In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra,[1] which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups.
which acts formally like the addition an Abelian group.
has a notion of commutativity, associativity, and an identity structure.
Although this seems like a rather lofty and abstract structure, there are several (very concrete) examples of Abelian 2-groups.
In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups.
(that is, a category in which every morphism is an isomorphism) with a bifunctor
and natural transformations which satisfy a host of axioms ensuring these transformations behave similarly to commutativity (
and morphisms are given by isomorphisms of line bundles.
since the only automorphisms of a line bundle are given by a non-vanishing function on
This makes is more clear why there should be natural transformations instead of equality of functors.
For example, we only have an isomorphism of line bundles but not direct equality.
This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformation switching the components.
Another source for Picard categories is from two-term chain complexes of Abelian groups which have a canonical groupoid structure associated to them.
We can write the set of objects as the abelian group
is the projection map and the target morphism
is Notice this definition implies the automorphism group of any object
Notice that if we repeat this construction for sheaves of abelian groups over a site
(or topological space), we get a sheaf of Abelian 2-groups.
It could be conjectured if this can be used to construct all such categories, but this is not the case.
[3] pg 88 One example is the cotangent complex for a local complete intersection scheme
There is a direct categorical interpretation of this Abelian 2-group from deformation theory using the Exalcomm category.
there is an associated Abelian 2-group of morphisms whose objects are given by functors between these two categories, and the arrows are given by natural transformations.
In order to classify abelian 2-groups, strict Picard categories using two-term chain complexes is not enough.
While studying an arbitrary Picard category, it becomes clear that there is additional data used to classify the structure of the category, it is given by the Postnikov invariant.
given by the commutativity arrow gives an element of the automorphism group
and this invariant induces a morphism from the isomorphism classes of objects in
, i.e. it gives a morphism which corresponds to the Postnikov invariant.
In particular, every Picard category given as a two-term chain complex has
because they correspond under the Dold-Kan correspondence to simplicial abelian groups with topological realizations as the product of Eilenberg–MacLane spaces For example, if we have a Picard category with
can only be given by a projection Instead this Picard category can be understood as a categorical realization of the truncated spectrum