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This entry is lacking a good textbook reference. It used to list, on the one extreme, an expository webpage, and on the other end a bunch of references which are citeable but way too over-specialized for any reader who just needs to look up what a torsor is.
I have now added pointer to
which comes closer, but still buries the simple idea under faithfully flat verbiage.
Do we maybe have a discrete group theory textbook that states the definition in a way useful for those readers who don’t already know it?
I don’t know about a textbook reference, but this paper by Breen cites the definition on page 2 here, and it doesn’t look too over-specialized for a quick look-up.
Okay, I have added a line saying “see also at principal bundle”.
But what I have in mind here is discussion of less than that: torsors over the point, preferably in discrete sets. Such as the default meaning when people speak of $\mathbb{Z}$-torsors. Such as in John Baez’s expository note but inside a citable textbook. Probably John wrote that note because such textbooks are not common.
Do such textbooks exist? I’ve just checked Artin, Lang, Dummit–Foote, Aluffi, and none of them mention torsors.
Thanks for checking! Maybe not.
Our Symmetry book mentions torsors (see current Sec. 4.6+4.7), and is meant to be an undergraduate textbook. But of course, the book is still far from finished, and we take as the main definition of a torsor that it’s any G-set merely equal to the principal G-torsor. (I don’t remember if we prove the equivalence with the standard definition yet, but it’ll be there in time—Ch. 4 is not very polished yet.)
Thanks, that sounds promising! But I don’t see a book behind that link. Do you have a pointer to a pdf for me?
Here you go — but as I said, it’s still quite preliminary.
In a model theoretic context of definable sets, principal homogeneous spaces are studied in
I found this page previously difficult to navigate; too verbose and unstructured, which made it difficult to quickly find what I wished to find. I have now heavily re-structured it. I have cut down the idea section and moved most of what was previously there either to remarks or to the beginning of the ’In general’ section. Apart from this and moving some material specific to sets, I have not touched the latter, but I have created an ’In sets’ section before it, and heavily rewritten this material, gathering things that had previously been strewn out in various places on the page.
I have also added a new section ’Functoriality (change of structure group)’ to explain how to transport a torsor along a group homomorphism. We should give a proof of the proposition I state there, but I have run out of time for the moment.
This proof was seriously fiddly, but I think it is right now! It would be very interesting if there were some nice construction of this ’push-forward’ of a torsor along a homomorphism as a colimit or something; I don’t mean just rewriting the proof/constructions diagrammatically, but actually exhibiting the actions of $G_1$ and $G_2$ that are involved by means of some canonical construction. The construction reminds me somewhat of the notion of a semidirect product, in case that provides any hints.
If anyone has a reference for the proof, or even the construction, this would be good to add too; I found a sketch of the construction in some notes (with a typo at one of the crucial points), but there was no proof that we actually obtain a torsor. As one can see, the proof is notationally tricky if nothing else, and there are a few places where one has to be careful.
Hi Richard,
looking at your latest edits here now, prompted by some curious sequence of events here.
I like how you removed the assumption that a torsor be inhabited! I had a mid-size crisis today with an apparent paradox in my 30 page proof I am working on, until I realized that this is the natural way to go about it. As usual with these trivial-seeming details, they are boring and irrelevant only as long as one does not internalize/enrich in fancy ways.
Also, I see now that we overlapped in feeling the need to write out some change-of-group adjunctions: I have added a pointer from your “functoriality”-proof section to my latest addition at: topological G-space – Change of equivariance groups.
(Of course, the change-of-grop adjoint triple is neither special to torsors, nor to topological G-actions, and we should really have this discussed on a more general page – maybe we need a page group action, to be splitt off from action.)
Perhaps we should mention that a natural way to define torsors is as algebras over an appropriate algebraic theory, namely, we have a ternary operation t satisfying t(x,x,y)=y=t(y,x,x) and t(v,w,t(x,y,z))=t(t(v,w,x),y,z). (Think of t(x,y,z) = “x y^{-1} z”.)
Then it is crystal clear that the empty set must be a torsor. And the category of torsors is a Malcev category, another important property.
Please do add that to the entry, if you have the energy!
