### On Weak Cokernels for 2-Groups

#### Posted by Urs Schreiber

In Detecting higher order necklaces I mentioned how

P. Carrasco A. R. Garzó́n and E. M. Vitale
*On categorical crossed modules*

Theory and Applications of Categories, Vol. 16, 2006, No. 22, pp 585-618

is related to

David Roberts, U.S.
*The inner automorphism 3-group of a strict 2-group*

arXiv:0708.1741

(html).

with the relation becoming obvious after drawing some diagram. This discussion, quite brief, but the picture provided there is useful to keep in mind, I have now prepared here:

This blog entry is hence mainly a private message to David Roberts (since it builds on our discussion of inner automorphism $(n+1)$-groups). And maybe to Todd Trimble (since it builds on our discussion of *tangent categories* (pdf, html). And to Jim Stasheff (since it is going to be applied to obstructions to lifts of $n$-Cartan connections (slides, BIG diagram)). And to John Baez of course, from whome I am hoping to receive more hints on how to think of the big picture . And to Bruce Bartlett, with whom I was talking about this here and by email.

Hence a blog post.

You’ll see that this issue of weak cokernels of 2-groups is the integral version of what I had started to discuss at the level of Lie $n$-algebras in

Obstructions and cokernels of Lie n-algebra morphisms

(pdf)

(html).

To whetten your appetite, a degenerate (low $n$) version of what I am trying to get at here is related to Schreier theory:

Given a sequence

$K \to G \to B$

of groups, we cannot, in general, find a splitting $G \leftarrow B$ which is a group homomorphism. But the failure is “coherent”.

One way to say this is:

denote by

$(K \to G)$

the weak cokernel of the map $K \to G$. This is nothing but the 2-group defined by the crossed module of the same name $K \to G$!

And this 2-group is equivalent to $B$

$(K \to G) \simeq B$

This means we always have morphisms

$B \hookrightarrow (K \to G)$

which are equivalences.

These are best thought of as pseudofunctors on the corresponding one-object 2-groupoids:

on morphisms these functors are nothing but lifts

$G \leftarrow B$

that may fail to respect composition, strictly. But the failure – the compositor – is something in $K$, indeed a 2-morphism in $\Sigma(K \to G)$.

Now, the 2-group $(K \to G)$ sits inside the 2-group $\mathrm{AUT}(K)$,

$(K\to G ) \to \mathrm{AUT}(K)$

of automorphic functors on $\Sigma K$

Notice how this is a nice way to understand lifting gerbes:

A $B$-bundle is given by a cocycle which can be thought of as lots of triangles labeled in $B$.

We may ask if we can lift these labels to labels in $G$. In general we
cannot. But what we can always do is lift them to labels in $(K \to G)$:
because that simply means that we choose any lifts, and then label the
*interior* of the triangle with the mistake in K that we made thereby.

So every $B$-cocycle is a $(K \toG)$- cocycle.

We may want to characterize the “mistakes in $K$” that we made by this attempted lift. To find these, we check which part of our failed lift doesn’t sit entirely in B, so we look at the cokernel of the canonical inclusion

$G \hookrightarrow (K \to G)$

(notice that we are using here that the “puffed up” version $(K \to G)$ of $B$ has the property that what used to be a surjection $G \to B$ now becomes an inclusion!)

which is simply $\Sigma K$

$G \to (K\to G ) \to \Sigma K$

Postcomposing our labelling of triangles with this morphism

$(K \to G) \to \Sigma K$

simply forgets the labels of the boundaries of the triangle and only remembers the $K$-labels in their interior. This is now the 2-cocycle describing a $\Sigma K$- 2-bundle. And this is the lifting 2-bundle (lifting gerbe).

If this 2-cocycle is nontrivial it means that the failures to lift that we made are real and cannot be undone by a more clever lift. If that 2-cocycle however is a 2-coboundary it means that there is a more clever choice of lift which produces no failure: the lifting gerbe trivializes.

You have maybe seen the pictures illustrating this on slides 322 onwards in my slides.

## Re: On Weak Cokernels for 2-Groups

I’m on it.

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