symmetric monoidal (∞,1)-category of spectra
Given a monoid $R$ in a monoidal category $(\mathcal{C}, \otimes)$, $R$Mod is the category whose objects are $R$-modules in $\mathcal{C}$ and whose morphisms are module homomorphisms.
Specifically if $(\mathcal{C}, \otimes)$ is the category Ab of abelian groups and $\otimes$ the tensor product of abelian groups, then $R$ is a ring.
We write just $Mod$ 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.
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 $Mod$ is a pair $(R,N)$ consisting of a commutative ring $R$ and an $R$-module $N$.
A morphism
is a pair consisting of a ring homomorphism $\phi : R \to R'$ and a morphism $\kappa : N \to \phi^* N'$ of $R$-modules, where $\phi^* N'$ is the tensor product $\phi^* N' := R \otimes_{\phi} N$.
Projecting out the first items in the pairs appearing in def. 1 yields a canonical functor
that exhibits $Mod$ as a bifibration over $R$.
The fiber of this projection over a ring $R$ is $Mod_R$, the category of $R$-modules.
In particular the fiber over the initial commutative ring $R = \mathbb{Z}$ is
the category Ab of abelian groups.
By an old observation of Quillen – reviewed at module – the bifibration $Mod \to CRing$ this is equivalent to the category of fiberwise abelian group object in the codomain fibration $[I,CRing] \to CRing$:
For a fixed ring $R$, the category $Mod_R$ of $R$-modules is canonically equivalent to $Ab(CRing/R)$, the category of abelian group objects in the overcategory $CRing/R$:
This says that $Mod \to Ring$ is the tangent category of $CRing$: 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 $Spec R$ that is given by $N$.
There is thus another natural projection from $Mod$ to rings, namely the functor that remembers these square-0 extensions
This functor has a left adjoint $\Omega : CRing \to Mod$ which is also a section: this is the functor that sends a ring to its module of Kähler differentials.
Let the ambient monoidal category be Ab equipped with the tensor product of abelian groups.
Let $R$ be a commutative ring. Then $R Mod$ is an abelian category.
We discuss now all the ingredients of this statement in detail.
Let $U : R Mod \to Set$ be the forgetful functor to the underlying sets.
$R Mod$ has a zero object, given by the 0-module, the trivial group equipped with trivial $R$-action.
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 $N$.
$R Mod$ 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.
$R Mod$ has all cokernels. The cokernel of a homomorphism $f : N_1 \to N_2$ is the quotient abelian group
of $N_2$ by the image of $f$.
The defining universal property of kernel and cokernels is immediately checked.
$U : R Mod \to Set$ preserves and reflects monomorphisms and epimorphisms:
A homomorphism $f : N_1 \to N_2$ in $R Mod$ is a monomorphism / epimorphism precisely if $U(f)$ is an injection / surjection.
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 $g_1 : N_2 \to \frac{N_2}{im(f)}$ 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 $\frac{N}{im(f)} = 0$ and hence that $N = im(f)$ and so that $f$ is surjective. The other direction is evident on elements.
For $N_1, N_2 \in R Mod$ two modules, define on the hom set $Hom_{R Mod}(N_1,N_2)$ the structure of an abelian group whose addition is given by argumentwise addition in $N_2$: $(f_1 + f_2) : n \mapsto f_1(n) + f_2(n)$.
With def. 2 $R Mod$ composition of morphisms
is a bilinear map, hence is equivalently a morphism
out of the tensor product of abelian groups.
This makes $R Mod$ 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 $R Mod$ 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:
$R Mod$ is an pre-additive category.
$R Mod$ 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.
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
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
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:
$R Mod$ is an additive category.
In $R Mod$
every monomorphism is the kernel of its cokernel;
every epimorphism is the cokernel of its kernel.
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.
Let $R$ be a commutative ring.
For $N_1, N_2 \in R Mod$, equip the hom-set $Hom_{R Mod}(N_1, N_2)$ with the structure of an $R$-module as follows: for all $f,g \in Hom_{R Mod}(N_1, N_2)$, all $n_1 \in N_1$ and all $r \in R$ set
$(f + g) \colon n_1 \mapsto f(n_1) + g(n_2)$
$r \cdot f \colon n_1 \mapsto r\cdot (f(n_1))$.
Write $[N_1,N_2] \in R Mod$ for the resulting $R$-module structure.
Equipped with the tensor product of modules, $R Mod$ 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.
Either by definition or by a basic property of the tensor product of modules, a module homomorphism $\phi \colon N_1 \otimes_R N_2 \to N_3$ 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
and
for the hom-adjuncts on the underlying sets. By the bilinearity of $\phi$ both of these are $R$-linear maps. The first being linear means that $\overline{\phi}$ is a function $\overline{\phi} \colon N_1 \to [N_2, N_3]$ to the set of module homomorphisms, and the second being linear says that it is itself a mododule homomorphisms by def. 3, since
The map $\phi \mapsto \overline{\phi}$ establishes a natural transformation
Conversely, every element of $Hom_{R Mod}(N_1, [N_2, N_3])$ 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 $(-) \otimes_R N_2 \vdash [N_2,-]$.
The Eilenberg-Watts theorem says that sufficiently exact functors between categories of modules are necessarily given by forming tensor products of modules.
Let $R$ be a ring.
Every $R$-module is the filtered colimit over is finite generated submodules.
See for instance (Kiersz, prop. 3).
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-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 |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-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 |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
Discussion of $R Mod$ in $(Ab, \otimes)$ being an abelian category is for instance in
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
Discussion of limits and colimits in $R Mod$ is in