an operad.


Of modules over a ring

Let RR be a commutative ring.


The ring RR is naturally a module over itself, by regarding its multiplication map RRRR \otimes R \to R as a module action RNNR \otimes N \to N with NRN \coloneqq R.


More generally, for nn \in \mathbb{N} the nn-fold direct sum of the abelian group underlying RR is naturally a module over RR

R nR nRRR nsummands. R^n \coloneqq R^{\oplus_n} \coloneqq \underbrace{R \oplus R \oplus \cdots \oplus R}_{n\;summands} \,.

The module action is componentwise:

r(r 1,r 2,,r n)=(rr 1,rr 2,rr n). r \cdot (r_1, r_2, \cdots, r_n) = (r \cdot r_1, r\cdot r_2, \cdot r \cdot r_n) \,.

Even more generally, for II \in Set any set, the direct sum iIR\oplus_{i \in I} R is an RR-module.

This is the free module (over RR) on the set SS.

The set II serves as the basis of a free module: a general element v iRv \in \oplus_i R is a formal linear combination of elements of II with coefficients in RR.

For special cases of the ring RR, the notion of RR-module is equivalent to other notions:


For R=R = \mathbb{Z} the integers, an RR-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=kR = k a field, an RR-module is equivalently a vector space over kk.

Every finitely generated free kk-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:SRf : S \to R a homomorphism of rings, restriction of scalars produces RR-modules f *Nf_* N from SS-modules NN and extension of scalars produces SS-modules f !Nf_! N from RR-modules NN.


For NN a module and {n i} iI\{n_i\}_{i \in I} a set of elements, the linear span

n i iIN, \langle n_i\rangle_{i \in I} \hookrightarrow N \,,

(hence the completion of this set under addition in NN and multiplication by RR) is a submodule of NN.


Consider example 8 for the case that the module is N=RN = R, the ring itself, as in example 1. Then a submodule is equivalently (called) an ideal of RR.


Let XX be a topological space and let

RC(X,) R \coloneqq C(X,\mathbb{C})

be the ring of continuous functions on XX with values in the complex numbers.

Given a complex vector bundle EXE \to X on XX, write Γ(E)\Gamma(E) for its set of continuous sections. Since for each point xXx \in X the fiber E xE_x of EE over xx is a \mathbb{C}-module (by example 6), Γ(X)\Gamma(X) is a C(X,)C(X,\mathbb{C})-module.

By the Serre-Swan theorem if XX is Hausdorff and compact, then Γ(X)\Gamma(X) is a projective C(X,)C(X,\mathbb{C})-module and indeed there is an equivalence between projective C(X,)C(X,\mathbb{C})-modules and complex vector bundles over XX.

More on this below in Vector bundle and modules.

Vector bundles and modules

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

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 XX 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)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 CC our (∞,1)-category of of test spaces (hence the opposite category C opC^{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

Mod:C op(,1)Cat Mod : C^{op} \to (\infty,1)Cat
Mod:UStab(C/U). Mod : U \mapsto Stab( C/U ) \,.

Then for XX any space regarded as an ∞-stack on CC, a “quasicorent \infty-stack of modules” on XX is a morphism

XMod. X \to Mod \,.



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)

Lectures notes on sheaves of modules / modules over a ringed space are in

Lecture notes with an eye towards physics are in

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 RR is equivalent to the category of abelian group objects in the overcategory CRing/RCRing/R was used by Quillen:

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

More ‘classical’ references for this 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

(∞,1)-modules over A-∞ algebras are discussed in section 4.2 of

Revised on January 6, 2015 22:12:43 by Adeel Khan (