symmetric monoidal (∞,1)-category of spectra
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.
Projecting out the first items in the pairs appearing in def. 1 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.
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
We discuss now all the ingredients of this statement in detail.
Let be the forgetful functor to the underlying sets.
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.
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.
With def. 2 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.
is an pre-additive category.
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.
is an additive category.
Using prop. 2 this is directly checked on the underlying sets: given a monomorphism , its cokernel is , The kernel of that morphism is evidently .
(e.g. 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.
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. 3, 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 .
Let be a ring.
See for instance (Kiersz, prop. 3).
|monoid/associative algebra||category of modules|
|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|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
Discussion of in being an abelian category is for instance in
Discussion of limits and colimits in is in