Given a monoid RR in a monoidal category (𝒞,)(\mathcal{C}, \otimes), RRMod is the category whose objects are RR-modules in 𝒞\mathcal{C} and whose morphisms are module homomorphisms.

Specifically if (𝒞,)(\mathcal{C}, \otimes) is the category Ab of abelian groups and \otimes the tensor product of abelian groups, then RR is a ring.

We write just ModMod for the category whose objects are pairs (R,N)(R,N) consisting of a monoid RR and an RR-module, and whose morphisms may also map between different monoids.

Definition of ModMod

We assume that the ambient monoidal category is Ab with the tensor product of abelian groups. But the definition works more generally


An object in ModMod is a pair (R,N)(R,N) consisting of a commutative ring RR and an RR-module NN.

A morphism

(ϕ,κ):(R,N)(R,N) (\phi,\kappa) : (R,N) \to (R',N')

is a pair consisting of a ring homomorphism ϕ:RR\phi : R \to R' and a morphism κ:Nϕ *N\kappa : N \to \phi^* N' of RR-modules, where ϕ *N\phi^* N' is the tensor product ϕ *N:=R ϕN\phi^* N' := R \otimes_{\phi} N.


ModMod as a bifibration

Projecting out the first items in the pairs appearing in def. 1 yields a canonical functor

p:ModCRing p :Mod \to CRing
(R,N)R. (R,N) \mapsto R \,.

that exhibits ModMod as a bifibration over RR.

The fiber of this projection over a ring RR is Mod RMod_R, the category of RR-modules.

In particular the fiber over the initial commutative ring R=R = \mathbb{Z} is

Mod =Ab Mod_{\mathbb{Z}} = Ab

the category Ab of abelian groups.

Tangents and deformation theory

By an old observation of Quillen – reviewed at module – the bifibration ModCRingMod \to CRing this is equivalent to the category of fiberwise abelian group object in the codomain fibration [I,CRing]CRing[I,CRing] \to CRing:

(ModCRing)Ab([I,CRing]]CRing). (Mod \to CRing) \simeq Ab([I,CRing]] \to CRing) \,.

For a fixed ring RR, the category Mod RMod_R of RR-modules is canonically equivalent to Ab(CRing/R)Ab(CRing/R), the category of abelian group objects in the overcategory CRing/RCRing/R:

Mod RAb(CRing/R). Mod_R \simeq Ab(CRing/R) \,.

This says that ModRingMod \to Ring is the tangent category of CRingCRing: the above equivalence regards an RR-module NN equivalently as the square-0 extension ring RNR \oplus N (with multiplication (r 1,n 1)(r 2,n 2)=(r 1r 2,r 1n 2+r 2n 1)(r_1,n_1) \cdots (r_2,n_2) = (r_1 r_2, r_1 n_2 + r_2 n_1)), which may be thought of as the ring of functions on the infinitesimal neighbourhood of the 0-section of the vector bundle (or rather quasicoherent sheaf) over SpecRSpec R that is given by NN.

There is thus another natural projection from ModMod to rings, namely the functor that remembers these square-0 extensions

f:ModCRing f : Mod \to CRing
(R,N)RN. (R,N) \mapsto R \oplus N \,.

This functor has a left adjoint Ω:CRingMod\Omega : CRing \to Mod which is also a section: this is the functor that sends a ring to its module of Kähler differentials.

(Ωf):ModfΩCRing. (\Omega \dashv f) : Mod \stackrel{\overset{\Omega}{\leftarrow}}{\underset{f}{\to}} CRing \,.

RModR Mod is an abelian category

Let the ambient monoidal category be Ab equipped with the tensor product of abelian groups.


Let RR be a commutative ring. Then RModR Mod is an abelian category.

In fact RModR Mod is a Grothendieck category.

We discuss now all the ingredients of this statement in detail.

Let U:RModSetU : R Mod \to Set be the forgetful functor to the underlying sets.


RModR Mod has a zero object, given by the 0-module, the trivial group equipped with trivial RR-action.


Clearly the 0-module 00 is a terminal object, since every morphism N0N \to 0 has to send all elements of NN to the unique element of 00, and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism 0N0 \to N always exists and is unique, as it has to send the unique element of 0, which is the neutral element, to the neutral element of NN.

  1. RModR Mod has all kernels. The kernel of a homomorphism f:N 1N 2f : N_1 \to N_2 is the set-theoretic preimage U(f) 1(0)U(f)^{-1}(0) equipped with the induced RR-module structure.

