Abelian 2-group

In Mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids formula_1 which have a bifunctor formula_2 which acts formally like the addition an Abelian group. Namely, the bifunctor formula_3 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.
Definition.
An Abelian 2-group is a groupoid formula_1 with a bifunctor formula_2 and natural transformationsformula_6which satisfy a host of axioms ensuring these transformations behave similarly to commutativity (formula_7) and associativity formula_8 for an Abelian group. One of the motivating examples of such a category comes from the Picard category of line bundles on a scheme (see below).
Examples.
Picard category.
For a scheme or variety formula_9, there is an Abelian 2-group formula_10 whose objects are line bundles formula_11 and morphisms are given by isomorphisms of line bundles. Notice over a given line bundle formula_11formula_13since the only automorphisms of a line bundle are given by a non-vanishing function on formula_9. The additive structure formula_3 is given by the tensor product formula_16 on the line bundles. 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 bundlesformula_17but not direct equality. This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformationformula_18switching the components. The associativity similarly follows from the associativity of tensor products of line bundles.
Two term chain complexes.
Another source for Picard categories is from two-term chain complexes of Abelian groupsformula_19which have a canonical groupoid structure associated to them. We can write the set of objects as the abelian group formula_20 and the set of arrows as the set formula_21. Then, the source morphism formula_22 of an arrow formula_23 is the projection mapformula_24and the target morphism formula_25 isformula_26Notice this definition implies the automorphism group of any object formula_27 is formula_28. Notice that if we repeat this construction for sheaves of abelian groups over a site formula_9 (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. In fact, this construction must be generalized to spectra to give a precise generalization pg 88.
Example of Abelian 2-group in algebraic geometry.
One example is the Cotangent complex for a local complete intersection scheme formula_9 which is given by the two-term complexformula_31for an embedding formula_32. There is a direct categorical interpretation of this Abelian 2-group from deformation theory using the Exalcomm category.
Note that in addition to using a 2-term chain complex, would could instead consider a chain complex formula_33 and construct an Abelian n-group (or infinity-group).
Abelian 2-group of morphisms.
For a pair of Abelian 2-groups formula_34 there is an associated Abelian 2-group of morphismsformula_35whose objects are given by functors between these two categories, and the arrows are given by natural transformations. Moreover, the bifunctor formula_36 on formula_37 induces a bifunctor structure on this groupoid, giving it an Abelian 2-group structure.
Classifying abelian 2-groups.
On order to classify abelian 2-groups, strict picard categories using two-term chain complexes is not enough. One approach is in stable homotopy theory using spectra which only have two non-trivial homotopy groups. 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.
Postnikov invariant.
For an Abelian 2-group formula_1 and a fixed object formula_39 the isomorphisms of the functors formula_40 and formula_41 given by the commutativity arrowformula_42gives an element of the automorphism group formula_43 which squares to formula_44, hence is contained in some formula_45. Sometimes this is suggestively written as formula_46. We can call this element formula_47 and this invariant induces a morphism from the isomorphism classes of objects in formula_1, denoted formula_49, to formula_43, i.e. it gives a morphismformula_51which corresponds to the Postnikov invariant. In particular, every Picard category given as a two-term chain complex has formula_52 because they correspond under the Dold-Kan correspondence to simplicial abelian groups with topological realizations as the product of Eilenberg-Maclane spacesformula_53 For example, if we have a Picard category with formula_54 and formula_55, there is no chain complex of Abelian groups giving these homology groups since formula_45 can only be given by a projectionformula_57Instead this Picard category can be understood as a categorical realization of the truncated spectrum formula_58 of the sphere spectrum where the only two non-trivial homotopy groups of the spectrum are in degrees formula_59 and formula_44.