symmetric monoidal (∞,1)-category of spectra
Given a monoid in a monoidal category , Mod is the category whose objects are -modules in and whose morphisms are module homomorphisms.
Specifically if is the category Ab of abelian groups and the tensor product of abelian groups, then is a ring.
We write just for the category whose objects are pairs consisting of a monoid and an -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 is a pair consisting of a commutative ring and an -module .
A morphism
is a pair consisting of a ring homomorphism and a morphism of -modules, where is the restriction of scalars.
Projecting out the first items in the pairs appearing in def. yields a canonical functor
that exhibits as a bifibration over .
The fiber of this projection over a ring is , the category of -modules.
In particular the fiber over the initial commutative ring is
the category Ab of abelian groups.
By an old observation of Quillen – reviewed at module – the bifibration is equivalent to the category of fiberwise abelian group objects in the codomain fibration :
For a fixed ring , the category of -modules is canonically equivalent to , the category of abelian group objects in the overcategory :
This says that is the tangent category of : the above equivalence regards an -module equivalently as the square-0 extension ring (with multiplication ), 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 that is given by .
There is thus another natural projection from to rings, namely the functor that remembers these square-0 extensions
This functor has a left adjoint 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.
We discuss now all the ingredients of this statement in detail.
Let be the forgetful functor to the underlying sets.
has a zero object, given by the 0-module, the trivial group equipped with trivial -action.
Clearly the 0-module is a terminal object, since every morphism has to send all elements of to the unique element of , and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism 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 .
The defining universal property of kernel and cokernels is immediately checked.
preserves and reflects monomorphisms and epimorphisms:
A homomorphism in is a monomorphism / epimorphism precisely if is an injection / surjection.
Suppose that is a monomorphism, hence that is such that for all morphisms such that already . Let then and be the inclusion of submodules generated by a single element and , respectively. It follows that if then already and so is an injection. Conversely, if is an injection then its image is a submodule and it follows directly that is a monomorphism.
Suppose now that is an epimorphism and hence that is such that for all morphisms such that already . Let then be the natural projection. and let be the zero morphism. Since by construction and we have that , which means that and hence that and so that is surjective. The other direction is evident on elements.
For two modules, define on the hom set the structure of an abelian group whose addition is given by argumentwise addition in : .
With def. composition of morphisms
is a bilinear map, hence is equivalently a morphism
out of the tensor product of abelian groups.
This makes 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 is even a closed category, see prop. below, but this we do not need for showing that it is abelian.
Prop. and prop. together say that:
is an pre-additive category.
has all products and coproducts, being direct products and direct sums .
The products are given by cartesian product of the underlying sets with componentwise addition and -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 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 . 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 , hence of finite formal sums of these.
Together cor. and prop. say that:
is an additive category.
In
every monomorphism is the kernel of its cokernel;
every epimorphism is the cokernel of its kernel.
Using prop. this is directly checked on the underlying sets: given a monomorphism , its cokernel is , The kernel of that morphism is evidently .
Now cor. and prop. imply theorem , by definition.
The operation of forming filtered colimits in is an exact functor.
(e.g. Weibel 1994, Lem. 2.6.14 Kiersz, prop. 4).
Let be a commutative ring.
For , equip the hom-set with the structure of an -module as follows: for all , all and all set
.
Write for the resulting -module structure.
Equipped with the tensor product of modules, becomes a monoidal category (in fact a distributive monoidal category). The tensor unit is regarded canonically as an -module over itself.
This is a closed monoidal category, the internal hom is given by the hom-modules of def. .
Either by definition or by a basic property of the tensor product of modules, a module homomorphism is precisely an -bilinear function of the underlying sets. For fixed elements and write
and
for the hom-adjuncts on the underlying sets. By the bilinearity of both of these are -linear maps. The first being linear means that is a function to the set of module homomorphisms, and the second being linear says that it is itself a mododule homomorphisms by def. , since
The map establishes a natural transformation
Conversely, every element of defines bilinear map, hence a homomorphism and this construction is inverse to the above, showing that it is a natural isomorphism. This exhibits the internal hom-adjunction .
The Eilenberg-Watts theorem says that sufficiently exact functors between categories of modules are necessarily given by forming tensor products of modules.
Let be a ring.
Every -module is the filtered colimit over its finite generated submodules.
See for instance (Kiersz, prop. 3).
For discussion of tiny objects in , see at Tiny object – In categories of modules over rings.
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
-algebra | -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 |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
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 |
---|---|
-2-algebra | -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 |
---|---|
-3-algebra | -4-module |
Discussion of in being an abelian category is for instance in
Discussion of limits and colimits in :
Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge University Press (1994) [doi:10.1017/CBO9781139644136, pdf]
Andy Kiersz, Colimits and homological algebra, 2006 (pdf)
Discussion of in the generality of module objects over a commutative monoid object internal to a bicomplete closed symmetric monoidal category and proof that it is itself bicomplete closed symmetric monoidal:
Mark Hovey, Brooke Shipley, Jeff Smith, Lemma 2.2.2 & 2.2.8 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]
Florian Marty, Prop. 1.2.14, 1.2.16, 1.2.17 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Toulouse (2009) [theses:2009TOU30071, pdf]
Martin Brandenburg, Prop. 4.1.10: Tensor categorical foundations of algebraic geometry [arXiv:1410.1716]
See also:
Lecture notes:
Discussion of homotopy theoretic modules via stabilization of slice model structures is 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
Formalization of abelian univalent categories of ring-modules, in homotopy type theory (univalent foundations of mathematics):
Last revised on October 6, 2023 at 14:23:56. See the history of this page for a list of all contributions to it.