strict 2-groups VS crossed modules

nLab defines a strict 2-groups in many different but equivalent ways, among them:

• an internal group object in Cat,
• an internal group object in Grpd

Also, it is known that strict 2-groups may be defined from crossed modules (see for example Baez). There is one thing I don't understand (I will consider the 2-group as a 2-category by the delooping process of the monoidal category that we obtain by one or the other of definitions):

• Seeing a 2-group as internal to Cat or to Grpd will give, at first glance slightly different structures, the difference is being that the second internalization will make 2-morphisms vertically invertible, but not the first one.

• Constructing a 2-group from a crossed-module (as done in the reference) will also make 2-morphisms vertically invertible.

Hence, I see that only when we define a 2-group as a group internal to Cat the 2-morphisms are not (a least directly) defined as vertically invertible. I suspect the functoriality of the group multiplication functor to be responsible for defining vertical inverses (this turned out to be false) but I do not see how! My question: how to prove that each 2-morphism of a 2-group (s̲e̲e̲n̲ ̲a̲s̲ ̲a̲ ̲g̲r̲o̲u̲p̲ ̲o̲b̲j̲e̲c̲t̲ ̲i̲n̲ ̲C̲a̲t̲) has a vertical inverse, knowing that the vertical composition is inherited from the category composition law in the object of Cat that, unlike an object in Grpd, does not define canonically inverses .

• Do you know what is called the Eckmann--Hilton argument? It might help shed a conceptual light on the somewhat ad-hoc answers below. Jan 20 '16 at 13:25
• Yes, if a group is equipped with two different products sharing the same unit, they must agree. Jan 20 '16 at 13:36
• Well it also applies to monoids, and the two products have to satisfy some interchange relation. I think basically here the idea is to see composition as the first product, and the product from the group law as the second product, and the interchange relation is the fact that the group law is a bifunctor. So the two laws "agree", and since everything is inversible wrt the group law, then everything is inversible w.r.t. composition. (Of course there are difficulties in applying this idea directly, I spent an hour on this yesterday and couldn't work out the details... ☹) Jan 20 '16 at 13:39
• Does "agree" mean agree? :-) because our two laws in the cat-group (that from the composition functor and that of category inner composition) do not agree! Jan 20 '16 at 13:49

Let us prove the analogous claim for weak 2-groups (this implies, in particular, the strict case). Let $\mathcal{C}$ be a monoidal category with unit $\mathbb{I} \in \mathcal{C}$.

Claim: Suppose there exists a functor $Inv: \mathcal{C} \to \mathcal{C}$ and natural isomorphisms $\psi_X: \mathbb{I} \stackrel{\cong}{\to} X \otimes Inv(X)$ and $\phi_X: Inv(X) \otimes X \stackrel{\cong}{\to} \mathbb{I}$. Then $\mathcal{C}$ is a groupoid.

Proof: Let $f: X \to Y$ be a map in $\mathcal{C}$. We need to show that $f$ is an isomorphism. Let $g: Y \to X$ be the composed map $$Y \stackrel{\psi_X \otimes Id_Y}{\to} X \otimes Inv(X) \otimes Y \stackrel{Id_X \otimes Inv(f) \otimes Id_Y}{\to} X \otimes Inv(Y) \otimes Y \stackrel{Id_X \otimes \phi_Y}{\to} X$$ We now claim that $g \circ f$ is an isomorphism. Indeed, $g \circ f$ is equal to the composition $$X \stackrel{\psi_X \otimes Id_X}{\to} X \otimes Inv(X) \otimes X \stackrel{Id_X \otimes Inv(f) \otimes f}{\to} X \otimes Inv(Y) \otimes Y \stackrel{Id_X \otimes \phi_Y}{\to} X$$ and the naturality of $\phi$ implies that $Inv(f) \otimes f$ is an isomorphism. A similar argument shows that $f \circ g$ is an isomoprhism. It follows that both $f$ and $g$ are isomorphisms.

• In conclusion, it is the inverse existence axiom that transforms the monoidal category to a groupoid, that's why it's equivalent to take an internal group object in Cat or in Grpd. Thank you. Jan 20 '16 at 12:09
• And more precisely the functoriality of these inverses. To illustrate this let $\mathcal{C}$ be the nerve of the poset of integers $(\mathbb{Z},\leq)$. The addition operation induces a monoidal product on $\mathcal{C}$ such that tensoring with $-n$ is inverse to tensoring with $n$, and yet $\mathcal{C}$ is not a groupoid. What happens here is that the operation which associates to an object $n \in \mathcal{C}$ the object $-n$ is contravariant, and not covariant, and this is enough to make the proof above fail. Jan 20 '16 at 12:48

Pedro, you seem to be making your life difficult! My first suggestion is to read the original sources on this and in particular:

R. Brown and C. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296 – 302, [pdf]

Tracking through from a 2-group to a crossed module and back again should give you a good intuition about what is going on. Something like this: Given a 2-cell $\alpha: g_1\Rightarrow g_2$ in your strict 2-group, you can write $\alpha$ as being $(\alpha \ast_0 g_1^{-1})\ast_0 g_1$ so a pre-whiskering of a 2-cell in the kernel of the source map. This is the sort of 2-cell that comes directly from a crossed module viewpoint, so look at that next.

If $\partial : C\to P$ is a crossed module, then the associated 2-group has the semi-direct product $C\rtimes P$ as its group of 2-cells. In Brown and Spencer you can find the formula for the vertical inverse of a 2-cell $(c,p)$. This gives $(c^{-1},\partial c.p)$, now push this formula back into your original setting multiply out the whiskering and you get a formula for the vertical inverse of $\alpha$.

Of course, this is just what Yonathan has given in non-strict case but relates things back to the original material.

It is worth mentioning once again that the Brown-Spencer method is nicely seen as being a mild but very neat generalisation of the proof that congruences in groups correspond to normal subgroups. It also uses the idea that often (and I do not want to be precise here) an A object in the category of B objects is the same as a B object in the category of A objects.

PS: A good reference for some of this is the short note by Magnus Forrester-Barker: http://arxiv.org/abs/math/0212065.

• "you seem to be making your life difficult" perhaps! But it itches me on the brain to see that the same group object gives the same structure either internal to Cat or Grpd. Now, I am relieved :-) Thank you for the original paper. Jan 20 '16 at 12:17
• Pedro, A lot of Ronnie Brown's papers are available on his personal website. He has had some of them retyped with comments which might be useful. Jan 20 '16 at 14:29

This material on strict group objects in groupoids is covered in the book partially titled Nonabelian Algebraic Topology (NAT)(pdf available), together with lots of history and intuition. See also this mathoverflow answer for a discussion of uses of different models.

My aim in this area from 1965 was towards higher versions of the Seifert-van Kampen Theorem, and this was achieved with Philip Higgins in 1974, utilising earlier work with Chris Spencer, through the use of double groupoids with connections and their equivalence with crossed modules (over groupoids!). So we can calculate some 2nd relative homotopy groups in terms of pushouts of crossed modules - see the book NAT- but this idea is not mentioned in this part of the n-lab.

For my aims, 2-groups lie between crossed modules and double groupoids (with connections) without the separate applications of either: thus crossed modules are useful for calculation and relation to classical homotopy theory, while double groupoids with connections are useful for intuition, conjectures and proofs. The equivalence of these two concepts allows one to hop between them at will, without in many cases worrying about the proof. I can't see these results being even conjectured globularly or simplicially.

The cubical methods are also very useful for discussing homotopies and higher homotopies, because of the rule $I^m \times I^n \cong I^{m+n}$.