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In [[Mathematics]], an Abelian 2-group is a higher dimensional analogue of an [[Abelian group]], in the sense of higher algebra<ref>Liquid error: wrong number of arguments (given 1, expected 2)</ref>, which were originally introduced by [[Alexander Grothendieck]] while studying abstract structures surrounding [[Abelian varieties]] and [[Picard group|Picard groups]]<ref></ref>. More concretely, they are given by groupoids <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 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> with a bifunctor <math>+:\mathbb{A}\times\mathbb{A} \to \mathbb{A}</math> and natural transformations<blockquote><math>\begin{align}
\tau: & X+Y \Rightarrow Y + X \\
\sigma: & (X+Y)+Z \Rightarrow X+(Y+Z)
\end{align}</math></blockquote>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 (see below).
== Examples ==
=== Picard category ===
For a [[Scheme (mathematics)|scheme]] or [[Algebraic variety|variety]] <math>X</math>, there is an Abelian 2-group <math>\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><blockquote><math>\text{End}(\mathcal{L}) = \text{Aut}(\mathcal{L}) \cong \mathcal{O}_X^*</math></blockquote>since the only automorphisms 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 functors. For example, we only have an isomorphism of line bundles<blockquote><math>\mathcal{L}\otimes\mathcal{L}' \cong \mathcal{L}'\otimes\mathcal{L}</math></blockquote>but not direct equality. This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformation<blockquote><math>\tau: (-\otimes -) \to (-\otimes -)</math></blockquote>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 complexes of Abelian groups<blockquote><math>A^{-1} \xrightarrow{d} A^0</math></blockquote>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<blockquote><math>s(a_{-1} + a_0) = a_0</math></blockquote>and the target morphism <math>t</math> is<blockquote><math>t(a_{-1}+a_0) = d(a_{-1}) + a_0</math></blockquote>Notice this definition implies the automorphism group of any object <math>a_0</math> is <math>\text{Ker}(d)</math>. Notice that if we repeat this construction for 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>Liquid error: wrong number of arguments (given 1, expected 2)</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<blockquote><math>\mathbf{L}_{X}^\bullet = i^*I/I^2 \to i^*\Omega_Y </math></blockquote>for an embedding <math>i:X \to Y</math>. There is a direct categorical interpretation of this Abelian 2-group from deformation theory using the [[Exalcomm]] category<ref>Liquid error: wrong number of arguments (given 1, expected 2)</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<blockquote><math>\text{Hom}(\mathbb{A},\mathbb{A}')</math></blockquote>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.
==== 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<blockquote><math>\tau : x + x \Rightarrow x+x</math></blockquote>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<blockquote><math>\varepsilon: \pi_0(\mathbb{A})\otimes\mathbb{Z}/2 \to \pi_1(\mathbb{A}) = \text{Aut}_{\mathbb {A} }(x)</math></blockquote>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 of [[Eilenberg–MacLane space|Eilenberg-Maclane spaces]]<blockquote><math>K(H^{-1}(A^\bullet), 1)\times K(H^0(A^\bullet),0)</math> </blockquote>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<blockquote><math>\mathbb{Z}\xrightarrow{\cdot 2} \mathbb{Z} \to \mathbb{Z}/2</math></blockquote>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)]]
*[[Gerbe]]
== References ==
<references />
* [https://ift.tt/2WtZEFF Thesis of Hoàng Xuân Sính (Gr Categories)]
*[[arxiv:1007.4138|On Abelian 2-categories and Derived 2-functors]]
* [[arxiv:1101.2918|Cohomology with coefficients in stacks of Picard categories]]
*[[arxiv:1702.02128|Cohomology with values in a sheaf of crossed groups over a site]] - gives techniques for defining sheaf cohomology with coefficients in a crossed module, or a Picard category
== Definition ==
An Abelian 2-group is a groupoid <math>\mathbb{A}</math> with a bifunctor <math>+:\mathbb{A}\times\mathbb{A} \to \mathbb{A}</math> and natural transformations<blockquote><math>\begin{align}
\tau: & X+Y \Rightarrow Y + X \\
\sigma: & (X+Y)+Z \Rightarrow X+(Y+Z)
\end{align}</math></blockquote>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 (see below).
== Examples ==
=== Picard category ===
For a [[Scheme (mathematics)|scheme]] or [[Algebraic variety|variety]] <math>X</math>, there is an Abelian 2-group <math>\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><blockquote><math>\text{End}(\mathcal{L}) = \text{Aut}(\mathcal{L}) \cong \mathcal{O}_X^*</math></blockquote>since the only automorphisms 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 functors. For example, we only have an isomorphism of line bundles<blockquote><math>\mathcal{L}\otimes\mathcal{L}' \cong \mathcal{L}'\otimes\mathcal{L}</math></blockquote>but not direct equality. This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformation<blockquote><math>\tau: (-\otimes -) \to (-\otimes -)</math></blockquote>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 complexes of Abelian groups<blockquote><math>A^{-1} \xrightarrow{d} A^0</math></blockquote>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<blockquote><math>s(a_{-1} + a_0) = a_0</math></blockquote>and the target morphism <math>t</math> is<blockquote><math>t(a_{-1}+a_0) = d(a_{-1}) + a_0</math></blockquote>Notice this definition implies the automorphism group of any object <math>a_0</math> is <math>\text{Ker}(d)</math>. Notice that if we repeat this construction for 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>Liquid error: wrong number of arguments (given 1, expected 2)</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<blockquote><math>\mathbf{L}_{X}^\bullet = i^*I/I^2 \to i^*\Omega_Y </math></blockquote>for an embedding <math>i:X \to Y</math>. There is a direct categorical interpretation of this Abelian 2-group from deformation theory using the [[Exalcomm]] category<ref>Liquid error: wrong number of arguments (given 1, expected 2)</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<blockquote><math>\text{Hom}(\mathbb{A},\mathbb{A}')</math></blockquote>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.
==== 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<blockquote><math>\tau : x + x \Rightarrow x+x</math></blockquote>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<blockquote><math>\varepsilon: \pi_0(\mathbb{A})\otimes\mathbb{Z}/2 \to \pi_1(\mathbb{A}) = \text{Aut}_{\mathbb {A} }(x)</math></blockquote>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 of [[Eilenberg–MacLane space|Eilenberg-Maclane spaces]]<blockquote><math>K(H^{-1}(A^\bullet), 1)\times K(H^0(A^\bullet),0)</math> </blockquote>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<blockquote><math>\mathbb{Z}\xrightarrow{\cdot 2} \mathbb{Z} \to \mathbb{Z}/2</math></blockquote>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)]]
*[[Gerbe]]
== References ==
<references />
* [https://ift.tt/2WtZEFF Thesis of Hoàng Xuân Sính (Gr Categories)]
*[[arxiv:1007.4138|On Abelian 2-categories and Derived 2-functors]]
* [[arxiv:1101.2918|Cohomology with coefficients in stacks of Picard categories]]
*[[arxiv:1702.02128|Cohomology with values in a sheaf of crossed groups over a site]] - gives techniques for defining sheaf cohomology with coefficients in a crossed module, or a Picard category
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