I found out that we already have it explained here: https://ncatlab.org/nlab/show/heap#heaps_and_torsors.
Do we need a separate article “heap” describing essentially the same thing as a torsor? Should the two articles be merged?
If you have energy for a major reorganization… But if you could just add prominent cross-links and import whatever is worth importing, it would already be an improvement.
Well, heap is not quite the same thing as torsor in general, at least it is different point of view, tradition and context. The usual term in universal algebra is associative Mal’cev structure.
For example, torsor is relative to a fixed group, and then you can think of a relative case (torsor for a sheaf of groups) etc. while heap is an algebraic structure.
Next, when we talk about category of torsors you usually mean a category for a given group $G$, while the category of heaps is category where morphisms may change the group in the sense that the morphism corresponds to a pair of an equivariant morphism of groups and a compatible map on the level of torsors.
Re #21: But this is like saying that we should have a separate article for a module over a fixed ring R, and a separate article for a module over an arbitrary ring, i.e., the category whose objects are pairs (R,M), R∈Ring, M∈Mod_R, and morphisms (R,M)→(R’,M’) are pairs (f:R→R’,M→f*M’).
Indeed, the situation for heaps and torsors is entirely analogous: a torsor is over a fixed group G and a heap is over an arbitrary group.
The article heap even says so explicitly in Section 4:
In fact, the category Heap is equivalent to the following category Tors: its objects are pairs (G,H) consisting of a group G and a G-torsor H, and its morphisms are pairs (ϕ,f):(G,H)→(G’,H’) consisting of a group homomorphism ϕ:G→G’ and a ϕ-equivariant map f:H→H’.
By the way, this statement is not quite correct: if a G-torsor is allowed to be empty, then we have many nonisomorphic empty torsors, one for each isomorphism class of groups. On the other hand, if a G-torsor is required to be nonempty, than the empty heap does not correspond to such a pair (G,H).
We have separate articles for presheaf and a functor, a separate article for action and for a module (and even representation) and a separate entry for principal bundle and for torsor. Edit: and groupoid versus Brandt groupoid (original algebraic equivalent of connected groupoid as partial binary structure). (This was all consciously made separate).
The point of view, generality and tradition is different. There are specifics of universal algebra for heaps while torsors are a more categorical and geometric topic. I think that mixing all the details from algebraic tradition in already complicated entry torsor will not do any good, nor will help people who want to consult only algebra.
I do not think that it is the same as your example with modules and rings. Module has its ring as part of the data, while for heap the group is determined only up to automorphism, or alternatively – if you take the underlying set – then up to a choice of unit. In (dual) noncommutative case is even worse as there are left and right automorphisms, and Grunspan has some example where this is essentially different.
while for heap the group is determined only up to automorphism, or alternatively – if you take the underlying set – then up to a choice of unit.
Up to a unique automorphism. Given a heap H, the corresponding group has as its underlying set all maps H→H of the form h↦t(h,a,b) for some a,b∈H. The identity map H→H is the identity element in the group. This construction is completely canonical, there are no choices involved.
In (dual) noncommutative case is even worse as there are left and right automorphisms, and Grunspan has some example where this is essentially different.
This is entirely analogous to how there are left and right modules over a noncommutative ring, which further reinforces the analogy.
or alternatively – if you take the underlying set – then up to a choice of unit.”The identity map…”
Yes, no choices for identity if you reconstruct it as an automorphism, but alternatively if you take a group and make the ternary operation then the forgetful functor forgets only the unit element. So if you reconstruct the group as the set itself then you need a choice of a (unit) element to make it into the group canonically. For torsor the group is given by the definition, while here you have to choose a point. This is the point of view of say, Bergman’s book cited under heap. Similarly in nc case, you have to add a character to get from a quantum heap to a Hopf algebra.
So if you reconstruct the group as the set itself then you need a choice of a (unit) element to make it into the group canonically.
The precise meaning of this statement is unclear to me, but under the most obvious interpretation, this does not seem to be the case.
The functor F from heaps to torsors sends a heap H to the torsor (LTrans(H),H), where LTrans(H) is a group whose elements are maps of sets f: H→H of the form h↦t(a,b,h) for some a,b∈H. The group LTrans(H) acts on the set H from the left: (f,h)↦f(h), turning H into an LTrans(H)-torsor.
