nLab
Mod

Idea

Given a monoid $R$ in a monoidal category $(𝒞,\otimes )$, $R$Mod is the category whose objects are $R$-modules in $𝒞$ and whose morphisms are module homomorphisms.

Specifically if $(𝒞,\otimes )$ is the category Ab of abelian groups and $\otimes$ the tensor product of abelian groups, then $R$ is a ring.

We write just $\mathrm{Mod}$ for the category whose objects are pairs $(R,N)$ consisting of a monoid $R$ and an $R$-module, and whose morphisms may also map between different monoids.

Definition of $\mathrm{Mod}$

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

Definition

An object in $\mathrm{Mod}$ is a pair $(R,N)$ consisting of a commutative ring $R$ and an $R$-module $N$.

A morphism

$$(\varphi ,\kappa ):(R,N)\to (R\prime ,N\prime )$$(\phi,\kappa) : (R,N) \to (R',N')

is a pair consisting of a ring homomorphism $\varphi :R\to R\prime$ and a morphism $\kappa :N\to {\varphi }^{*}N\prime$ of $R$-modules, where ${\varphi }^{*}N\prime$ is the tensor product ${\varphi }^{*}N\prime :=R{\otimes }_{\varphi }N$.

Properties

$\mathrm{Mod}$ as a bifibration

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

$$p:\mathrm{Mod}\to \mathrm{CRing}$$p :Mod \to CRing

$$(R,N)\mapsto R.$$(R,N) \mapsto R \,.

that exhibits $\mathrm{Mod}$ as a bifibration over $R$.

The fiber of this projection over a ring $R$ is ${\mathrm{Mod}}_R$, the category of $R$-modules.

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

$${\mathrm{Mod}}_{\mathbb{Z}}=\mathrm{Ab}$$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 $\mathrm{Mod}\to \mathrm{CRing}$ this is equivalent to the category of fiberwise abelian group object in the codomain fibration $[I,\mathrm{CRing}]\to \mathrm{CRing}$:

$$(\mathrm{Mod}\to \mathrm{CRing})\simeq \mathrm{Ab}([I,\mathrm{CRing}]]\to \mathrm{CRing}).$$(Mod \to CRing) \simeq Ab([I,CRing]] \to CRing) \,.

For a fixed ring $R$, the category ${\mathrm{Mod}}_R$ of $R$-modules is canonically equivalent to $\mathrm{Ab}(\mathrm{CRing}/R)$, the category of abelian group objects in the overcategory $\mathrm{CRing}/R$:

$${\mathrm{Mod}}_R\simeq \mathrm{Ab}(\mathrm{CRing}/R).$$Mod_R \simeq Ab(CRing/R) \,.

This says that $\mathrm{Mod}\to \mathrm{Ring}$ is the tangent category of $\mathrm{CRing}$: the above equivalence regards an $R$-module $N$ equivalently as the square-0-extension ring $R\oplus N$ (with producte $(r_1,n_1)\cdots (r_2,n_2)=(r_1{r}_2,r_1{n}_2+r_2{n}_1)$), 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 $\mathrm{Spec}R$ that is given by $N$.

There is thus another natural projection from $\mathrm{Mod}$ to rings, namely the functor that remembers these square-0-extensions

$$f:\mathrm{Mod}\to \mathrm{CRing}$$f : Mod \to CRing

$$(R,N)\mapsto R\oplus N.$$(R,N) \mapsto R \oplus N \,.

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

$$(\Omega ⊣f):\mathrm{Mod}\stackrel{\stackrel{\Omega }{\leftarrow }}{\underset{f}{\to }}\mathrm{CRing}.$$(\Omega \dashv f) : Mod \stackrel{\overset{\Omega}{\leftarrow}}{\underset{f}{\to}} CRing \,.

$R\mathrm{Mod}$ is an abelian category

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

Theorem

Let $R$ be a commutative ring. Then $R\mathrm{Mod}$ is an abelian category.

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

Let $U:R\mathrm{Mod}\to \mathrm{Set}$ be the forgetful functor to the underlying sets.

Proposition

$R\mathrm{Mod}$ has a zero object, given by the 0-module, the trivial group equipped with trivial $R$-action.

Proof

Clearly the 0-module $0$ is a terminal object, since every morphism $N\to 0$ has to send all elements of $N$ to the unique element of $0$, and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism $0\to N$ 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 $N$.

Proposition

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

2. $R\mathrm{Mod}$ has all cokernels. The cokernel of a homomorphism $f:N_1\to N_2$ is the quotient abelian group

   $$\mathrm{coker}f=\frac{N_2}{\mathrm{im}(f)}$$coker f = \frac{N_2}{im(f)}

   of $N_2$ by the image of $f$.

Proof

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

Proposition

$U:R\mathrm{Mod}\to \mathrm{Set}$ preserves and reflects monomorphisms and epimorphisms:

A homomorphism $f:N_1\to N_2$ in $R\mathrm{Mod}$ is a monomorphism / epimorphism precisely if $U(f)$ is an injection / surjection.

