Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
On the nLab page for semidirect product, it says:
If a group acts on a group on the left, then there is a semidirect product group whose underlying set is but whose multiplication is
for .
What is ?
It is the result of acting with on the left on .
Thanks. I added your explanation to the definition.
Nope, it means conjugation by h, i.e. . If not, , but is the identity of the semidirect product.
As far as that though, I’d be interested about why that is the formula. I have some vague notions of why, but I’d be interested to listen.
I was wondering about that too. But if it is conjugation, then would need to act on on both the left and right?
I rolled back my minor change.
In any case, the statement could probably use some clarification on the nLab.
There is a canonical right action assigned to every left action given by the formula for , , with a fixed left action , if I remember correctly. Categorically, a right action is the same as giving a contravariant left action.
Let’s test it out: . So yes, this is the right one.
@ Harry. You are both right and wrong! It depends on the interior or exterior viewpoint. If, as stated in the n-Lab, the group is given, together with the action then what I said was correct. You construct a group as described, with the given formula as DEFINING the multiplication. You now have a group and it comes with a neat epimorphism to with kernel ISOMORPHIC to . There is thus a copy of in there and the epimorphism is split so there is a copy of in there as well. An element of sits in there as and an element of as , now if you look at conjugating the copy of by the copy of you get
so the semidirect product CONVERTS the EXTERIOR action of on into the INTERIOR action, by conjugation, of the copy of on the copy of , hence you are both wrong (exteriorly) and right (after the construction of the semidirect product has been done).
This sort of point used to be made in books on group theory (I remember one by Rotman) and was very confusing.
Another point that may be worth making is that an action need not be given by a morphism . In fact, in some settings there is no automorphism thing around so you use
satisfying certain obvious rules. (With profinite groups, the group of automorphisms is not always profinite, although it can be topologised in a sensible way.) Of course, you also have, as you pointed out, the idea of an action as a functor from as a category, and that is also very useful.
Aha, I can see possibly where the initial confusion might be. The action is an action on the GROUP not on the underlying set of , i.e. an action by automorphisms. Saying it is on the left does not mean left multiplication in any sense, it does mean left action!
If this was not the source of confusion, ignore this comment!
For the record, when I said
I was wondering about that too.
it was referring to a softer question Harri asked, but he then edited his comment to a more strongly worded disagreement.
Thanks Tim! I’m headed to bed and will have a look at the details tomorrow, but your nice description definitely needs a home on the nLab.
Edit: Also, when I said the statement could use some clarification, I was referring to the original statement on the nLab and not Tim’s comment :)
Don’t worry. I find formulae with semidirect products give me trouble. I remember them correctly (usually) but do not trust my memory. Try working out . The answer is very sensible, but worth doing. (Being told it is not the same.)
Sometimes I find an n-lab entry assumes too much too early, and perhaps a double level entry i.e. semi-direct products for ‘les nuls’, with a second entry giving more interpretations etc. for once the dummy stage has been passed!!! :-)
Surely this entry was originally written with the intention of defining the concept for groupoids, assuming that one already knew it for groups.
I have re-reverted the explanation of , added some structure, and briefly described internal and right semidirect products.
The notation unambiguously means conjugation. If you want to denote a left action, a proper notation is .
Actually, the notation is not infrequently used for a left action, especially in the context of semidirect products and crossed modules where the usual actions considered are closely related to conjugation (and become conjugation once you construct the semidirect product group).
a proper notation is…
it’s too late to rewrite my paper with Urs, then :). Also, better tell Ronnie Brown and his collaborators, from whom I picked up the notation.
By the way, I was reading semidirect product because I’m hoping to understand the relation between functors and forms, which means I need to know something about 2-groups.
Re Sdps and actions (Sdp = semi-direct product)
I get a bit worried when someone writes ‘unambiguously’. would parse / type incorrectly if it was only used as conjugation. , it should be countered also means multiplication! :-)
Some people write and then write . (I note that João Faria Martins does this in his papers especially when looking at crossed modules.) That is a good unambiguous notation especially when there may be two different actions of on being considered, but I am a bit lazy and do not use it! :-)
The idea of action as functor gives another good careful notation: F: BG\to GroupsF(h)(\gamma)$. That is very useful and, Eric, if you are looking at sdps to understand functors and forms, first rewrite sdps in the functor notation and then the multiplication. (Then look at the Grothendieck construction as Ehresmann thought of it i.e. as a generalisation of sdp. Each step is fairly easy but you can bootstrap your intuition nicely by going around this ‘loop’ rewriting things in another way.)
