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In some situations I have seen reference to coextensions as being the epimorphisms. Perhaps this needs a mention if it is a current usage (most of the sources I have seen are moderately old OR on Hopf algebras!)
Well I do not remember those things quite well. But it is a point of view on extensions in which one takes a subalgebra and looks at its extension and there is a Hopf algebra of symmetry which helps extension. It is usually sort of replacing a Galois group. Regarding that subobject is an algebra and the other datum is different, i.e. Hopf, the dualization makes sense. For usual extension saying A extends A” or A” extends A is matter of convention, but with Hopf tools it may be a different algebraic recipe, so one talks coextensions. Maybe not the best choice of words: it is rather etxension in coGalois sense I think, but should check.
My only point was that there is a use of the term coextension. (I found an old reference Jonathan Leech, H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975.
and more recently in a Hopf context: www.emis.de/journals/HOA/IJMMS/Volume2003_69/4345.pdf
Charlie Wells wrote a paper on extensions of categories and mentioned the use of the term. A coextension of ’thing’ is just an epimorphism with ’thing’ as its codomain
No! This is backwards! It becomes substantially clearer in the following case:
Let R be a commutative ring, and let M be an R-module. Then an extension of R by M is a map of commutative rings $S\to R$ and a short exact sequence in the category of $\mathbf{Z}$-modules (U is the forgetful functor and $res$ is the restriction of scalars to $\mathbf{Z}):
$0\to res(M)\to U(S)\to U(R)\to 0$
(This is true over an arbitrary base ring, but I didn’t feel like appropriately relativizing everything).
For groups, the same idea applies. Given a group G and another group H, we want to find a new group for which H is a normal subgroup that projects back down to G when we kill H.
I think that the case of ring and algebra extensions is the most compelling, since their kernels are never subalgebras.
@Harry. As Zoran says both conventions are used extensively in the literature. What you say is one of the two, and Leech uses coextension for that. A field extension is a LARGER field. That is almost certainly one of the origins of the convention that an extension is a larger group (or whatever). Your point is in fact very good but for a different reason than you might want! The sort of situation you are considering is really derived from algebraic geometry where one extends Spec(R) to Spec(S) using a nilpotent ’fuzz’ given by M.
I am not saying that either convention is somehow right. Wells mentions that Leech uses coextension but that he will use extension for the same situation. The terminology is largely determined by the context, which is the more fundamental object the lefthand one or the right. Even in group theory there are situations where one wants to add in additional elements, and of course, there is the question of normality to look at.
On a slightly different tack are HNN-extensions which add in elements so as to realise an automorphism between two subgroups of the base group.
(The Wikipedia article is moderately good on these.)
I am sort of hoping that discussing this a bit we can come to some remark on the page that gives examples of usage, warns that the terminology can be different in different sources in the literature (so always check which is being used), and to mention the term ’coextension’ as well.
@Tim: By the way, since this is exactly on point, do you know any way to explain how 1-extensions $0\to A\to B\to C\to 0$ are related to loops around the basepoint (taken to be the image of the zero map) in the DK-function complexes (gotten, say, by means of hammock localization) that we assign to the pair of simplicial modules $L_H(sR-Mod,W)(C,A)$?
That is, I’m trying to think about a direct relationship between the classical Yoneda categories $Yext^n(C,A)$ and $n$-loops around the basepoint.
What would be pretty spectacular is if we could show something like “every n-exact sequence arises from some sort of loop in the function complex”, then show something vice-versa, then show that up to homotopy, everything actually works.
It’s just really interesting to think of the very suggestive picture you get if you attach both ends of the exact sequence together, you get a high-dimensional loop-looking thing.
Some instant reactions i.e. off the cuff!!!!
There is a paper by Friedlander in which he gives a neat description of a classical fibration (so in topology) in terms of the homotopy coherent crossed module structure (that means that you look at the original proof that the inclusion of a fibre into a fibration yields a crossed module on $\pi_1$s and you extract the essence! I don’t know that it is absolutely directly relevant but have a look (and keep the viewpoint classical until you see what is happening). The proof is essentially : if $F\to E\to B$ is a pointed fibration sequence, we need a loop in $E$ to act on one in $F$, you conjugate, then map down to $B$ to get a null homotopy that you lift to $E$ (using ’fibration’) and the other end is the desired ’action’. Very neat. Of course you need to check that things are independent of choices made up to homotopy coherence etc. )Now look in the hammocks!)
Second point is that Huebschmann showed a result on n-fold crossed extensions and cohomology classes. This might give a further indication of what you suggest (which I would think likely).
Of course, the Deligne idea works very neatly at the simplicial group level and if on looking at it in more detail you want to hear more of that just holler!!!
@Tim: Some other information that might be helpful regarding this question: Someone (I forget who at the moment) proved that pi_i of the nerve of Yoneda’s ext^n category is exactly Ext^{n-i} for 0<=i<=n and 0 for i>n.
