## April 24, 2020

### Crossed Homomorphisms

#### Posted by John Baez

While reading Gille and Szamuely’s Central Simple Algebras and Galois Cohomology I’m finding myself frustrated by my poor understanding of $H^1$ in group cohomology.

Roughly speaking, $H^2(G,A)$ classifies group extensions of a group $G$ by an abelian group $A$ on which it acts. $H^3(G,A)$ classifies 2-group extensions of $G$ by an abelian group $A$ on which it acts. And so on —this continues on up forever. I love this story: I call it the layer-cake philosophy of cohomology. But I never figured out how $H^1$ or $H^0$ fit into this story!

If you blindly follow the pattern, $H^1(G,A)$ should classify ways of extending a group $G$ by a group $A$ on which it acts to get a 0-group. But what does that mean? Is there any way to make it make sense? There must be.

(I won’t even try to think about $H^0$ this way. Not today anyway.)

What everyone says is this. Suppose we are given a group $G$ and a not-necessarily abelian group $A$ on which $G$ acts. Let’s write $g \!\triangleright\! a$ for the result of acting on $a \in A$ by $g \in G$. Then $H^1(G,A)$ consists of ‘cohomology classes’ of ‘crossed homomorphisms’ $\phi \colon G \to A$. Here a crossed homomorphism $\phi \colon G \to A$ is a function such that

$\phi(g h) = \phi(g) \;\; g \!\triangleright\! \phi(h)$

for all $g, h \in G$, and two crossed homomorphisms $\phi, \psi \colon G \to A$ count as cohomologous if for some $a \in A$ we have

$a \; \; \phi(g) = \psi(g) \; \; g \!\triangleright\! a$

for all $g \in G$.

All this reminds me a bit of semidirect products and crossed modules. Basically, a crossed homomorphism obeys the homomorphism rule $\phi(g h) = \phi(g) \phi(h)$ up to a fudge factor involving the action of $G$ on $A$. Two crossed homomorphisms are equivalent if they’re conjugate up to a fudge factor involving the action of $G$ on $A$:

$\psi(g) = a \; \phi(g) \; (g \!\triangleright\! a)^{-1}$

But for some reason I’ve never studied crossed homomorphisms, so I don’t see how they’re connected to topology… or anything else.

Well, that’s not completely true. Gille and Szamuely introduce them with an example. Stripping all specific details from this example, here’s what I seem to get. It helps a bit.

Suppose we have a group $G$ acting as endo-transformations, not necessarily natural, of the identity functor on some category $X$.

So, for any $g \in G$ and any object $x \in X$ we get an isomorphism

$g_x \colon x \to x$

and these obey

$(g h)_x = g_x h_x$

but these isomorphisms are not necessarily natural. So, for any morphism

$f: x \to y$

in our category we get a not-necessarily-commutative square:

$f g_x \ne g_y f$

Now let’s fix an invertible morphism $f \colon x \to y$; we’ll get a crossed homomorphism from this. Let $\phi(g)$ be the automorphism of $x$ that we get from going all the way around our not-necessarily-commutative square:

$\phi(g) = f^{-1} g_y f g_x^{-1}$

(This would be so much better with pictures!)

So, we get a map

$\phi \colon G \to \mathrm{Aut}(x)$

Now, it turns out that

$\phi(g h) = \phi(g)\, g \, \phi(h)\, g^{-1} \qquad \qquad (\star)$

Showing this requires a little calculation; I won’t do it here.

But note, $G$ acts on the group $\mathrm{Aut}(x)$ via conjugation! We can write this action as follows:

$g \!\triangleright\! a = g_x \, a \, g_x^{-1}$

With this notation, $(\star)$ becomes a special case of the equation in the definition of “crossed homomorphism”:

$\phi(g h) = \phi(g) \; \; g \!\triangleright\! \phi(h)$

In short:

Theorem. Suppose we are given a group $G$ acting as not-necessarily-natural transformations of the identity functor on a category $X$, and an isomorphism $f \colon x \to y$ in $X$. Then we get an action of $G$ on the group $\mathrm{Aut}(x)$ and a crossed homomorphism $\phi \colon G \to \mathrm{Aut}(x)$. If $G$ acts as natural transformations, this crossed homomorphism is trivial.

The hypotheses of this theorem seem a bit awkward to me, but I swear I’ve lifted them straight out of Gille and Szamuely, who do an example.

So, I’m wondering if there’s some better way to think about what’s happening here, or about crossed homomorphisms in general. For a topological understanding we might as well let $X$ be a groupoid. But what does it mean to have a group acting as not-necessarily-natural transformations of the identity functor? Well, in terms of topology, I guess it’s something like a choice of loop for each point.

If you want to see Gille and Szamuely’s example, go to Central Simple Algebras and Galois Cohomology and look at Section 2.3, “Galois descent”.