Another useful source is the Magnus Forrester-Barker’s nice note at http://www.maths.bangor.ac.uk/research/preprints/00/algtop00.html#00.29
Although you may not need that if you have gone through the connections between 2-groups etc, and crossed modules before. It is anyway a useful thing to have online. (Edit: fixed some typos.)
I have added to action a brief section on actions of a group object. I have not mentioned notation :-)
The idea of action as functor gives another good careful notation: would yield . That is very useful and, Eric, if you are looking at sdps to understand functors and forms, first rewrite sdps in the functor notation and then the multiplication. (Then look at the Grothendieck construction as Ehresmann thought of it i.e. as a generalisation of sdp. Each step is fairly easy but you can bootstrap your intuition nicely by going around this ‘loop’ rewriting things in another way.)
That sounds like a good homework assignment. Maybe a bit over my head though. I wouldn’t mind if someone spoiled the fun and showed me how to rewrite sdps in functor notation :)
By the way, the idea of relating functors and forms this way is so pretty. Is it really in Urs and Konrad’s 2008 paper it was first written down?
Have a look at the menagerie (http://ncatlab.org/nlab/files/menagerie10a.pdf) and its discussion of the Grothendieck construction (p. 274). That may help. That is really the same thing as your ’homework’. (I hope the train is not too noisy and you have a smooth ride when trying this!)
One thought that does not always occur to people. In the semidirect product the underlying set is the product, but you can also think of this as the disjoint union of a family of copies of indexed by the elements of . Now look at the ’usual’ way of building a coproduct of sets, i.e. look at that disjoint union and record as well as each element the index of the set in which you are ’seeing’ it. Sort of gives elements where , doh! That looks like the product if each is the same set!. You can think of this as a set of fibres over the set of indices. All the fibres are the same.
Now look at the Grothendieck construction. (I keep saying that don’t I!) There are links between the objects so you get links between the fibres over them. That is the action / functor coming in.
That looks like the product if each is the same set!
In other words: a product is a repeated sum; multiplication is repeated addition.
EGGSACTLY!!
John had a nice version of that somewhere. Anyone have a link. I remember an array of 1’s. Ronnie used it several times.
Still haven’t had a chance to read through this and am running out the door to catch a train to another day of mayhem, but had a thought last night while sleeping.
If we think of a group as a one-object category instead, then we could form a functor category . It would probably follow that so each functor would have an inverse and it has identities by definition and it is associative, so this would make the functors in a group. Or something…
Am I close? Gotta run!
No.
A functor from to is the same as a homomorphism from to (although, interestingly, two such functors can be naturally isomorphic even if the corresponding group homomorphisms are not equal). So as with homomorphisms, we have , but plenty of them are not invertible.
If you restrict to the invertible homomorphisms, then you get a group, but that’s nothing special, any category has automorphism groups.
If you restrict to the invertible homomorphisms / functors and the natural transformations between them you get a well known 2-group.
No.
Ok. But I think I at least deserve some points for recognizing that functors are the same as homomorphisms . That is kind of cute :)
Still in the vortex. Gotta run!
Happy spinning. :-)
Now think about the natural transformations between those homomorphisms. That is even cuter!
Lemma 6 For groupoids and , the functor category, , is a groupoid.
That is cute :)
So for each element of the group , i.e. morphism in , and functors , the commuting square coming from the natural transformation gives conjugation by , i.e.
PS: This now makes me think that whenever we see a conjugation, there is secretly some natural transformation hiding somewhere.
@ Eric
Actually, you should ignore my comment #23.
When you wrote , I assumed that you wanted to be an inverse functor; and all that.
But now I realise that you were defining some other group structure on . In fact, you want a group structure on .
If is commutative, then this works, defining products pointwise, and we do have (because ). But in general, the pointwise product of two group homomorphisms is not a homomorphism.
More generally, if is any set, then becomes a group by pointwise multiplication. But (when is a group) is a subset of but not in general a subgroup.
@Eric: Now even more cuter!!!! is a monoid (under composition) and a groupoid and the structures satisfy interchange. Looking just at the automorphisms and the nat trans between them gives you a 2-group. It thus gives you a crossed module. (Easy exercise in group theory to work out what it is.)
Thanks Tim and Toby.
This stuff is pretty cool. I’m basically trying to understand “kindergarten 1-transport” or “kindergarten 2-forms as 2-functors”. By “kindergarten”, I mean I’m not concerned about a continuum so do not need to worry about Lie 2-groups and all that. I just want to think of space as a finite diagram, i.e. a finite diagram perhaps representing a discretization or finitary model of some continuum (directed) space. Something like a mesh for computational fluid dynamics, electromagnetics, etc.