  2. RModR Mod has all cokernels. The cokernel of a homomorphism f:N 1N 2f : N_1 \to N_2 is the quotient abelian group

    cokerf=N 2im(f) coker f = \frac{N_2}{im(f)}

    of N 2N_2 by the image of ff.


The defining universal property of kernel and cokernels is immediately checked.


U:RModSetU : R Mod \to Set preserves and reflects monomorphisms and epimorphisms:

A homomorphism f:N 1N 2f : N_1 \to N_2 in RModR Mod is a monomorphism / epimorphism precisely if U(f)U(f) is an injection / surjection.


Suppose that ff is a monomorphism, hence that f:N 1N 2f : N_1 \to N_2 is such that for all morphisms g 1,g 2:KN 1g_1, g_2 : K \to N_1 such that fg 1=fg 2f \circ g_1 = f \circ g_2 already g 1=g 2g_1 = g_2. Let then g 1g_1 and g 2g_2 be the inclusion of submodules generated by a single element k 1Kk_1 \in K and k 2Kk_2 \in K, respectively. It follows that if f(k 1)=f(k 2)f(k_1) = f(k_2) then already k 1=k 2k_1 = k_2 and so ff is an injection. Conversely, if ff is an injection then its image is a submodule and it follows directly that ff is a monomorphism.

Suppose now that ff is an epimorphism and hence that f:N 1N 2f : N_1 \to N_2 is such that for all morphisms g 1,g 2:N 2Kg_1, g_2 : N_2 \to K such that fg 1=fg 2f \circ g_1 = f \circ g_2 already g 1=g 2g_1 = g_2. Let then g 1:N 2N 2im(f)g_1 : N_2 \to \frac{N_2}{im(f)} be the natural projection. and let g 2:N 20g_2 : N_2 \to 0 be the zero morphism. Since by construction fg 1=0f \circ g_1 = 0 and fg 2=0f \circ g_2 = 0 we have that g 1=0g_1 = 0, which means that Nim(f)=0\frac{N}{im(f)} = 0 and hence that N=im(f)N = im(f) and so that ff is surjective. The other direction is evident on elements.


For N 1,N 2RModN_1, N_2 \in R Mod two modules, define on the hom set Hom RMod(N 1,N 2)Hom_{R Mod}(N_1,N_2) the structure of an abelian group whose addition is given by argumentwise addition in N 2N_2: (f 1+f 2):nf 1(n)+f 2(n)(f_1 + f_2) : n \mapsto f_1(n) + f_2(n).


With def. 2 RModR Mod composition of morphisms

:Hom(N 1,N 2)×Hom(N 2,N 3)Hom(N 1,N 3) \circ : Hom(N_1, N_2) \times Hom(N_2, N_3) \to Hom(N_1,N_3)

is a bilinear map, hence is equivalently a morphism

Hom(N 1,N 2)Hom(N 2,N 3)Hom(N 1,N 3) Hom(N_1, N_2) \otimes Hom(N_2,N_3) \to Hom(N_1, N_3)

out of the tensor product of abelian groups.

This makes RModR Mod into an Ab-enriched category.


Linearity of composition in the second argument is immediate from the pointwise definition of the abelian group structure on morphisms. Linearity of the composition in the first argument comes down to linearity of the second module homomorphism.


In fact RModR Mod is even a closed category, see prop. 8 below, but this we do not need for showing that it is abelian.

Prop. 1 and prop. 4 together say that:


RModR Mod is an pre-additive category.


RModR Mod has all products and coproducts, being direct products iIN i\prod_{i \in I} N_i and direct sums iIN i\oplus_{i \in I} N_i.

The products are given by cartesian product of the underlying sets with componentwise addition and RR-action.

The direct sum is the submodule of the direct product consisting of tuples of elements such that only finitely many are non-zero.


The defining universal properties are directly checked. Notice that the direct product iIN i\prod_{i \in I} N_i consists of arbitrary tuples because it needs to have a projection map

p j: iIN iN j p_j : \prod_{i \in I} N_i \to N_j

to each of the modules in the product, reproducing all of a possibly infinite number of non-trivial maps {KN j}\{K \to N_j\}. On the other hand, the direct sum just needs to contain all the modules in the sum

ι j:N j iIN i \iota_j : N_j \to \oplus_{i \in I} N_i

and since, being a module, it needs to be closed only under addition of finitely many elements, so it consists only of linear combinations of the elements in the N jN_j, hence of finite formal sums of these.

Together cor. 2 and prop. 5 say that:


RModR Mod is an additive category.


In RModR Mod


Using prop. 2 this is directly checked on the underlying sets: given a monomorphism KNK \hookrightarrow N, its cokernel is NNKN \to \frac{N}{K}, The kernel of that morphism is evidently KNK \hookrightarrow N.

