Abelian 2-group

For a scheme or variety ${\displaystyle X}$, there is an Abelian 2-group ${\displaystyle {\textbf {Pic}}(X)}$ whose objects are line bundles ${\displaystyle {\mathcal {L}}}$ and morphisms are given by isomorphisms of line bundles. Notice over a given line bundle ${\displaystyle {\mathcal {L}}}$

${\displaystyle {\text{End}}({\mathcal {L}})={\text{Aut}}({\mathcal {L}})\cong {\mathcal {O}}_{X}^{*}}$

since the only automorphisms of a line bundle are given by a non-vanishing function on ${\displaystyle X}$. The additive structure ${\displaystyle +}$ is given by the tensor product ${\displaystyle \otimes }$ 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

${\displaystyle {\mathcal {L}}\otimes {\mathcal {L}}'\cong {\mathcal {L}}'\otimes {\mathcal {L}}}$

but not direct equality. This isomorphism is independent of the line bundles chosen and are functorial hence they give the natural transformation

${\displaystyle \tau$ :(-\otimes -)\to (-\otimes -)}

switching the components. The associativity similarly follows from the associativity of tensor products of line bundles.

Another source for Picard categories is from two-term chain complexes of Abelian groups

${\displaystyle A^{-1}\xrightarrow {d} A^{0}}$

which have a canonical groupoid structure associated to them. We can write the set of objects as the abelian group ${\displaystyle A^{0}}$ and the set of arrows as the set ${\displaystyle A^{-1}\oplus A^{0}}$. Then, the source morphism ${\displaystyle s}$ of an arrow ${\displaystyle (a_{-1},a_{0})}$ is the projection map

${\displaystyle s(a_{-1}+a_{0})=a_{0}}$

and the target morphism ${\displaystyle t}$ is

${\displaystyle t(a_{-1}+a_{0})=d(a_{-1})+a_{0}}$

Notice this definition implies the automorphism group of any object ${\displaystyle a_{0}}$ is ${\displaystyle {\text{Ker}}(d)}$. Notice that if we repeat this construction for sheaves of abelian groups over a site ${\displaystyle X}$ (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}.

One example is the Cotangent complex for a local complete intersection scheme ${\displaystyle X}$ which is given by the two-term complex

${\displaystyle \mathbf {L} _{X}^{\bullet }=i^{*}I/I^{2}\to i^{*}\Omega _{Y}}$

for an embedding ${\displaystyle i:X\to Y}$. 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 ${\displaystyle A^{\bullet }\in Ch^{\leq 0}({\text{Ab}})}$ and construct an Abelian n-group (or infinity-group).

For a pair of Abelian 2-groups ${\displaystyle \mathbb {A} ,\mathbb {A} '}$ there is an associated Abelian 2-group of morphisms

${\displaystyle {\text{Hom}}(\mathbb {A} ,\mathbb {A} ')}$

whose objects are given by functors between these two categories, and the arrows are given by natural transformations. Moreover, the bifunctor ${\displaystyle +'}$ on ${\displaystyle \mathbb {A} '}$ induces a bifunctor structure on this groupoid, giving it an Abelian 2-group structure.

For an Abelian 2-group ${\displaystyle \mathbb {A} }$ and a fixed object ${\displaystyle x\in {\text{Ob}}(\mathbb {A} )}$ the isomorphisms of the functors ${\displaystyle x+(-)}$ and ${\displaystyle (-)+x}$ given by the commutativity arrow

${\displaystyle \tau :x+x\Rightarrow x+x}$

gives an element of the automorphism group ${\displaystyle {\text{Aut}}_{\mathbb {A} }(x)}$ which squares to ${\displaystyle 1}$, hence is contained in some ${\displaystyle \mathbb {Z} /2}$. Sometimes this is suggestively written as ${\displaystyle \pi _{1}(\mathbb {A} )}$. We can call this element ${\displaystyle \varepsilon }$ and this invariant induces a morphism from the isomorphism classes of objects in ${\displaystyle \mathbb {A} }$, denoted ${\displaystyle \pi _{0}(\mathbb {A} )}$, to ${\displaystyle {\text{Aut}}_{\mathbb {A} }(x)}$, i.e. it gives a morphism

${\displaystyle \varepsilon$ :\pi _{0}(\mathbb {A} )\otimes \mathbb {Z} /2\to \pi _{1}(\mathbb {A} )={\text{Aut}}_{\mathbb {A} }(x)}

which corresponds to the Postnikov invariant. In particular, every Picard category given as a two-term chain complex has ${\displaystyle \varepsilon =0}$ because they correspond under the Dold-Kan correspondence to simplicial abelian groups with topological realizations as the product of Eilenberg-Maclane spaces

${\displaystyle K(H^{-1}(A^{\bullet }),1)\times K(H^{0}(A^{\bullet }),0)}$

For example, if we have a Picard category with ${\displaystyle \pi _{1}(\mathbb {A} )=\mathbb {Z} /2}$ and ${\displaystyle \pi _{0}(\mathbb {A} )=\mathbb {Z} }$, there is no chain complex of Abelian groups giving these homology groups since ${\displaystyle \mathbb {Z} /2}$ can only be given by a projection

${\displaystyle \mathbb {Z} \xrightarrow {\cdot 2} \mathbb {Z} \to \mathbb {Z} /2}$

Instead this Picard category can be understood as a categorical realization of the truncated spectrum ${\displaystyle \tau _{\leq 1}\mathbb {S} }$ of the sphere spectrum where the only two non-trivial homotopy groups of the spectrum are in degrees ${\displaystyle 0}$ and ${\displaystyle 1}$.