I also read somewhere that the function complex between two simplicial abelian groups is naturally a spectrum. How does that work?
OOPS I am allergic to spectra…. there were too many students of Frank Adams around when I was a PG!!! They always seemed to (i) know such a lot that I did not and (ii) not be interested in what I was interested in. :-) Result: I only really know the area superficialy. Even so, perhaps something (almost certainly false) along the following lines may help. Replace the simplicial abelian groups by chain complexes then prove that the original function complex is weakly equiv to the inverse Dold-Kan of the chain complex function complex. (This goes wrong I know but seeing exactly how it goes wrong should help. I feel that the negative dimensional stuff is the problem and the solution somehow. (as I said this is off the top of my head!))
For the first part that looks something related to a Duskin result. Back in his Memoir AMS or shortly afterwards, but I may be wrong. (You need to note these things down somewhere … such as here! :-))
Tim, the definition with epimorphisms is extremely special. The true meaning for algebraic context is in categories with some notion of exact sequences, I mean the third member by which one extends is part of the data. Of course, there is also the usual notion of extension of maps and extension problem in algebraic topology which does not ask for the )co)kernel or Galois group. It is more complicated I see now to write this entry properly.
@Tim: No, it was this paper, but the trouble is: The authors of that paper were not familiar with Dwyer and Kan’s work on simplicial localization, so everything is done in some kind of crazy notion of loop for chain complexes.
I mean we should distinguish normal extensions and extensions for example, already in algebraic cases.
I have extended the entry extension to have both the extensions of maps and the extensions of objects.
@Zoran I think that extension is an even more general idea than short exact sequence, for instance, Baues and Wirshing talk about linear extensions of categories, and the ’kernel’ term is not even in the same category as the others. Their idea is almost purely that of an epimorphism to the known term with information about a kernel type object. (It is already in the paper/preprint by Wells that I mentioned earlier.) When you get to extensions of groups, for instance, then you have a short exact sequence of groups (and really nothing more to start with), then you identify a thing extended by another thing. and that is probably best seen as a refined concept as it identifies the ends, (but which is the ‘by’ applied to is then open for debate and I would suggest that we mention there are two conventions and we have chosen one with ’coextension’ as the dual convention perhaps). My feeling is that we will end up with a larger entry than you originally envisaged, but that is to the good as it indicates there is some need for that length, as we have been seeing.
@Harry. Thanks for the paper. There is a short cut that may not be obvious. Alan Robinson’s way of doing things was shown to be a homotopy coherent Kan extension by Cordier and someone whose name I sometimes forget…. in the paper on homotopy coherent homotopy theory in the TAMS. I have not checked through Amnon’s use of that idea but this is just a total derived functor analog in this setting. That makes it almost certain that what you suggested is true.
There should be some sort of splitting, or cleavage involved, where the sitting of the first part which is extended is somehow controlled either by symmetry or by the presence of complement. This is common to all the examples above.
I wonder if that is always the case. If one has an extension of algebraic groups, or of sheaves of groups in a topos, there will usually be a local splitting but not necessarily a global one. You seem to be thinking of Schreier theory as the model. (I have seen references to Schreier extensions but I am not sure that that is what is being referred to.) There is a Schreier theory for 2-groups and that does not forget back to sets and so I doubt it has splittings as you suggest. In any case it is not really something that is invariably there, rather in the cases where it is the theory is more amenable!
How are extensions mentioned in semi-abelian theory, as I do not have enough of the main sources with me.
In any case I think you mean ’section’ not ’splitting’ as a split extension is always a semidirect product’
I mean intutively splitting and cleavage as suggestion that a piece sits in the whole in rigid way, separated from the rest which can be somehow nicely described by a symmetry or alike. I do not mean the splitting in the very special sense like in group theory. Surely not always one can reduce such generalized splittings of object into subobject and other data just to more trivial splitting in some simpler category downstairs plus some algebraic data (“cocycle”) but it is often the case. The fact that it is not always cocycle is intersting from the point of view of generalizing nonabelian homological algebra.
I misunderstood what you meant by ’splitting’, but I’m not sure I really understand what you do mean. (Recall split epi is a recognised term in category theory so it is not just in group theory that the terminology used.)
By splitting again I mean that we have two data comprising the whole extension. The thing which we extend and some other thing which is informationally complementary. Base and fiber for example. Subalgebra and Galois group. Normal subgroup and quotient. Algebra of coinvariants and Hopf algebra making the extension Hopf-Galois. Now the two data are nontrivially put together, so there is additional freedom to classify, which may or may not be of the nature of section of one of the maps in the game, taken in some underlying category (after forgetting). The latter situation is not general. One may have cohomological classification when the sections do nnot make sense. Maybe decomposition is here better than word splitting, as splitting is often here of technical meaning, like in group theory, and semiabelian setup in general.
