# nLab Platonic 2-group

Contents

group theory

### Cohomology and Extensions

#### Higher category theory

higher category theory

# Contents

## Idea

A Platonic 2-group (Epa 10) is a 2-group higher extension of a “Platonic group” in the ADE classification, i.e. of a finite subgroup of SO(3) or else of its double cover, hence a finite subgroup of SU(2).

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

## Properties

### Relation to the string 2-group

###### Proposition

The universal Platonic 2-group extensions $\mathcal{G}_{uni}[i]$ of $G_{ADE}$ finite subgroups of SU(2) by $U(1)$ (Epa-Ganter 16, def.2.11) are equivalently the restrictions of the string 2-group extension $String(SU(2)) \to SU(2)$ of $SU(2)$, hence the classifying cocycle is the restriction of the smooth second Chern class:

$\array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\mathbf{B}^3 i}} \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) } \,,$

where $i$ is the canonical inclusion of roots of unity, $i: \mathbb{Z}/{|G|} \hookrightarrow U(1)$.