nLab 2-group

Idea

The notion of 2-group is a vertical categorification of the notion of group.

It is the special case of an n-group for $n=2$, equivalently an ∞-group which is 1-truncated. Under the looping and delooping-equivalence, 2-groups are equivalent to pointed connected homotopy 2-types.

Somewhat more precisely, a $2$-group is a group object in the (2,1)-category of groupoids. Equivalently, it is a monoidal groupoid in which the tensor product with any object has an inverse up to isomorphism. Also equivalently, by the looping and delooping-equivalence, it is a pointed 2-groupoid with a single equivalence class of objects.

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

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

  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,

  $$\array{ C_1 \times_{t,s} C_1 &\to& C_1 \\ \downarrow && \downarrow^{t} \\ C_1 &\stackrel{s}{\to}& C_0 }$$

  there is, in addition, a homomorphism

  $$C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1$$

  “respecting $s$ and $t$”;

• such that the composition $comp$ 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 where every morphism is invertible and 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 where every morphism is invertible and 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 HDA V (see references) shows that every weak $2$-group may be made coherent. For purposes of internalization, one probably wants to use the coherent version.

Definition

Definition

The (2,1)-category $2Grp$ of 2-groups is equivalently

• the full sub-(2,1)-category of that of monoidal categories and strong monoidal functors on those that are groupoids and whose tensor product has weak inverses for each object;

• the full sub-(∞,1)-category of that of ∞-groups on the 1-truncated objects;

• the full sub-(∞,1)-category of that of group objects in ∞Grpd on the 1-truncated objects;

• the full sub-(∞,1)-category

  $$\infty Grpd_{1 \leq \bullet \leq 2}^{*/} \hookrightarrow \infty Grpd^{*/}$$

  of ∞Grpd$^{*/}$ on those objects which are both connected as well as 2-truncated.

Remark

The last equivalent characterization is related to the previous ones by the looping and delooping-equivalence

$$\array{ Grp(\infty Grpd) &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}& \infty Grpd^{*/}_{\geq 1} \\ \uparrow^{\mathrlap{full\;inc.}} && \uparrow^{\mathrlap{full\;inc.}} \\ 2 Grp &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}& 2Grpd_{\geq 1}^{*/} } \,.$$

Here $(-)^{*/}$ denotes taking pointed objects, hence the slice under the point, and $(-)_{\geq}$ denotes the full full inclusion on connected objects.

By replacing in the last of these equivalent characterizations the ambient (∞,1)-topos ∞Grpd with any other one, to be denoted $\mathbf{H}$, obtains notions of 2-groups with extra structure. For instance for $\mathbf{H} =$ Smooth∞Grpd the $(\infty,1)$-topos of smooth ∞-groupoids one obtains:

Definition

The (2,1)-category $Smooth2Grp$ of smooth 2-groups is

$$\array{ Grp(Smooth \infty Grpd) &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}& Smooth\infty Grpd^{*/}_{\geq 1} \\ \uparrow^{\mathrlap{full\;inc.}} && \uparrow^{\mathrlap{full\;inc.}} \\ Smooth 2 Grp &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}& Smooth2Grpd_{\geq 1}^{*/} } \,.$$

Below in presentation by crossed modules are discussed more explict presentations of $2Grp$ and $Smooth2Grpd$ etc. by explicit algebraic data.

Properties

Presentation by crossed modules

By the discussion there, every ∞-group has a presentation by a simplicial group. More precisely, the (∞,1)-category, $\infty Grp$, is presented by the model structure on simplicial groups (for instance under simplicial localization)

$$\infty Grpd \simeq L_W Grp^{\Delta^{op}} \,.$$

Moreover, if $G \in Grp^{\Delta^{op}}$ is an n-group, then it is equivalent to a n-coskeletal simplicial group. For $n = 2$ one finds that these are naturally identified with crossed modules of groups (see there for more details).

In conclusion, this means that

Proposition

The (2,1)-category $2Grp$ of 2-groups is equivalent to the simplicial localization of the category with weak equivalences whose

• objects are crossed modules

• morphisms are homomorphisms of crossed modules;

• weak equivalences are those morphisms of crossed modules which correspond to weak homotopy equivalences of the corresponding simplicial groups.

A straightforward analysis shows that

Proposition

For $(G_1 \stackrel{\delta}{\to} G_0, G_0 \stackrel{\alpha}{\to} Aut(G_1))$ a crossed module, the homotopy groups of the corresponding 2-group/simplicial group are

• $\pi_0 = G_0 / im(\delta)$ (the quotient of $G_0$ by the image of $\delta$, which is necessarily a normal subgroup of $G_0$);

• $\pi_1 = ker(\delta)$ (the kernel of $\delta$).