Proof

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

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

Definition

For $N_1,N_2\in R\mathrm{Mod}$ two modules, define on the hom set ${\mathrm{Hom}}_{R\mathrm{Mod}}(N_1,N_2)$ the structure of an abelian group whose addition is given by argumentwise addition in $N_2$: $(f_1+f_2):n\mapsto f_1(n)+f_2(n)$.

Proposition

With def. 2 $R\mathrm{Mod}$ composition of morphisms

$$\circ :\mathrm{Hom}(N_1,N_2)\times \mathrm{Hom}(N_2,N_3)\to \mathrm{Hom}(N_1,N_3)$$\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

$$\mathrm{Hom}(N_1,N_2)\otimes \mathrm{Hom}(N_2,N_3)\to \mathrm{Hom}(N_1,N_3)$$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 $R\mathrm{Mod}$ into an Ab-enriched category.

Proof

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.

Remark

In fact $R\mathrm{Mod}$ is even a closed category, see prop. 7 below, but this we do not need for showing that it is abelian.

Prop. 1 and prop. 4 together say that:

Corollary

$R\mathrm{Mod}$ is an pre-additive category.

Proposition

$R\mathrm{Mod}$ has all products and coproducts, being direct products ${\prod }_{i\in I}{N}_i$ and direct sums ${\oplus }_{i\in I}{N}_i$.

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

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

Proof

The defining universal properties are directly checked. Notice that the direct product ${\prod }_{i\in I}{N}_i$ consists of arbitrary tuples because it needs to have a projection map

$$p_j:\prod _{i\in I}{N}_i\to N_j$$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 $\{K\to N_j\}$. On the other hand, the direct sum just needs to contain all the modules in the sum

$${\iota }_j:N_j\to {\oplus }_{i\in I}{N}_i$$\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 $N_j$, hence of finite formal sums of these.

Together cor. 2 and prop. 5 say that:

Corollary

$R\mathrm{Mod}$ is an additive category.

Proposition

In $R\mathrm{Mod}$

• every monomorphism is the kernel of its cokernel;

• every epimorphism is the cokernel of its kernel.

Proof

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

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

$R\mathrm{Mod}$ is a closed monoidal category

Let $R$ be a commutative ring.

Definition

For $N_1,N_2\in R\mathrm{Mod}$, equip the hom-set ${\mathrm{Hom}}_{R\mathrm{Mod}}(N_1,N_2)$ with the structure of an $R$-module as follows: for all $f,g\in {\mathrm{Hom}}_{R\mathrm{Mod}}(N_1,N_2)$, all $n_1\in N_1$ and all $r\in R$ set

• $(f+g):n_1\mapsto f(n_1)+g(n_2)$

• $r\cdot f:n_1\mapsto r\cdot (f(n_1))$.

Write $[N_1,N_2]\in R\mathrm{Mod}$ for the resulting $R$-module structure.

Proposition

Equipped with the tensor product of modules, $R\mathrm{Mod}$ becomes a monoidal category. The tensor unit is $R$ regarded canonically as an $R$-module over itself.

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

Proof

Either by definition or by a basic property of the tensor product of modules, a module homomorphism $\varphi :N_1{\otimes }_R{N}_2\to N_3$ is precisely an $R$-bilinear function of the underlying sets. For fixed elements $n_1\in N_1$ and $n_2\in N_2$ write

$$\overline{\varphi }(n_1)≔\varphi (n_1,-):N_2\to N_3$$\overline{\phi}(n_1) \coloneqq \phi(n_1, -) \colon N_2 \to N_3

and

$$\varphi (-,n_2):N_1\to N_3$$\phi(-,n_2) \colon N_1 \to N_3

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

$$\overline{\varphi }(r\cdot n_1)=(n_2\mapsto \varphi (r\cdot n_1,n_2)=r\varphi (n_1,n_2))=r\cdot (\overline{\varphi }(n_1)).$$\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 $\varphi \mapsto \overline{\varphi }$ establishes a natural transformation

$${\mathrm{Hom}}_{R\mathrm{Mod}}(N_1{\otimes }_R{N}_2,N_3)\stackrel{}{\to }{\mathrm{Hom}}_{R\mathrm{Mod}}(N_1,[N_2,N_3]).$$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 ${\mathrm{Hom}}_{R\mathrm{Mod}}(N_1,[N_2,N_3])$ defines bilinear map, hence a homomorphism $N_1{\otimes }_R{N}_2\to N_3$ and this construction is inverse to the above, showing that it is a natural isomorphism. This exhibits the internal hom-adjunction $(-){\otimes }_R{N}_2⊢[N_2,-]$.

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 $R$ be a ring.

Proposition

Every $R$-module is the filtered colimit over is finite generated submodules.

See for instance (Kiersz, prop. 3).

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category	3-category of module 2-categories
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Related concepts

• Vect

• 2Mod

• nMod?

• 2-category of module categories

References

Discussion of $R\mathrm{Mod}$ in $(\mathrm{Ab},\otimes )$ being an abelian category is for instance in

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

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

• Jacob Lurie, Deformation Theory

Discussion of limits and colimits in $R\mathrm{Mod}$ is in

• Andy Kiersz, Colimits and homological algebra (pdf)

category: category