Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nforum nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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?

    v1, current

    • 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.

    diff, v2, current

    • 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

    added pointer to:

    diff, v5, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2021

    added pointer to:

    diff, v5, current

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

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

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

    diff, v6, current

    • 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,Γ) adΓFnctr /BG(BG,B(ΓG)) CrsHom(G,\Gamma) \sslash_{\!\! ad} \Gamma \;\; \simeq \;\; Fnctr_{{}_{/\mathbf{B}G}} \big( \mathbf{B}G ,\, \mathbf{B}(\Gamma \rtimes G) \big)

    diff, v7, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2021
    • (edited Aug 31st 2021)

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

    G^ΓG,such thatpr 2(G^)GandG^i(Γ)={e}. \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.

    diff, v8, current

    • 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.

    diff, v9, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2021

    added an Idea-section (here)

    diff, v9, current

    • 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 𝒢\mathcal{G} the strict 2-group corresponding to a crossed module ΓδG\Gamma \xrightarrow{\delta} G, then in the stict (2,1)-category of 2-groups, the set of 2-morphisms out of id 𝒢id_{\mathcal{G}} is in bijection to crossed homomorphisms GΓG\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)

    Added reference to Whitehead’s combinatorial homotopy 2, and added Fox derivatives, etc., to related concepts.

    diff, v13, current

    • 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.

    diff, v14, current

    • 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

    added pointer to:

    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.)

    diff, v15, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2021

    added (here) the example mentioned in #19.

    diff, v16, current

    • 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 HGH \subset G, the crossed homomorphisms HΓH \to \Gamma with conjugations between them are equivalently the HH-fixed loci of the functor groupoid from EG\mathbf{E}G to BΓ\mathbf{B}\Gamma:

    CrsHom(H,Γ) adΓFnctr(EG,BΓ) H CrsHom(H,\Gamma) \sslash_{\!\!ad} \Gamma \;\simeq\; Fnctr \big( \mathbf{E}G ,\, \mathbf{B}\Gamma \big)^H

    diff, v17, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

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

    diff, v23, current

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2021

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

    diff, v25, current

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2021

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

    diff, v27, current

    • 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

    diff, v28, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)