In this case, we can parallel transport abstract edges across abstract faces and our groups do not need to even be smooth. Any 2-groups would suffice I think. I’m still working on my intuition for semidirect products and crossed modules, but this is the direction I hope to go.
You may appreciate the approach in http://arxiv.org/abs/math-ph/0203056 - it treats AUT(G) 2-transport over a simplicial complex, but not in those terms. The ’group-like category’ in the article is equivalent to the 2-group arising from the crossed module .
Thanks David :)
I’m hoping to generalize this material on lattice Yang-Mills (from some mystery authors) to the case of higher transport so this paper definitely looks helpful.
One thing to keep an eye out for is the relation between BF theory and surface transport (John and/or Urs has some stuff on this). I don’t know of any other physics-like theory that comes up at this ’easy’ level of higher gauge theory - otherwise my PhD would have been on it :). One thing that is important is that curvature for 2-transport is valued in a 3-group (see if you can guess which other paper by one of those mystery authors I’m thinking of). And strict functoriality might have to be relaxed in the assignment of (2-)group elements to links/plaquettes, unlike in those early papers on surface transport, else things can become what is known as ’fake flat’.
Just for the record, can we write down the numbering conventions here?
Ordinary parallel transport. Is that 0-transport or 1-transport?
Curvature arising from -transport. Is that -curvature or -curvature?
I looked at Attal’s stuff when I was trying to interpret Turaev’s HQFTs in terms of crossed modules, gerbes etc. It looked nice, but I am not really happy with the forms / curvature stuff so did not go further. I think David and I chatted about this one evening at Ross Street’s birthday meeting in Sydney.
Another author who may be worth looking at for related stuff is Roger Picken. He keeps things fairly down to earth, although he is looking at the non-discrete case mostly. Joao Faria-Martins has useful stuff on this as well.
Ordinary parallel transport. Is that 0-transport or 1-transport?
for me this is 1-transport: transport along 1-dimensional objects.
Curvature arising from k-transport. Is that k-curvature or (k+1)-curvature?
-curvature: bundle gerbes have, for example, surface holonomy and 3-curvature.
Ok. Thanks. Just to make sure I can count, regular curvature is 2-curvature, right? (Seems a little unfortunate)
Well it is a 2-form…
and actually curvature of 1-transport with values in G has values in the 2-group INN(G) (nice, eh?)
Well it is a 2-form…
Ok ok. Good point :)
and actually curvature of 1-transport with values in G has values in the 2-group INN(G) (nice, eh?)
It might be if I knew what was :)
So for those mysterious authors, they turned into a group algebra. Can be turned into a group algebra? Or should it be something “higher”?
Regarding “discrete” stuff, I will probably try this with directed cubes. Has anyone done this before?
I’m slowly learning to love bigons (or trying anyway).
I often stare at this diagram
and wonder if the 2-arrow should go from one path to the parallel path or an initial edge to a final edge. I’m guessing this crowd would suggest the former, but I can’t give up the latter for some reason.
The arrows represent a “flow of time” to me and a bigon from one path to the other parallel path would not “flow”. I’d like my 2-morphisms to represent time as well if possible.
Rule of thumb: 2-arrows live between 1-arrows with the same source and target (and this is still true for n-arrows and n-1 arrows).
I read (not absorbed) most of Attal’s paper on the train ride home.
One thought I had, which probably applies to most similar papers, was that I think parallel transport should come with a “cost”. That cost can be interpreted as some notion of time.
So instead of a path groupoid, I’d try to work things out on a path category, i.e. a directed space. For example, in the square
we can think of time as being directed diagonally up and to the right. As an aid, I think of the morphisms as rays of light, i.e. light-like paths. With this interpretation in mind, it is difficult to reconcile with the usual notion of 2-morphism. Maybe there is a way.
So I would tend to think that the bottom light-like edge should be 2-transported upward toward the top light-like edge like:
Similarly, the left light-like edge should be 2-transported to the right light-like edge:
Anything else doesn’t seem causal.
Is there a way to reconcile this with the usual way to think of 2-morphisms?
It might be if I knew what was :)
Haven’t absorbed it yet and I’m headed to bed now, but a very very nice looking reference can be found here (by some mystery authors :))
I’m guessing this crowd would suggest the former, but I can’t give up the latter for some reason.
