homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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.
The earliest version studied is that of strict 2-groups.
A strict $2$-group consists of:
a collection of group homomorphisms of the form
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,
there is, in addition, a homomorphism
“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.
A weak $2$-group, or simply $2$-group, is a (weak) monoidal category such that:
A coherent $2$-group is a monoidal category equipped with:
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.
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;
of ∞Grpd$^{*/}$ on those objects which are both connected as well as 2-truncated.
The last equivalent characterization is related to the previous ones by the looping and delooping-equivalence
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:
The (2,1)-category $Smooth2Grp$ of smooth 2-groups is
Below in presentation by crossed modules are discussed more explict presentations of $2Grp$ and $Smooth2Grpd$ etc. by explicit algebraic data.
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)
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
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
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.)
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
corresponding to the crossed module $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary automorphism group of $H$.
See inner automorphism 2-group.
See string 2-group.
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.
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.
The morphism $(N \hookrightarrow G) \to G/N$ is a weak equivalence of crossed modules, prop. 1. Accordingly, it presents an equivalence of 2-groups.
The canonical morphism in question is given by the commuting diagram of groups
By prop. 2 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.
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
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:
For smooth 2-groups, useful examples of the above are smooth refinements of various universal characteristic classes:
the second Stiefel-Whitney class
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
induced from the central extension $\mathbb{Z} \to \mathbb{R} \to U(1)$.
2-group, crossed module, differential crossed module
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex
2-groups were introduced, under the name gr-categories, in
supervised by Grothendieck.
Exposition and discussion of 2-groups as special monoidal categories is in
Discussion of structured 2-groups (e.g. smooth 2-groups) is in sections 2.6.5 and 3.4.2 of