The functor G from torsors to heaps sends a torsor (G,H) to the heap H (with the obvious ternary operation).
The composition GF is the identity functor.
The composition FG is the functor from torsors to torsors that sends a torsor (G,H) to the torsor (LTrans(H),H).
The identity functor is naturally isomorphic to FG via the isomorphism (G,H)→(LTrans(H),H) that is given by the identity map on H and the isomorphism of groups G→LTrans(H) that sends an element g to the map of sets H→H given by h↦gh.
Thus, the categories of heaps and torsors are equivalent. There are no noncanonical choices of unit elements needed here.
It’s really not important whether the articles are separate or not, there is never an objective way to make such design decisions. Important would be, instead, that each article makes it crystal clear that there are different notions, and how, and to point to the other article – and any other related article – as need be.
Maybe a bit of philosophy can help; to paraphrase Quine: “No entity without (a notion of) identity”. That is, a (mathematical) notion comes with a notion of identity/isomorphism/equivalence. So two notions are the same if they present the same homotopy type. I think we agree on that.
The point is that the type of $G$-torsors does not embed into the type of heaps, which as noted above is equivalent to the total type of torsors, $Tors = \sum_{G:Group}G\text{-}Tors$. Just like the type of complex vector spaces doesn’t embed into the total type of modules, $Mod = \sum_{R:Ring}R\text{-}Mod$, because we pick up isomorphisms from the base in either case.
Now, groups are (equivalently) pointed, connected groupoids, while heaps are bipointed, connected groupoids. The underlying set of a heap is the set of isomorphisms from the first to the second point. There are two canonical maps from heaps to groups, the left and right automorphism group, and these are only merely isomorphic (unless they are abelian). So when we think of a heap as a group $G$ together with a $G$-torsor, there’s a bias built in: we have two $G$-torsors and one is deemed “untwisted” and the other “twisted”, but we could have picked the opposite convention.
Now, as also mentioned above, the type of heaps embeds into the algebraic type of associative Mal’cev structures. The empty such arises as the isomorphisms between disconnected objects in a groupoid.
I hope somebody finds the time to work this good material into the entry!
While that is underway, I see that long ago on MO here people suggested that “pseudo torsor” should be used for not-necessarily-inhabited torsors, to avoid confusion. The StacksProject agrees (here) and apparently this terminology goes back to EGA IV 16.5.15, though I have not checked.
But, yeah, some such terminological distinction would be good to retain, for clarity.
Added mentioning that non-inhabited torsors are also called pseudo-torsors.
Hi Urs, thanks for taking a look! Regarding the empty torsor, I probably should have drawn explicit notice to this, apologies! I made the change partly because it is obviously makes sense and is more elegant/can be formulated in a simpler logic without the inhabitedness condition, leading to a nicer category; and also because in recovering a group making a choice of point is important, and any choice is as good as any other, so requiring inhabitedness is sort of misleading from one perspective, especially if one has a constructive mindset, because it draws attention to one point over the others.
Yeah, but there is a conflict with convention. I do think now one should say “pseudo torsor” for the possibly empty version. But all is good, I have added the pointer to the entry.
Yes absolutely, it looks good.
Re #30:
Now, groups are (equivalently) pointed, connected groupoids, while heaps are bipointed, connected groupoids. The underlying set of a heap is the set of isomorphisms from the first to the second point. There are two canonical maps from heaps to groups, the left and right automorphism group, and these are only merely isomorphic (unless they are abelian). So when we think of a heap as a group G together with a G-torsor, there’s a bias built in: we have two G-torsors and one is deemed “untwisted” and the other “twisted”, but we could have picked the opposite convention.
Yes, but this is entirely similar to the situation with rings and modules: every abelian group A has a canonical structure of a left End(A)-module and a right End(A)^op-module. The left action is privileged here because endomorphisms are composed from right to left.
Likewise, every heap H has a canonical action of LTrans(H) on the left and RTrans(H)^op on the right. Again, the left action is privileged here because endomorphisms are composed from right to left.
But I agree with the mathematical content.