Now cor. 2 and prop. 6 imply theorem 1, by definition.


The operation of forming filtered colimits in RModR Mod is an exact functor.

(e.g. Kiersz, prop. 4)

RModR Mod is a closed monoidal category

Let RR be a commutative ring.


For N 1,N 2RModN_1, N_2 \in R Mod, equip the hom-set Hom RMod(N 1,N 2)Hom_{R Mod}(N_1, N_2) with the structure of an RR-module as follows: for all f,gHom RMod(N 1,N 2)f,g \in Hom_{R Mod}(N_1, N_2), all n 1N 1n_1 \in N_1 and all rRr \in R set

  • (f+g):n 1f(n 1)+g(n 2)(f + g) \colon n_1 \mapsto f(n_1) + g(n_2)

  • rf:n 1r(f(n 1))r \cdot f \colon n_1 \mapsto r\cdot (f(n_1)).

Write [N 1,N 2]RMod[N_1,N_2] \in R Mod for the resulting RR-module structure.


Equipped with the tensor product of modules, RModR Mod becomes a monoidal category. The tensor unit is RR regarded canonically as an RR-module over itself.

This is a closed monoidal category, the internal hom is given by the hom-modules of def. 3.


Either by definition or by a basic property of the tensor product of modules, a module homomorphism ϕ:N 1 RN 2N 3\phi \colon N_1 \otimes_R N_2 \to N_3 is precisely an RR-bilinear function of the underlying sets. For fixed elements n 1N 1n_1 \in N_1 and n 2N 2n_2 \in N_2 write

ϕ¯(n 1)ϕ(n 1,):N 2N 3 \overline{\phi}(n_1) \coloneqq \phi(n_1, -) \colon N_2 \to N_3


ϕ(,n 2):N 1N 3 \phi(-,n_2) \colon N_1 \to N_3

for the hom-adjuncts on the underlying sets. By the bilinearity of ϕ\phi both of these are RR-linear maps. The first being linear means that ϕ¯\overline{\phi} is a function ϕ¯:N 1[N 2,N 3]\overline{\phi} \colon N_1 \to [N_2, N_3] to the set of module homomorphisms, and the second being linear says that it is itself a mododule homomorphisms by def. 3, since

ϕ¯(rn 1)=(n 2ϕ(rn 1,n 2)=rϕ(n 1,n 2))=r(ϕ¯(n 1)). \overline{\phi}(r\cdot n_1) = (n_2 \mapsto \phi(r\cdot n_1, n_2) = r \phi(n_1, n_2)) = r \cdot \left(\overline{\phi}(n_1)\right) \,.

The map ϕϕ¯\phi \mapsto \overline{\phi} establishes a natural transformation

Hom RMod(N 1 RN 2,N 3)Hom RMod(N 1,[N 2,N 3]). Hom_{R Mod}(N_1 \otimes_R N_2, N_3) \stackrel{}{\to} Hom_{R Mod}(N_1, [N_2, N_3]) \,.

Conversely, every element of Hom RMod(N 1,[N 2,N 3])Hom_{R Mod}(N_1, [N_2, N_3]) defines bilinear map, hence a homomorphism N 1 RN 2N 3N_1 \otimes_R N_2 \to N_3 and this construction is inverse to the above, showing that it is a natural isomorphism. This exhibits the internal hom-adjunction () RN 2[N 2,](-) \otimes_R N_2 \vdash [N_2,-].

Exact functors between categories of modules

The Eilenberg-Watts theorem says that sufficiently exact functors between categories of modules are necessarily given by forming tensor products of modules.

Limits and colimits

Let RR be a ring.


Every RR-module is the filtered colimit over its finite generated submodules.

See for instance (Kiersz, prop. 3).

Tiny objects

For discussion of tiny objects in ModMod, see at Tiny object – In categories of modules over rings.

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


Discussion of RModR Mod in (Ab,)(Ab, \otimes) being an abelian category is for instance in

  • Rankeya Datta, The category of modules over a commutative ring and abelian categories (pdf)

Discussion of limits and colimits in RModR Mod is in

  • Andy Kiersz, Colimits and homological algebra, 2006 (pdf)

Discussion of homotopy theoretic modules via stabilization of slice model structures is in

  • Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104

A summary of the discussion in Mod as a bifibration and Tangents and deformation theory together with their embedding into the bigger picture of tangent (∞,1)-categories is in

category: category

Revised on November 29, 2016 03:18:42 by Urs Schreiber (