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In [[mathematics]], an '''Abelian 2-group''' is a higher dimensional analogue of an [[Abelian group]], in the sense of [[higher-dimensional algebra|higher algebra]],<ref>{{cite arXiv|last1=Jibladze|first1=Mamuka|last2=Pirashvili|first2=Teimuraz|date=2011-06-28|title=Cohomology with coefficients in stacks of Picard categories|class=math.AT|eprint=1101.2918}}</ref> which were originally introduced by [[Alexander Grothendieck]] while studying abstract structures surrounding [[Abelian varieties]] and [[Picard group]]s.<ref>{{Cite book|last=Grothendieck|first=Alexandrel|title=SGA 4|pages=29–30|chapter=Expose XVIII|chapter-url=http://www.normalesup.org/~forgogozo/SGA4/18/18.pdf}}</ref> More concretely, they are given by [[groupoid]]s <math>\mathbb{A}</math> which have a [[bifunctor]] <math>+:\mathbb{A}\times\mathbb{A} \to \mathbb{A}</math> which acts formally like the addition an Abelian group. Namely, the bifunctor <math>+</math> has a notion of [[commutativity]], [[associativity]], and an [[identity element|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-group (category theory)|''n''-groups]].

== Definition ==
An Abelian 2-group is a groupoid <math>\mathbb{A}</math> (that is, a [[category (mathematics)|category]] in which every [[morphism]] is an [[isomorphism]]) with a bifunctor <math>+: \mathbb{A}\times\mathbb{A} \to \mathbb{A}</math> and [[natural transformation]]s
:<math>\begin{align}
\tau: & X+Y \Rightarrow Y + X \\
\sigma: & (X+Y)+Z \Rightarrow X+(Y+Z)
\end{align}</math>
which satisfy a host of axioms ensuring these transformations behave similarly to commutativity (<math>\tau</math>) and associativity <math>(\sigma)</math> for an Abelian group. One of the motivating examples of such a category comes from the Picard category of line bundles on a [[scheme (mathematics)|scheme]] (see below).

== Examples ==

=== Picard category ===
For a [[scheme (mathematics)|scheme]] or [[algebraic variety|variety]] <math>X</math>, there is an Abelian 2-group <math>\operatorname{\textbf{Pic}} X</math> whose objects are line bundles <math>\mathcal{L}</math> and morphisms are given by isomorphisms of line bundles. Notice over a given line bundle <math>\mathcal{L}</math>
:<math>\text{End}(\mathcal{L}) = \text{Aut}(\mathcal{L}) \cong \mathcal{O}_X^*</math>
since the only [[automorphism]]s of a line bundle are given by a non-vanishing function on <math>X</math>. The additive structure <math>+</math> is given by the tensor product <math>\otimes</math> on the line bundles. This makes is more clear why there should be natural transformations instead of equality of [[functor]]s. For example, we only have an isomorphism of line bundles
:<math>\mathcal{L}\otimes\mathcal{L}' \cong \mathcal{L}'\otimes\mathcal{L}</math>
but not direct equality. This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformation
:<math>\tau: (-\otimes -) \to (-\otimes -)</math>
switching 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 complex]]es of Abelian groups
:<math>A^{-1} \xrightarrow{d} A^0</math>
which have a canonical groupoid structure associated to them. We can write the set of objects as the abelian group <math>A^0</math> and the set of arrows as the set <math>A^{-1}\oplus A^0</math>. Then, the source morphism <math>s</math> of an arrow <math>(a_{-1},a_0)</math> is the projection map
:<math>s(a_{-1} + a_0) = a_0</math>
and the target morphism <math>t</math> is
:<math>t(a_{-1}+a_0) = d(a_{-1}) + a_0</math>
Notice this definition implies the [[automorphism group]] of any object <math>a_0</math> is <math>\operatorname{Ker} d</math>. Notice that if we repeat this construction for [[sheaf (mathematics)|sheaves]] of abelian groups over a [[Grothendieck topology|site]] <math>X</math> (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 [[spectrum (topology)|spectra]] to give a precise generalization.<ref>{{cite journal|last1=Hopkins|first1=M. J.|last2=Singer|first2=I. M.|date=2005-08-24|title=Quadratic functions in geometry, topology, and M-theory|journal=J. Differ. Geom.|volume=70|issue=3|pages=329–452|doi=10.4310/jdg/1143642908|arxiv=math/0211216|s2cid=119170140}}</ref> <sup>pg 88</sup>

==== Example of Abelian 2-group in algebraic geometry ====
One example is the [[cotangent complex]] for a local complete intersection scheme <math>X</math> which is given by the two-term complex
:<math>\mathbf{L}_{X}^\bullet = i^*I/I^2 \to i^*\Omega_Y </math>
for an embedding <math>i:X \to Y</math>. There is a direct [[category theory|categorical]] interpretation of this Abelian 2-group from deformation theory using the [[Exalcomm]] category.<ref>{{Cite web|last=Olsson|first=Martin|title=Tangent and Obstruction Theories|url=https://math.berkeley.edu/~molsson/MSRISummer07.pdf|pages=13–18}}</ref>

Note that in addition to using a 2-term chain complex, would could instead consider a chain complex <math>A^\bullet \in Ch^{\leq 0}(\text{Ab})</math> and construct an Abelian ''n''-group (or infinity-group).