@Harry. Thanks for the paper. There is a short cut that may not be obvious. Alan Robinson’s way of doing things was shown to be a homotopy coherent Kan extension by Cordier and someone whose name I sometimes forget…. in the paper on homotopy coherent homotopy theory in the TAMS. I have not checked through Amnon’s use of that idea but this is just a total derived functor analog in this setting. That makes it almost certain that what you suggested is true.
@Tim: If you wouldn’t mind explaining a little closer to the beginning, I’d really appreciate it.
Alan Robinson (then at Warwick) had a student (Patricia Gardener, but her thesis was not published) who worked on modelling derived functors using methods ultimately derived from Segal. Cordier and myself wrote a paper about coherent ends (yes them again!!!!) and on p. 36 or there abouts of the paper we compared Gardener’s version of Robinson’s construction with our left coherent Kan extension. (The best thing is to look it up in the paper as the background for it is quite complex (from our point of view) and has been simplified since I think. (probably by Mike in math/0610194.)
The H.c. left Kan extension is given by a h.c. coend formula and so is a generalised (enriched/weighted) homotopy colimit (don’t worry about that look at the formula take the bits and think, glue them together correctly… I don’t think you need the mechanism just the concept which you can handle in your own way.) Gardener’s method aimed to generalise the André-Barr-Beck generalised derived functors. This used ‘models’ (forming a subcategory $\mathbf{A}$ on which a functor $F: \mathbf{A}\to \mathbf{pointed discrete spaces}$ was defined and then constructed a nerve of an overcategory $\tau(B,F)$. The derived functors came out as the homotopy groups of the result. The $\pi_0$ was just the left Kan extension of $F$ along the inclusion of $\mathbf{A}$ into the bigger category $\mathbf{B}$. (That was fine and dandy but did not do much.)
We replaced $F$ by the constant simplicial set valued functor $K(F,0)$ and showed (our proposition 6.4) that the nerve of her category was exactly our coherent left Kan extension.
I have not examined the paper that you sent me in detail (and don’t think we need to) but I think they are using a similar idea and that their methods can now be bipassed because the n-Lab perspective gives more powerful tools.
I must stop here for the moment. I have a copy of Cordier-Porter and can send it to you but it was scanned and you can probably get a better looking one at your university.
In the disambiguation blurb it used to say:
This entry is about extension of morphisms, dual to lift. For extensions in the sense of “added structure”, see at group extension, Lie algebra extension, infinitesimal extension etc..
We should have a dedicated page to this other sense of extension, so I changed this to
This entry is about extension of morphisms, dual to lift. For extensions in the sense of “added structure”, such as group extension, Lie algebra extension, infinitesimal extension etc., see at algebra extension.
and will now go and create some minimum at algebra extension.
Re #23, but this page does contain a section on extension in the second sense, so the blurb isn’t right.
I quite like the idea of having simple descriptions of both on the same page though.
I don’t expect anyone is waiting to be convinced. The content in question seems all rather trivial and tautological to me. You could go ahead and clean up the entry.
David 24, Urs 25
I am extremely busy till Monday, but soon I will follow the advice in 25 and make new changes to extension (the part which refers to extension of objects, not morphisms) and algebra extension and related items. I am now putting a notice as somebody might want to correct my point of view below which will reflect my edit. Namely the thing is a bit tricky. Let me explain.
Long ago I contributed much to Lie algebra extension and group extension, the cases where the point of view, now taken in this entry, that extension of objects is considered when we have some sort of notion of short exact sequence in that category (I also raised this point in the discussion above in 2010, while claiming that this holds in some more primitive underlying category, after forgetting some structure). Then a version of this POV s taken at algebra extension where one talks of extension as epimorphisms with remark that the kernel is usually in the same category. Now the prime example, epimorphism of unital associative algebras has kernel which is ideal, hence not a subalgebra in the sense of unital subalgebras; moreover for fields (see stub field extension), it is __mono__morphism rather than epi and the cokernel is certainly not a field. Also often one has a forgetful functor from algebras of some type to a category which has less structure and one has something like short exact sequence there, and not in the original category.
Finally there is some historical connection between extension of morphisms and extensions of objects which the disambiguation entry treats as totally distinct categories of notions. Namely, one has extension (monomorphism) of spaces and one wants to extend a function or mapping on a space to the extension what gave rise to an extension of mappings; later this was semantically generalized and the problem of extension of mapping was considered not only in the original (still most important) case of extending along monomorphism.
How to make the entries compatible with all this usual usage and make it theoretically sound I do not know, I will try to think through next week and make appropriate changes. If somebody has ideas extending my remarks or corrections as well, you are very welcome to respond to this announcement in the meantime. I will also carefully reread our old discussion above in the thread to try to be as inclusive as possible (now it is easier after all contributions and grown related entries to do this).
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