They actually have several examples. They always start by fixing a field $k$ and a finite Galois extension $K$ of $k$. Then here’s one of their categories $X$: it has vector spaces over $k$ as objects, but $K$-linear transformations $f \colon K \otimes V \to K \otimes W$ as morphisms from $V$ to $W$. Their group $G$ is the Galois group $\mathrm{Gal}(K|k)$. Any $g \in G$ gives, for any vector space $V$ over $k$, a linear transformation $g_V \colon K \otimes V \to K \otimes V$. But this is not natural because it does not commute, in the obvious sense, with all $K$-linear transformations.

By the way, the application they’re leading up to — which is what got me into this mess in the first place — is this: there’s a one-to-one correspondence between:

• isomorphism classes of central simple algebras over $k$ that become isomorphic to $n \times n$ matrix algebras when tensored with $K$,

and

• elements of $H^1(G, \mathrm{PGL}_n(K))$, where $G$ acts on the projective general linear group $\mathrm{PGL}_n(K)$ in the obvious way.

In this example we have the same group $G = \mathrm{Gal}(K|k)$, but a different category $X$. Now I guess this has central simple algebras over $k$ that become isomorphic to $n \times n$ matrix algebras as objects, and $K$-algebra morphisms $f \colon K \otimes A \to K \otimes B$ as morphisms from an algebra $A$ to an algebra $B$. If we fix an object $A$ in this category, we get $\mathrm{Aut}(A) = \mathrm{PGL}_n(K)$.

I was just trying to understand this… not just follow the argument step-by-step, but actually understand it. I think I understand it a bit better after writing this blog article. But I want to understand crossed homomorphisms and $H^1$ in group cohomology from many points of view now… a glaring hole in my education.

Posted at April 24, 2020 9:02 PM UTC

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One of the things measured by $H^1(G; A)$ is the set of conjugacy classes of splittings of the split extension $A \to A \rtimes G \to G$. If you stretch enough, then splittings of the split $(n+1)$-group extension $A[n] \to A[n] \rtimes G \to G$ should be the same as $n$-group extensions $A[n-1] \to E \to G$. (Me, I haven’t done yoga in weeks.) So a splitting of the split 1-group extension should be a “0-group extensions”.

Whether this is helpful to your application is, of course, for you to decide.

Posted by: Theo Johnson-Freyd on April 25, 2020 12:36 AM | Permalink | Reply to this

### Re:

Thanks — “conjugacy classes of splittings of $A \ltimes G$” is nice and memorable.

I have a particular thing I’m trying to understand, but also I want to understand crossed homomorphisms in every way possible.

Posted by: John Baez on April 25, 2020 1:04 AM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Crossed homomorphisms are also called ‘derivations’ and as such are related to Fox derivatives and other lovely things coming from Knot Theory. (The derivations terminology is used more in the Abelian case.)

Another related idea is that a crossed homomorpism is a ‘homotopy’ or 2-cell as in a lax functor. If you use the groupoid version of the definitions then it does become clearer. (I won’t go into it here as I know you prefer to worry things out yourself.;-)

Posted by: Tim Porter on April 25, 2020 7:38 AM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Thanks, but I actually asked for help here! Can someone explain how a crossed homomorphism is a homotopy or 2-cell in a lax functor?

Posted by: John Baez on April 25, 2020 5:56 PM | Permalink | Reply to this

### Re: Crossed Homomorphisms

I may be confused here, but isn’t a 2-cocycle a homotopy 2-cell of a pseudofunctor? (I’m doing everything invertible.)

If $G$ is a group, form the Grothendieck construction of a pseudofunctor $BG\to Grp \subseteq Grpd \subseteq Cat$ we get a non-split Grothendieck opfibration, which in this case correponds to a non-split extension. The compositor 2-cells should therefore be in correspondence with factor sets (at least for central extensions), not with crossed homomorphisms.

(Again, I may be the one making the off-by-one mistake.)

Posted by: Paolo Perrone on April 25, 2020 11:01 PM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Paolo, it is more a question of `I may be the one making the off-by-one mistake’not you! There still is a link with homotopies of crossed complexes but I think I replied too quickly to John’s post and added in the other part. I hope to have time to write down something about the homotopies later.

Posted by: Tim Porter on April 26, 2020 9:30 AM | Permalink | Reply to this

### Re: Crossed Homomorphisms

So now I’m thinking the topological way of understanding $H^1(G,A)$ may be the best route to a deeper algebraic understanding (as often the case).

Please correct anything wrong you see here:

If $A$ is a discrete group (not necessarily abelian) on which the discrete group $G$ acts trivially, the group cohomology $H^1(G,A)$ is the set of homomorphisms $\phi \colon G \to A$ mod conjugation:

$\phi \sim \psi$

iff there exists $a \in A$ with

$\psi(g) = a \phi(g) a^{-1}$

for all $g \in G$. But $H^1(G,A)$ is naturally isomorphic to the singular cohomology $H^1(K(G,1), A)$. It’s also naturally isomorphic to the set of pointed homotopy classes $[K(G,1), K(A,1)]$. But, most conceptually, it’s naturally isomorphic to the set of isomorphism classes of principal $A$-bundles over $K(G,1)$.