27
The precise meaning of this statement is unclear to me, but under the most obvious interpretation, this does not seem to be the case
I was not claiming that there is no equivalence of categories at all, but taking the point of view of Bergman’s book and most of the heap tradition, that a heap is canonically obtained by a forgetful functor from groups, while the group for a torsor is tipically external, set theoretically not sharing the underlying set with the heap. As you know, to get the group back with the same underlying set you have to specify the unit, it is like choosing a section of a principal bundle. The corollary is that every identity for the ternary operation of the heap follows from the 3 relations in the definition. It is less obvious to see that from the perspective of reconstructing the automorphisms as a functor to groups (which never reconstructs identically the identical set).
To see that it is less obvious to see there are no additional relations (less obvious for those who did not see it yet), let me tell you a personal story for the quantum case. In the quantum case, I observed the analogue of the theorem Hopf algebras are equivalent to copointed quantum heaps, and some related things, in 2000. Grunspan put his definition of “quantum torsor” (with a definition relative over a base, but the rest was over the field like in my case) a year or two later with an additional relation in the definition. He was working with an automorphism point of view and wrote 4 axioms in quantum case and proved reconstruction of a Hopf algebra of automorphisms, left or right. For his definition, it took Peter Schauenburg to write a separate paper to show that the relation 4 is not needed. From the point of view of choosing a character in my picture it is obvious and I never had an idea of a need for the strange 4th relation of Grunspan. Though this was a research of one evening (prompted by a lecture of an algebraic geometer in 1999 on moduli of bundles, mentioning some problem with automorphisms) I was a little bit disappointed that neither of them cited my result although I had correspondence with both immediately when their papers appeared and my thesis had been defended earlier. T. Booker and R. Street had a paper later on the wider context
In my memory, in the principal bundle community, Anders Kock spent much time on various axiomatics of division/translation operation for principal bundles in synthetic approach, working with $a^{-1}b$ notation and $a\backslash b$ notation, but still the fact that if one focuses on the ternary relation instead, only 3 relations are enough is not discussed I think.
Example. Given two isomorphic objects $X$ and $Y$ in any category $C$, all isomorphisms between $X$ and $Y$ form a torsor (both for $Aut(X)$ and for $Aut(Y)$, which are mutually isomorphic but not canonically). This is an insight used in (M. Kontsevich, Operads and motives in deformation quantization, Lett.Math.Phys.48:35-72 (1999) arXiv:math/9904055 doi) explaining period matrices from the point of view of a coordinate ring of an affine torsor.
Added the following new subsection:
It is possible to define torsors using a single-sorted algebraic theory. This is entirely analogous to how affine spaces can be defined either as sets with a free and transitive action of a vector space, or, equivalently, as sets equipped with operations that take arbitrary affine combinations with coefficients in a given ring.
More precisely, a torsor (also known as a heap when stated in a single-sorted form) is a set $T$ equipped with a ternary operation
$t\colon T^3 \to T$such that
$t(a,a,b)=t(b,a,a)=b,\qquad t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).$A homomorphism of torsors is a map of sets that preserves this operation.
The equivalence with the two-sorted definition is demonstrated as follows.
Given a $G$-torsor $T$, we send it to the set $T$ equipped with the ternary operation $t(a,b,c)=g(a,b)c$, where $g(a,b)$ is the unique element of $G$ such that $g(a,b)b=a$.
Given a torsor $(T,t)$, we send it to the pair $(LTrans(T),T)$, where $LTrans(T)$ is a subgroup of the group of bijections on the set $T$ comprising precisely the bijections of the form $c\mapsto t(a,b,c)$ for some elements $a,b\in T$. The group $G$ acts on $T$ by evaluation: $gt=g(t)$.
Mapping $(T,t)\mapsto (LTrans(T),T)\mapsto (T,t)$ gives back the same torsor $(T,t)$ that we started from.
Mapping $(G,T)\mapsto (T,t)\mapsto (LTrans(T),t)$ produces a torsor $(LTrans(T),t)$ that is naturally isomorphic to $(G,T)$ via the isomorphism
$(G,T)\to(LTrans(T),T),\qquad g\mapsto (t\mapsto gt),\qquad t\mapsto t.$39 looks good to me.
added pointer to:
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