That is certainly the usual thing, as Harry’s rule of thumb suggests. That is what we have in a -category. It’s important there that the square is just a special case; you can take any polygon and divide its boundary into any two contiguous pieces (which are oriented oppositely) so that the composite of each piece is a -arrow from a common source to a common target. Then the -arrow goes from one composite to the other. In fact, it’s misleading to think about these polygons (including the original squares) at all; really -arrows in a -category always and only fill bigons. (You can break up each side of the bigon into a composite of several -arrows if you like, but that’s irrelevant to the -arrow.)
However, you can also work in a double category. There we have two classes of -arrows, one called (and usually drawn) ‘horizontal’ and the other ‘vertical’. Then instead of a bigon, you must use a square, and the -arrow goes between the horizontal -arrows in the square. (Again, you could break up each side of the square into a composite of several -arrows if you like, but that’s irrelevant to the -arrow.)
Second rule of thumb:
Let bigons be bigons.
Meanwhile, apparently there are a lot of complete idiots on the internet.
Harry 4,6 and 12: The definition should not be circular, right ? The claim in 12 is such, impatient.
One starts with a left action of on . So at the point of definition there is no conjugation action whatsoever used, nor it makes really sense for elements in two different abstract groups. Then is the result of the left action of on . This helps defining the product in the semidirect product. In this new product, new copies of and sit, and the left action will be, after the identifications equal to the conjugation action. The same idea is for the smash product of Hopf algebras, see the entry crossed product algebra.
@Zoran: Regarding #12, I was merely stating that most algebraists use the left exponent to denote conjugation.
One thought I had, which probably applies to most similar papers, was that I think parallel transport should come with a “cost”. That cost can be interpreted as some notion of time.
and this makes me think of categories enriched in the rig , if you are interested in having ’costs’ associated to paths (this has to be done in a certain way, not just any old enriched category will do here). Or perhaps your category wants to live over a monoid (say ) interpreted as a category with one object, so that paths are assigned costs, and concatenating paths adds the costs.
and this makes me think of categories enriched in the rig
Thanks David. This comes back to my old idea about “extruding” groupoids. You and Mike helped me out a lot on this subject before. I need to collect all that wisdom in one place so I can remember everything.
I thought of that when I read Attal’s paper and he discusses how he keeps “thin homotopies” as distinct paths. For example is not the same path as . When I read that, I said, “Of course they are not the same path. One is three (time) steps and the other is one (time) step.”
If you extrude everything, it is pretty obvious (although extruding isn’t really necessary, I like to use it as a conceptual crutch).
apparently there are a lot of complete idiots on the internet.
Who knew?
it’s misleading to think about these polygons (including the original squares) at all; really 2-arrows in a 2-category always and only fill bigons.
I think that’s a matter of opinion, or maybe a choice of presentation for a theory. When you say that, you’re probably thinking of a 2-category as structure on an underlying 2-quiver (= 2-globular set). But 2Cat is also monadic over the category of 2-computads, so the “underlying data” of a 2-category can just as well be thought of as consisting of 2-cells from strings of n arrows to strings of m arrows, for all nonnegative integers n and m.
Double categories are funny because you have these 2-cells that sit in a square, but it’s entirely your choice about what to call their “source” and “target”. You can say that they go from one horizontal arrow to the other, with the two vertical arrows being the “frame” or something “along which” they go, or vice versa, or you can say that they go from the “formal composite” of a horizontal and vertical arrow to the formal composite of a vertical and horizontal arrow. I think one way of thinking makes more sense for some double categories, but the other makes more sense for others. For instance, if horizontal arrows are bimodules and vertical arrows are ring homomorphisms, I really want to think of a 2-cell as a map of bimodules that happens to be “equivariant” along a pair of ring homomorphisms. But if horizontal arrows are lax monoidal functors and vertical ones are colax monoidal functors, I think it makes much more sense to think of the source of a 2-cell as a formal composite of monoidal functors of opposite laxity.
However, you can also work in a double category.
I wonder if there is a way to interpret a 2-form as a smooth functor to a Lie double groupoid instead of a Lie 2-group?
@Mike: you say ’Double categories are funny because you have these 2-cells that sit in a square, but it’s entirely your choice about what to call their “source” and “target”.’
Surely the point of double categories is that they do not have a source and target as such. They do have a direction 1 source and a direction 2 source etc for each square. (Horizontal and vertical do not generalise to higher triple categories etc.)
Horizontal and vertical do not generalise to higher triple categories etc.
Of course not, but that doesn’t give me any qualms about using those words in the case of double categories. For triple categories I like to speak about “horizontal,” “vertical,” and “depthwise.”
They do have a direction 1 source and a direction 2 source etc for each square.