=== Abelian 2-group of morphisms ===
For a pair of Abelian 2-groups <math>\mathbb{A},\mathbb{A}'</math> there is an associated Abelian 2-group of morphisms
:<math>\text{Hom}(\mathbb{A},\mathbb{A}')</math>
whose objects are given by functors between these two categories, and the arrows are given by natural transformations. Moreover, the bifunctor  <math>+'</math> on <math>\mathbb{A}'</math> induces a bifunctor structure on this groupoid, giving it an Abelian 2-group structure.

== Classifying abelian 2-groups ==
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 [[trivial group|non-trivial]] [[homotopy group]]s. 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 <math>\mathbb{A}</math> and a fixed object <math>x \in \text{Ob}(\mathbb{A})</math> the isomorphisms of the functors <math>x+(-)</math> and <math>(-)+x</math> given by the commutativity arrow
:<math>\tau : x + x \Rightarrow x+x</math>
gives an element of the automorphism group <math>\text{Aut}_\mathbb{A}(x)</math> which squares to <math>1</math>, hence is contained in some <math>\mathbb{Z}/2</math>. Sometimes this is suggestively written as <math>\pi_1(\mathbb{A})</math>. We can call this element <math>\varepsilon</math> and this invariant induces a morphism from the isomorphism classes of objects in <math>\mathbb{A}</math>, denoted <math>\pi_0(\mathbb{A})</math>, to <math>\text{Aut}_\mathbb{A}(x)</math>, i.e. it gives a morphism
:<math>\varepsilon: \pi_0(\mathbb{A})\otimes\mathbb{Z}/2 \to \pi_1(\mathbb{A}) = \text{Aut}_{\mathbb {A} }(x)</math>
which corresponds to the [[Postnikov system|Postnikov invariant]]. In particular, every Picard category given as a two-term chain complex has <math>\varepsilon = 0</math> because they correspond under the Dold-Kan correspondence to simplicial abelian groups with topological realizations as the [[product topology|product]] of [[Eilenberg–MacLane space]]s
:<math>K(H^{-1}(A^\bullet), 1)\times K(H^0(A^\bullet),0)</math>
For example, if we have a Picard category with <math>\pi_1(\mathbb{A}) = \mathbb{Z}/2</math> and <math>\pi_0(\mathbb{A}) = \mathbb{Z}</math>, there is no chain complex of Abelian groups giving these homology groups since <math>\mathbb{Z}/2</math> can only be given by a projection
:<math>\mathbb{Z}\xrightarrow{\cdot 2} \mathbb{Z} \to \mathbb{Z}/2</math>
Instead this Picard category can be understood as a categorical realization of the truncated spectrum <math>\tau_{\leq 1} \mathbb{S}</math> of the [[sphere spectrum]] where the only two non-trivial homotopy groups of the spectrum are in degrees <math>0</math> and <math>1</math>.

== See also ==

* [[∞-groupoid]]
* [[N-group (category theory)|''n''-group]]
* [[Gerbe]]

== References ==
<references />
* [https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html Thesis of Hoàng Xuân Sính (Gr Categories)]
*{{cite arXiv|eprint=1007.4138|last1=Pirashvili |first1=Teimuraz |title=On abelian 2-categories and derived 2-functors |year=2010 |class=math.CT }}
* {{cite arXiv| eprint=1101.2918 | last1=Jibladze | first1=Mamuka | last2=Pirashvili | first2=Teimuraz | title=Cohomology with coefficients in stacks of Picard categories | year=2011 | class=math.AT }}
*{{cite arXiv|eprint=1702.02128|last1=Debremaeker |first1=Raymond |title=Cohomology with values in a sheaf of crossed groups over a site |year=2017 |class=math.AG }} - gives techniques for defining sheaf cohomology with coefficients in a crossed module, or a Picard category
*{{cite arXiv|eprint=1201.2686|last1=Johnson |first1=Niles |last2=Osorno |first2=Angélica M. |title=Modeling Stable One-Types |year=2012 |class=math.AT }} - exposition of stable 1-types containing relation with picard categories
*{{cite journal|arxiv=1606.07032|doi=10.2140/agt.2017.17.2763 |title=Stable Postnikov data of Picard 2–categories |year=2017 |last1=Gurski |first1=Nick |last2=Johnson |first2=Niles |last3=Osorno |first3=Angélica |last4=Stephan |first4=Marc |journal=Algebraic & Geometric Topology |volume=17 |issue=5 |pages=2763–2806 |s2cid=119324062 }}

[[Category:Abelian group theory|2-group]]
[[Category:Algebraic topology]]
[[Category:Homological algebra]]