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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

starting an entry, for the moment mainly in order to record the fact that “crossed homomorphisms” are equivalently homomorphic sections of the corresponding semidirect product group projection. This is obvious, but is there a reference that makes it explicit?

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2021
• (edited Aug 30th 2021)

As some people call these ‘derivative’ we already have some material on this under Fox derivatives and also at derivation on a group. Good references would include Ken Brown’s Cohomology of Groups, in Chapter IV section 2. Another linked entry would be under group extension.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

That’s the abelian case. I have added a remark.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

Ah, I see that K. Brown states the identification of #1 on p. 88. Will add the pointer now.

Will be also adding this to derivation on a group, where it didn’t used to be cited…

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2021

The non-abelian case is already present in the section on ‘Split extensions and semidirect product groups’ in group extensions. The term derivation is often used in combinatorial group theory as being equivalent to crossed homomorphism.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

If you have another reference using the term “crossed homomorphism” for the general non-abelian case, then let’s add it.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2021
• (edited Aug 30th 2021)

The three terms crossed homomorphism, derivation and 1-cocyle are used, for example, in a paper by Daniel Guin here, but the usage is so common by then so I do not know of a good ‘old’ source. That was just found by Googling!. Crossed homomorphisms do arise as such with homotopies between crossed modules and of course, 1-cocycles again as such are common in non-abelian cohomology.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

I do not know of a good ‘old’ source

No problem, it just sounded in #5 like you meant to allude to one.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2021

Most of the time I use derivation or 1-cocycle so forget the other name!

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

I don’t want to belabor the point further, as I am sensing there is none. Just to highlight that the term “derivation” for crossed homomorphisms is at best weird, as it clashes without need or purpose with an innocent standard term. That’s the reason why over at “derivation on a group” the definition needs to be appended right away with the disclaimer that it doesn’t define what it sounds like it’s defining.

When its codomain is abelian, then you might call a homomorphism a logarithm (as then it takes “products” to “sums”) but that is true generally and has nothing to do with the crossed property characteristic of crossed homomorphisms. But when the codomain is allowed to be non-abelian, as is the case here in the situation under discussion, it seems outright crazy to speak of homomorphisms (crossed or not) as derivations. It’s like the guy saying that black is white, of which Douglas Adams famously knew that he “got himself killed on the next zebra crossing.”

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 30th 2021
• (edited Aug 30th 2021)

added statement (here) of the following example/theorem:

For well-behaved $G$-equivariant classifying spaces of $\Gamma$-principal bundles, the connected components of their $H$-fixed loci are in bijection to the conjugacy classes (properly understood) of crossed homomorphisms from $H$ to $\Gamma$.

This is, after a little reformulation, the content of Lashof & May 1986, Thm. 10.

This statement becomes more transparent using the Murayama-Shimakawa-model for the equivariant classifying spaces. May add this next, but not tonight.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

added (here) statement and proof that the groupoid of crossed homomorphisms is isomorphic to the sliced functor groupoid:

$CrsHom(G,\Gamma) \sslash_{\!\! ad} \Gamma \;\; \simeq \;\; Fnctr_{{}_{/\mathbf{B}G}} \big( \mathbf{B}G ,\, \mathbf{B}(\Gamma \rtimes G) \big)$
• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeAug 31st 2021
• (edited Aug 31st 2021)

Added (here) the characterization of the graphs of crossed homomorphisms as the subgroups

$\widehat G \;\subset\; \Gamma \rtimes G \,, \;\;\;\; \text{such that} \;\;\;\; \mathrm{pr}_2\big(\widehat G\big) \simeq G \;\;\; \text{and} \;\;\; \widehat{G} \cap i(\Gamma) \;=\; \{\mathrm{e}\} \,.$

together with a remark that this is how crossed homomorphisms implicitly appear in the articles by Peter May on equivariant bundle theory.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 31st 2021

added (here) a full-blown diagram showing how crossed homomorphisms with crossed conugations between them are isomorphic to sliced functors and transformations between them.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 31st 2021

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2021

Looking good. What is the relationship between crossed homomorphism and crossed module of groups?

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 31st 2021

Not sure yet what an interesting statement might be that relates the two. This “crossed”-terminology is not systematic.