But what if $G$ acts on $A$ nontrivially? That’s where I’m missing the final ‘most conceptual’ understanding of $H^1(G,A)$. Algebraically it’s the set of crossed homomorphisms $\phi \colon G \to A$ modulo a twisted version of conjugation. That is,

$\phi \sim \psi$

iff there exists $a \in A$ with

$\psi(g) = a \phi(g) (\rho(g) a)^{-1}$

where $\rho$ is the action of $G$ on $A$. We can also think of $H^1(G,A)$ as the first cohomology of the space $K(G,1)$ with coefficients in the locally constant sheaf given by the action of the fundamental group of this space (namely $G$) on $A$. But what sort of entities resembling principal $A$-bundles over $K(G,1)$ are being classified by this cohomology?

Oh, I guess it’s $A$-bundles that aren’t principal, but ‘twisted principal’ in a manner governed by the action of $G$ on $A$. Can someone point me to a nice description of these ‘twisted principal bundles’? I now think I’ve seen it somewhere, once upon a time.

Posted by: John Baez on April 25, 2020 7:24 PM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Let me try to sketch out what these ‘twisted principal bundles’ are suppose to be. We’re given a connected pointed topological space $X$ and an action of its fundamental group on the (not necessarily abelian) discrete group $A$, say

$\rho \colon \pi_1(X) \to \mathrm{Aut}(A)$

We want a concept of ‘$\rho$-twisted principal $A$-bundle’ on $A$. When $X = K(G,1)$, the set of isomorphism classes of these is supposed to be $H^1(G,A)$: the first cohomology of $G$ with coefficients in the group $A$ on which it acts via $\rho$.

Since I like bundles better than sheaves, I’ll avoid the language of ‘local systems’. Instead I’ll claim that $\rho$ determines (up to isomorphism) a covering space

$\pi \colon E \to X$

whose fibers $E_x$ are groups isomorphic to $A$, with the fiber over the basepoint equals to $A$, and such that lifting any based loop $\gamma$ gives the map from $A$ to itself that equals $\rho([\gamma]) \in \mathrm{Aut}(A)$. If I need $X$ to be locally connected, so be it.

Then, I’ll say a $\rho$-twisted principal $A$-bundle $P \to X$ is a covering space of $X$ where the fiber over $x \in X$ is an $E_x$-torsor.

This is a bit rough but I hope it gets across the idea: instead of the fiber at $x$ being a torsor of a fixed group $A$, as in a principal bundle, they’re torsors of the variable group $E_x$. All these groups $E_x$ are isomorphic to $A$, but not in a canonical way!

Posted by: John Baez on April 25, 2020 10:39 PM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Every topologist, of course, knows of a bunch of bundles-of-groups natural enough, given reasonable spaces $X$, $(x\in X) \mapsto \pi_n (X, x)$ of which the “twisted-principal bundles” are probably well worth considering.

Posted by: Jesse McKeown on May 1, 2020 2:54 PM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Hi John

David pointed your posts out to me. Danny, David, Raymond and I have thought a bit about the following generalisation of principal bundles which seems to be what you have here.

Replace a Lie group $G$ by a locally trivial bundle of Lie groups all homomorphic to $G$. Like your $E$. The category of these is equivalent to the category of $Aut(G)$ principal bundles. I never knew what to call these. I started calling $H$ a $G$-group if $H$ is homomorphic to $G$ then I could talk about a bundle of $G$-groups.

Then you can consider locally trivial principal bundles acted on by a bundle of $G$-groups. The category of these is equivalent to the category of $G \rtimes Aut(G)$ principal bundles in the usual sense. I just called these generalised principal $G$-group bundles which seemed alike a mouthful.

There is a theory of connections as well but now the bundle of groups needs a connection as well as the principal bundle. (This comes up also in https://arxiv.org/abs/hep-th/0312154) We were aiming at non-abelian bundle gerbes with these sorts of objects replacing abelian principal bundles. So we spent some time developing a theory of generalised bibundles which have different bundles of groups acting on left and right, biconnections etc. This is related to what is in https://arxiv.org/abs/math/0511696. The only thing that has ever appeared by us is an abelian version of this we used in http://arxiv.org/abs/1112.1752 where we replaced the usual correspondence between central extensions and lifting bundle gerbes with one for abelian extensions and lifting generalised bundle gerbes.

Regards Michael

Posted by: Michael Murray on April 26, 2020 4:38 AM | Permalink | Reply to this

### Re: Crossed Homomorphisms

Nice, this puts an interesting more geometric spin on the subject. I should have known that you would have thought about these things!

Posted by: John Baez on April 28, 2020 7:46 AM | Permalink | Reply to this

### Re: Crossed Homomorphisms

I’ve just made a lot more progress understanding the issue raised in this post — thanks to a series blog articles by Qiaochu Yuan, who was clearly trying to straighten out the exact same issues!

For details, go here.

Posted by: John Baez on April 26, 2020 6:31 PM | Permalink | Reply to this

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