strict 2-groups VS crossed modules

Question

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 .

Answer 1

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.

Answer 2

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.

Answer 3

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 $I^m \times I^n \cong I^{m+n}$.

See also my preprint page for other expositions (Paris, Galway, Aveiro).

I also add a warning that Mac Lane's "Categories for the Working Mathematician" 2nd edition has a section on crossed modules and group objects in groupoids, but he fails to state or use the second axiom for a crossed module.

November 26, 2016: See also this preprint on Modelling and Computing Homotopy Types: I.