That’s one way to think about it, which is perhaps closest to the formal definition, if you define double categories to be internal categories in Cat. But I think the intuitions I described are also valid ways to think about it (and not really very different, either).
@ Mike I hope I was not implying that I though your way was ‘invalid’ in some way, no I was trying to say that often with double categories it is useful to keep the two directions very separate for some way in their study as they may have different geometric meaning / content. Then taking the double nerve of the double category and working with that as a bisimplicial set, before applying diag or the Artin-Mazur codiag to get a usable simplicial set leads to something that measures the changes in both directions. I did some work some years ago on ordered groupoids which correspond to inverse semigroups and the two directions are very different in feeling, one is a partial order, the other consists of ’local’ transformations / partial bijections.
often with double categories it is useful to keep the two directions very separate for some way in their study as they may have different geometric meaning / content
I definitely agree with that. I think there are lots of different “flavors” of double categories which often come with different intuitions.
A beautiful beautiful description of semidirect product is given on Page 20 in:
Yes. That is a good description. It essentially dates back to Ehresmann and Grothendieck, but not so neatly expressed. Have you also glanced and Forrester-Barker’s description of the process of going between crossed modules and 2-groups (due to Ronnie in his early work with Spencer)?
Have you also glanced and Forrester-Barker’s description of the process of going between crossed modules and 2-groups (due to Ronnie in his early work with Spencer)?
Not yet, but I’ll put it on the list of reading material. I still haven’t made the conceptual leap to understanding 2-categories yet, but this paper helps a lot.
My problem is that I learned from watching John lecture in both meatspace and cyberspace that 2-morphisms are maps between 1-morphisms, but I never bothered looking too carefully at the drawings, so all this time, I’ve thought a 2-morphism should go between 1-morphisms regardless of whether the 1-morphisms share the same source and target.
The picture in my head is like a directed graph. Directed 1-cells go between directed 0-cells. Directed 2-cells go between directed 1-cells, but they are not constrained to go between directed 1-cells sharing the same source and target. Therefore, is a directed graph in the traditional sense. However, I also expect to also be a quite general directed graph where are thought of as vertices. The way 2-categories are usually defined in terms of bigons form a particular boring kind of directed 2-graph.
I’m probably just thinking about things the wrong way, but this is a mental stumbling block for me.
I suspect that my way of thinking about things is essentially equivalent to the standard way because my 2-morphisms would probably have to come with a pair of component 1-morphisms (similar to a natural transformation) that turn the “square” into a bigon anyway.
For example, if we had two 1-morphisms that do not share source and target such as
I’d still want a 2-map (using different notation to avoid a clash), but it might need to come with component 1-morphisms and so it would look like
Then we could probably think of as a typical 2-morphism
Then my 2-map would correspond to standard 2-morphisms when the components are identities, but would otherwise be more natural in my opinion (understanding that “natural” is a personal opinion not necessarily shared by others, but that is OK).
Ok. Since I am thinking out loud…
Now I am tempted to introduce a 1-morphism
for any 2-map such that
Note: This is intentionally designed to look like a boundary.
For horizontal composition, this leads to a bunch of interesting looking relations among , , and that I’m still working out. For example, with and for and 2-maps and we have
Edit: I’m also thinking there should be some kind of “gluing” 2-morphism to glue the two parallel components together such that
One thought is that the 2-category model of spaces using bigons is derived from situations where you are mapping a square into the space but two opposite sides of the square are mapped to points, so a homotopy rel endpoints. The picture you seem to like is the free homotopy. Unfortunately they are more difficult to handle in that spatial situation. You are using a double category rather than a 2-category. These can be useful as you can constrain the sides to have some special property, instead of merely being constant.
There seems to be a similarity between what you are thinking of an certain diagrams that Larry Breen uses when describing non-Abelian cohomology classes.
Interesting. Thanks Tim. That was a lot more encouraging than the “He’s nuts!” I was expecting :)
By the way, in the case of a 2-groupoid where all component 1-morphisms are identities, i.e. standard 2-groupoids as usually considered, my boundary constraint for horizontal composition becomes
Does that look familiar? I like this idea of taking the boundary of a 2-morphism resulting in a 1-morphism that makes the diagram commute.
Edit: Note also that when dealing with typical 2-groupoids the boundary of the 2-morphism is even nicer, i.e.
Reminiscent of Stokes theorem, eh? :)
I didn’t realize it at the time I wrote it, but
looks like a semidirect product :)
In other words,
Edit: A little time later (not wanting to flood with even more posts)
Let and , then let denote the fact that , then we can define a product
The term on the right-hand side refers to horizontal composition with
such that
from which it follows that
1 to 60 of 60