Accordingly, a weak equivalence of crossed modules $f : G \to H$ is a morphism of crossed modules which induces an isomorphism of kernel and cokernel of $\delta_G$ with that of $\delta_H$.

Similar statements hold for 2-groups with extra structure. For instance the $(2,1)$-category $Smooth2Grp$ of smooth 2-groups is equivalent to the simplicial localization of the category whose

• objects are sheaves of crossed modules on CartSp${}_{smooth}$;

• weak equivalences are those morphisms of sheaves of crossed modules which on every stalk induce weak equivalences of crossed modules as above.

(See the discussion at Smooth∞Grpd for more on this.)

Examples

Specific examples

Picard 2-group

• Picard 2-group

Automorphism 2-groups

For $C$ any 2-category and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ 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 $C = Grp_2 \subset Grpd$ is the 2-category of group obtained by regarding groups as one-object groupoids, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$AUT(H) := Aut_{Grp_2}(H)$$

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

Inner automorphism 2-groups

See inner automorphism 2-group.

String 2-group

See string 2-group.

Platonic 2-group

See Platonic 2-group

Equivalences of 2-groups

We discuss some weak equivalences in the category with weak equivalences of crossed modules and crossed module homomorphisms, which presents $2Grp$ by the discussion above.

From inclusions of normal subgroups

Let $G$ be a group and $N \hookrightarrow G$ the inclusion of a normal subgroup. Equipped with the canonical action of $G$ on $N$ by conjugation, this inclusion constitutes a crossed module. There is a canonical morphism of crossed modules from $(N \hookrightarrow G)$ to $(1 \to G/N)$, hence to the ordinary quotient group, regarded as a crossed module.

Observation

The morphism $(N \hookrightarrow G) \to G/N$ is a weak equivalence of crossed modules, prop. . Accordingly, it presents an equivalence of 2-groups.

Proof

The canonical morphism in question is given by the commuting diagram of groups

$$\array{ N &\stackrel{f_1}{\to}& 1 \\ \downarrow && \downarrow \\ G &\stackrel{f_0}{\to}& G/N } \,.$$

By prop. we need to check that this induces an isomorphism on the kernel and cokernel of the vertical morphisms.

The kernel of the left vertical morphism is the trivial group, because $N \hookrightarrow G$ is an inclusion, by definition. Clearly also the kernel of the right vertical morphisms is the trivial group. Hence $f_1$ restricted to the kernels is the unique morphism from the trivial group to itself, hence is an isomrphism.

Moreover, the cokernel of the left vertical morphism is of course the quotient $G/N$ and $f_0$, being the quotient map, is manifestly an isomorphism on cokernels.

This class of weak equivalence plays an important role as constituting 2-anafunctors that exhibit long fiber sequence extensions of short exact sequences of central extensions of groups.

Observation

Let $A \to \hat G$ be the inclusion of a central subgroup, exhibiting a central extension $A \to \hat G \to G$ with $G := \hat G/A$. Then this short exact sequence of groups extends to a long fiber sequence of 2-groups

$$A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A \,,$$

where $\mathbf{B}A$ denotes the 2-group given by the crossed module $(A \to 1)$, and similarly for the other cases.

Here the connecting homomorphism $G \to \mathbf{B}A$ is presented in the category of crossed modules by a zig-zag / anafunctor whose left leg is the above weak equivalence:

$$(1 \to G) \stackrel{\simeq}{\leftarrow} (A \to \hat G) \to (A \to 1) \,.$$

Example

For smooth 2-groups, useful examples of the above are smooth refinements of various universal characteristic classes:

• the second Stiefel-Whitney class

  $$w_2 : \mathbf{B}SO \to \mathbf{B}^2\mathbb{Z}_2$$

  is induced this way from the central extension $\mathbb{Z}_2 \to Spin \to SO$ of the special orthogonal group by the spin group;

• the first Chern class

  $$c_1 : \mathbf{B}U(1) \to \mathbf{B}^2 \mathbb{Z}$$

  induced from the central extension $\mathbb{Z} \to \mathbb{R} \to U(1)$.

