nLab
2-group

Idea

A $2$-group is a vertical categorification of the idea of group.

It is the special case of an n-group for $n=2$.

Definition

A $2$-group is a groupal groupoid, that is a groupoid whose objects have been made into a group. Equivalently, it is a monoidal category in which each every object and morphism is invertible. Also equivalently, it is a $2$-groupoid with one object, a very basic case of a k-tuply groupal n-groupoid.

Like other notions of higher category theory, $2$-groups come in weak and strict forms, depending on how you interpret the above.

Strict $2$-groups

The earliest version studied is that of strict 2-groups.

A strict $2$-group consists of:

• a collection of group homomorphisms of the form

  $$C_1\stackrel{s,t}{\to }{C}_0\stackrel{i}{\to }{C}_1$$C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1

  such that the composites $s\cdot i$ and $t\cdot i$ are the identity morphisms on $C_0$, and such that, writing $C_1{\times }_{t,s}{C}_1$ for the pullback,

  $$\begin{array}{ccc}{C}_1{\times }_{t,s}{C}_1& \to & C_1\\ \downarrow & & {\downarrow }^t\\ C_1& \stackrel{s}{\to }& C_0\end{array}$$\array{ C_1 \times_{t,s} C_1 &\to& C_1 \\ \downarrow && \downarrow^{t} \\ C_1 &\stackrel{s}{\to}& C_0 }

  there is, in addition, a homomorphism

  $$C_1{\times }_{t,s}{C}_1\stackrel{\mathrm{comp}}{\to }{C}_1$$C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1

  “respecting $s$ and $t$”;

• such that the composition $\mathrm{comp}$ is associative and unital with respect to $i$ “in the obvious way”.

See strict 2-group for further discussion and examples.

Weak $2$-groups

A weak $2$-group, or simply $2$-group, is a (weak) monoidal category such that:

• given any object $x$, there exists an object $x^{-1}$ such that the monoidal products $x\otimes x^{-1}$ and $x^{-1}\otimes x$ are each isomorphic to the monoidal unit $1$.

A coherent $2$-group is a monoidal category equipped with:

• for each object $x$ a specific object $x^{-1}$ and specific isomorphisms from $x\otimes x^{-1}$ and $x^{-1}\otimes x$ to $1$ which form an adjoint equivalence.

A theorem in HDA5 shows that every weak $2$-group may be made coherent. For purposes of internalization, one probably wants to use the coherent version.

We can also write this out in detail … later.

Examples

Strict 2-groups

Since strict 2-groups are equivalent to crossed modules, see the examples listed there.

Automorphism 2-groups

For $C$ any 2-category and $c\in C$ any object of it, the category ${\mathrm{Aut}}_C(c)\subset {\mathrm{Hom}}_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a 2-group, whose group product comes from the horizontal composition in $C$.

If $C$ is a strict 2-category there is the notion of strict automorphism 2-group. See there for more details on that case.

For instance if $C={\mathrm{Grp}}_2\subset \mathrm{Grpd}$ is the 2-category of group obtained by regarding groups as one-object groupoids, then for $H\in \mathrm{Grp}$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$\mathrm{AUT}(H):={\mathrm{Aut}}_{{\mathrm{Grp}}_2}(H)$$AUT(H) := Aut_{Grp_2}(H)

corresponding to the crossed module $(H\stackrel{\mathrm{Ad}}{\to }\mathrm{Aut}(H))$, where $\mathrm{Aut}(H)$ is the ordinary automorphism group of $H$.

Inner automorphism 2-groups

See inner automorphism 2-group.

String 2-group

See string 2-group.

References

• John Baez and Aaron Lauda, HDA V: 2-Groups (arXiv).