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.[2] More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor 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

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An Abelian 2-group is a groupoid   (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 associativity   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

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Picard category

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For a scheme or variety  , there is an Abelian 2-group   whose objects are line bundles   and morphisms are given by isomorphisms of line bundles. Notice over a given line bundle  

 

since the only automorphisms of a line bundle are given by a non-vanishing function on  . The additive structure   is given by the tensor product   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 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. The associativity similarly follows from the associativity of tensor products of line bundles.

Two term chain complexes

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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   and the set of arrows as the set  . Then, the source morphism   of an arrow   is the projection map

 

and the target morphism   is

 

Notice this definition implies the automorphism group of any object   is  . 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. In fact, this construction must be generalized to spectra to give a precise generalization.[3] pg 88

Example of Abelian 2-group in algebraic geometry

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One example is the cotangent complex for a local complete intersection scheme   which is given by the two-term complex

 

for an embedding  . There is a direct categorical interpretation of this Abelian 2-group from deformation theory using the Exalcomm category.[4]

Note that in addition to using a 2-term chain complex, would could instead consider a chain complex   and construct an Abelian n-group (or infinity-group).

Abelian 2-group of morphisms

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For a pair of Abelian 2-groups   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. Moreover, the bifunctor   on   induces a bifunctor structure on this groupoid, giving it an Abelian 2-group structure.

Classifying abelian 2-groups

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In 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

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For an Abelian 2-group   and a fixed object   the isomorphisms of the functors   and   given by the commutativity arrow

 

gives an element of the automorphism group   which squares to  , hence is contained in some  . Sometimes this is suggestively written as  . We can call this element   and this invariant induces a morphism from the isomorphism classes of objects in  , denoted  , to  , 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   and  , there is no chain complex of Abelian groups giving these homology groups since   can only be given by a projection

 

Instead this Picard category can be understood as a categorical realization of the truncated spectrum   of the sphere spectrum where the only two non-trivial homotopy groups of the spectrum are in degrees   and  .

See also

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References

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  1. ^ Jibladze, Mamuka; Pirashvili, Teimuraz (2011-06-28). "Cohomology with coefficients in stacks of Picard categories". arXiv:1101.2918 [math.AT].
  2. ^ Grothendieck, Alexandrel. "Expose XVIII" (PDF). SGA 4. pp. 29–30.
  3. ^ Hopkins, M. J.; Singer, I. M. (2005-08-24). "Quadratic functions in geometry, topology, and M-theory". J. Differ. Geom. 70 (3): 329–452. arXiv:math/0211216. doi:10.4310/jdg/1143642908. S2CID 119170140.
  4. ^ Olsson, Martin. "Tangent and Obstruction Theories" (PDF). pp. 13–18.
  • Thesis of Hoàng Xuân Sính (Gr Categories)
  • Pirashvili, Teimuraz (2010). "On abelian 2-categories and derived 2-functors". arXiv:1007.4138 [math.CT].
  • Jibladze, Mamuka; Pirashvili, Teimuraz (2011). "Cohomology with coefficients in stacks of Picard categories". arXiv:1101.2918 [math.AT].
  • Debremaeker, Raymond (2017). "Cohomology with values in a sheaf of crossed groups over a site". arXiv:1702.02128 [math.AG]. - gives techniques for defining sheaf cohomology with coefficients in a crossed module, or a Picard category
  • Johnson, Niles; Osorno, Angélica M. (2012). "Modeling Stable One-Types". arXiv:1201.2686 [math.AT]. - exposition of stable 1-types containing relation with picard categories
  • Gurski, Nick; Johnson, Niles; Osorno, Angélica; Stephan, Marc (2017). "Stable Postnikov data of Picard 2–categories". Algebraic & Geometric Topology. 17 (5): 2763–2806. arXiv:1606.07032. doi:10.2140/agt.2017.17.2763. S2CID 119324062.