Related concepts

• group

  • groupoid, monoidal groupoid

  • 2-group, crossed module, differential crossed module

    • braided 2-group, symmetric 2-group

    • quantum 2-group

  • 3-group, 2-crossed module / crossed square, differential 2-crossed module

  • n-group

  • ∞-group, simplicial group, crossed complex, hypercrossed complex

  • group stack

    smooth 2-group

• higher gauge symmetry

• generalized global symmetry

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid

References

Original articles

The notion of 2-groups first appears in

• Alexandru Solian, Groupe dans une catégorie, C. R. Acad. Sc. Paris 275 (1972) [ark:/12148/bpt6k57310477/f7]

• Alexandru Solian, Coherence in categorical groups, Communications in Algebra 9 10 (1980) 1039-1057 [doi:10.1080/00927878108822631]

and in

• Hoàng Xuân Sính, Gr-catégories, PhD thesis, Hanoi (1973, 1975) [web, pdf, pdf]

Sinh (1973) was supervised by Alexander Grothendieck and showed that 2-groups are classified, up to equivalence, by quadruples consisting of:

• a group $G$
• an abelian group $A$
• an action of $G$ as automorphisms of $A$
• an element of $H^3(G,A)$, often called the Sinh invariant.

She later published two papers on the subject:

• Hoàng Xuân Sính, Gr-catégories strictes, Acta Mathematica Vietnamica 3 2 (1978) 47-59 [web, pdf]

• Hoàng Xuân Sính, Catégories de Picard restreintes, Acta Mathematica Vietnamica 7 1 (1983) 117-122 [pdf, web]

Sinh 1978 appears to prove that every 2-group is equivalent to a strict 2-group arising from a crossed module. The second calls a symmetric 2-group (i.e. a symmetric monoidal category with all objects and morphisms invertible) a Picard category, and calls a Picard category restrained if the braiding $B_{x,x} \colon x \otimes x \to x \otimes x$ is the identity for all objects $x$. The article then proves that every Picard category is equivalent to one arising from a 2-term chain complex of abelian groups.

Computational enumeration of geometrically discrete 2-groups using the computer program XMod:

• Murat Alp, Christopher Wensley, Enumeration of $Cat^1$-groups of low order, Int. J. Algebra Comput. 10, 407 (2000) [doi:10.1142/S0218196700000170]

• Graham Ellis, Luyen Van Le, Homotopy 2-types of Low order, Experimental Mathematics 23 4 (2014) [doi:10.1080/10586458.2014.912059, pdf]

Review

An early textbook account on strict 2-groups and explaining the relation to crossed modules:

• Saunders MacLane, §XII.8 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

Exposition of general 2-groups as monoidal categories with all objects and morphisms invertible (sometimes called Picard 2-groups):

• John Baez, Aaron Lauda, HDA V: 2-Groups, Theory and Applications of Categories 12 14 (2004) 423-491 [arXiv:math.QA/0307200, tac:12-14]

Geometric 2-groups

Beware that most of the above discussion is about geometrically discrete 2-groups.

Discussion of geometrically structured 2-groups (notably smooth 2-groups, hence “Lie 2-groups”):

• Chris Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group Geometry & Topology 15 (2011) 609-676 [arXiv:0911.2483, doi:10.2140/gt.2011.15.609]

and via group stacks:

• Urs Schreiber, §2.6.5 & §3.4.2 of: differential cohomology in a cohesive topos [arXiv:1310.7930]

For more on this see the references at string 2-group.

Examples

A key example due to its universality for higher central extensions of compact Lie groups by the circle 2-group to smooth 2-groups is

• the string 2-group – see there for references.

Further on 2-group-extensions by the circle 2-group:

of tori (see also at T-duality 2-group):

• Nora Ganter, Categorical Tori, SIGMA 14 (2018), 014, 18 (arXiv:1406.7046)

of finite subgroups of SU(2) (to Platonic 2-groups):

• Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, Higher Structures 1(1):122-146, 2017 (arXiv:1605.09192, hs:30)

Arguments that 2-groups play a role in symmetry protected topological phases of matter:

• Anton Kapustin, Ryan Thorngren, Higher symmetry and gapped phases of gauge theories, in Algebra, Geometry, and Physics in the 21st Century, Progress in Mathematics, 324 (2017) 177-202 [arXiv:1309.4721, doi:10.1007/978-3-319-59939-7_5]

• Maissam Barkeshli, Parsa Bonderson, Meng Cheng, Zhenghan Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, Phys. Rev. B 100 115147 (2019) [arXiv:1410.4540, doi:10.1103/PhysRevB.100.115147, talk pdf]

In this vein, 2-groups are now discussed by a vocal group of physicists under the term “generalized symmetries” or similar.