nLab
Mod

Idea

That category $\mathrm{Mod}$ is the category of all modules over all commutative rings.

Definition

• An object 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 ${\varphi }^{*}N\prime :=R{\otimes }_{\varphi }N$.

As a bifibration

Projecting out the first items in these pairs 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 \,.

References

A summary of these classical facts together with their embedding into the bigger picture of tangent (∞,1)-categories is in

• Jacob Lurie, Deformation Theory

category: category