One thing that comes to mind:

Given $\mathcal{G}$ the strict 2-group corresponding to a crossed module $\Gamma \xrightarrow{\delta} G$, then in the stict (2,1)-category of 2-groups, the set of 2-morphisms out of $id_{\mathcal{G}}$ is in bijection to crossed homomorphisms $G\to \Gamma$.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2021

Let’s see. From here, p. 12, given two morphisms between two 2-groups, then a transformation between these morphisms is a pair whose second element is a crossed homomorphism.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2021
• (edited Aug 31st 2021)

So, much as you say.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeAug 31st 2021

True, good catch, that’s a more general version of #19.

Might be worth recording in the entry. But myself, I am done for tonight.

• CommentRowNumber23.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021
• (edited Sep 1st 2021)

I found a ‘good old source’ for crossed homomorphisms with possibly non-abelian codomain. It is Combinatorial Homotopy II, of course. More precisely:

J. H. C. Whitehead, Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55, (1949), 453 – 496.

They are mentioned early on (section 3, p. 457) and also are important in section 8, p. 468.

The transformations mentioned by DavidC correspond to homotopies, in much the same way that degree 1 maps between chain complexes are ‘chain homotopies’.

Fox derivatives (and yes that term is the usual one in this context since the first two axioms of their definition are typical partial derivative formulae and the third one is a modification of the product rule for the same), are an essential tool for the study of presentations of groups by generators and relations. Any group presentation gives a free crossed module, namely the fundamental crossed module of the 2-complex defined by the generators and relations. This is already mentioned at Fox derivative. If we ’abelianise’ the crossed module, (so, essentially, passing to the universal cover of the 2-complex) the ’boundary map’ of the free crossed module becomes the Alexander matrix of the presentation. This is already very well handled at Fox derivative. Fox’s paper is from 1948 but I do not think he uses the term ‘crossed homomorphism’ as such.

• CommentRowNumber24.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021
• (edited Sep 1st 2021)

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeSep 1st 2021

John Baez wrote a post on crossed homomorphisms, pointing to some link with Galois descent, discussed here.

• CommentRowNumber26.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021

There is a link to Knots and Primes, and thus to Arithmetic Topology. This goes further than the book John Baez was using. It links things to profinite Fox derivatives, and Alexander-Fox theory, which corresponds to Iwasawa Theory (which I do not understand). There are several introductions online.

There should be some way of going further with all this, but I do not have a good enough knowledge of the Number Theory side of things. I have a lot of stuff on profinite group presentations etc., and on profinite homotopy theory, but really that should be up-dated to Pyknotic or perhaps Condensed homotopy theory, as that then starts to be nearer the Arithmetic Number Theory ideas.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeSep 1st 2021

That reference by Whitehead is good! I have added pointer to it also from the main text, now a remark here.

• CommentRowNumber28.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021

I wonder if the notion might be even older. Does it go back to Reidemeister’s work in the 1930s?

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeSep 1st 2021

where the characterization of the graphs of non-abelian crossed homomorphisms (not under that name, though) is already proven.

(This is a completely elementary and easy proof, which is probably why tom Dieck gives it in-line inside a definition. And yet, making this explicit goes a long way, as Murayama-Shimakawa must have finally realized in 1995.)

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeSep 1st 2021

added (here) the example mentioned in #19.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeSep 1st 2021
• (edited Sep 1st 2021)

Finally I have added (here) what I had set out to discuss:

statement and proof that, for subgroups $H \subset G$, the crossed homomorphisms $H \to \Gamma$ with conjugations between them are equivalently the $H$-fixed loci of the functor groupoid from $\mathbf{E}G$ to $\mathbf{B}\Gamma$:

$CrsHom(H,\Gamma) \sslash_{\!\!ad} \Gamma \;\simeq\; Fnctr \big( \mathbf{E}G ,\, \mathbf{B}\Gamma \big)^H$
• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021

added (here) characterization of the automorphism groups of crossed homomorphisms under crossed conjugation

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeSep 5th 2021

added (here) observation and proof that for $H \subset G$ a subgroup, its Weyl group acts on crossed conjugacy classes of crossed homomorphisms, hence on $H^1_{Grp}(H,\,\Gamma)$.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeSep 5th 2021

added (here) statement and proof that this $W_G(H)$-action on $H^1_{Grp}(H, \Gamma)$ is the one induced from the canonical Weyl group action on the $H$-fixed functor groupoid $Fnctr(\mathbf{E}G, \mathbf{B}\Gamma)^H$.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeSep 6th 2021

Oh, wow, I only just discovered that numbered equation-refs work from inside \text-environments inside maths environments. That offers a neat way to write proofs.

Have improved the writeup of the previous one accordingly, here