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Re #3: For the record, Malcev spelled his name as Malcev in his non-Russian papers, so it would make sense to use this spelling to make it easier to find his papers.
Redirects herd, associative Malcev algebra.
Malcev spelled his name as Malcev in his non-Russian papers, so it would make sense to use this spelling
I agree that it makes sense so I agreed, thank you. On the other hand, I do not consider it a completely sensitive choice from personal point of view. All people with native signs not canonical in short version of Latin alphabet are under pressure to simplify. Traditional Croatian (and many other) library catalogues do not simplify Russian and have strict rules which make certain standard bijection with diacritics. I had registered many times as Škoda in various web forms and the form did not pass or gave errors because of some technological glitch. Sometimes it is worse for indexing and so on. It is hard to type Š, and if I type it say in the html source of my webpage it will not parse correctly. Some editors do not like if you are from small university, strange country or you have a hard to spell name. So sometimes we yield to pressures. Many of my accounts have Skoda although it is possible on those platforms to make Škoda, for example my facebook page has unfortunately no diacritics, making it wrong in my native language (and at the different place in alphabetic order). It is not that I like it and I hope that nobody after 50 years says it was my choice having informed knowledge and ready alternatives. It is the choice I am forced to do sometimes, in order not to have too many troubles in my functioning.
Re #5: At least in your case there is a canonical spelling in the Latin alphabet (with diacritics). For the Russian language, there is a Wild West of various Romanizations. The Russian government used at least 4 different spellings of my first name (for different people; fortunately only 2 spellings in my case). This is really annoying when it happens in official documents, like passports etc.
Concerning the nLab, that’s why we can write Мальцев if the original spelling is important, e.g., we are referring to his Russian papers.
Added:
cross-linked with pseudo-torsor
Added definition of abelian heap and the fact the hom-sets of the resulting category inherit an abelian heap structure.
I guess this makes the category $AbHeap$ a closed category, and I’ve checked some of the conditions, but not all of the extranaturality etc. I don’t know about this being closed monoidal, but if abelian heaps form an commutative algebraic theory, then it will.
I have added some formatting and touched wording and hyperlinking of the idea-section.
I couldn’t make sense of this passage:
shifting $a$ via the (right) translation in the group which translates $b$ into $c$.
(apparently trying to narrate the expression $a \cdot b^{-1} \cdot c$)
and have removed it for the time being. If it is felt to be necessary, let’s clarify what it is trying to say and then add it back in.
(It’s clear that group multiplication is a kind of “shifting by translation”, if you wish, but is there more to be said here?)
It’s clear that group multiplication is a kind of “shifting by translation”
Quite a few people in finite geometries use this terminology when talking about “parallelogram spaces”, as well as some differential/synthetic geometers like Anders Kock. Translation is nothing else then affine group action and for some people noncommutative version is subject to the same intuition and terminology. The fact that there is a unique element of a group which sends any fixed torsor element $a$ to another fixed torsor element $b$ makes the translation translating $a$ to $b$ a well defined automorphism. In geometry we say translation by a “translation vector” from $a$ to $b$ (wikipedia: “it makes sense to subtract two points of the space, giving a translation vector”).
Moreover, in the general theory of torsors, people talk about left translations and right translations (e.g. Breen in his papers o torsors, gerbes etc.) and if $P$ is a free right $G$-space where $G$ is a topological group then the map which to a pair $(a,b)$ of points in the same $G$-orbit assigns the unique element of $G$ which sends $a$ to $b$ is called translation map $\tau: P^*\to G$ (where $P^*\subset P\times P$ is the subspace of all pairs which are in the same orbit) or by some other people a division map. By a classical definition from 1950s $P$ is a principal $G$-bundle if $\tau$ is continuous. This is automatic if the bundle $P\to P/G$ is locally trivial, however in the topological category there are principal bundles in this sense which are not locally trivial.
My opinion is that the translation terminology is an important and well attested viewpoint in this subject.
As I said, its clear that one can think of group multiplication as translation, but what it is that this passage meant to say on top of this basic fact?
I suspect the passage was meant to give an intuition for how to think of $t(-,-,-)$ via one of the three constructions in this Prop.. But it needs to be said better in order to be informative.
but what it is that this passage meant to say on top of this basic fact?
No, I never said that it says that any multiplication is a translation (although this metaphor is used elsewhere, I am not talking this here at all). It is more specific in the context of this ternary operation: the statement says that multiplying $a$ by $b^{-1}c$ is translating “point” $a$ by “vector” from $b$ to $c$. Translation is operation between two objects of different type: a point and a vector, and the vector is itself determined by two other points, hence ternary operation on single sort of points is derived.
The real use is that this point of view gives the basic intuition on this ternary operation, for example to figure out (rather than to verify) the axioms. If one (a newcomer, not seasoned researcher on principal bundles like you or me) just works mechanically with expression $a b^{-1} c$ it is even not obvious why this operation (or any model satisfying the axioms) determines the group up to a choice of unit element.
15 I think one should assert at least that $b^{-1} c$ is the element which sends $b$ to $c$, hence one applies to $a$ the same right multiplication which sends $b$ to $c$; this as a measure of the action is more pertinent to the intuition than saying what if $b$ is identity.
Added:
The map $[0,1]\to[0,1]$ that sends $0\mapsto 0$, $1/3\mapsto 1$, $2/3\mapsto 0$, $1\mapsto 1$, and interpolates linearly between these points yields an associative Malcev operation on $[\Sigma X,Y]$, where $X$ and $Y$ are (unpointed) spaces, $\Sigma$ is the suspension functor, and $[-,-]$ denotes the set of morphisms in the homotopy category.
Thus, $[\Sigma X,Y]$ is a (nonabelian) heap. Likewise, the full mapping space $Map(\Sigma X,Y)$ can be turned into an (∞,1)-heap, defined as an (∞,1)-algebra (in spaces) over the algebraic theory of heaps.
See Vokřínek {#Vokrinek} for more information.
On the category theory community server, Eric Downes was confused because he read about the “automorphism group” of a heap on the nLab and believed that this referred to the group of automorphisms of the heap - i.e., the group of invertible maps from the heap to itself, preserving the heap operations. As it turns out, the nLab article uses “automorphism group of a heap” and “Aut(H)” to mean something completely different! I don’t like this.
I believe the category of heaps is equivalent to the category of pairs consisting of a torsor $G$ and a $G$-torsor $X$, with a fairly obvious notion of morphism between such pairs. Taking this viewpoint, the so-called “automorphism group” of a heap $(G,X)$ is the group $G$. So I would be inclined to call it the “underlying group” of the heap.
Spelling it out a bit: any heap $H$ has a ternary operation $t: H^3 \to H$, and the set of maps $t(h, h', -): H \to H$ forms a group $G$ under composition, which I’d call the underlying group of $H$. Then the underlying set of the heap becomes a $G$-torsor, which I’m calling $X$ above.
Conversely given a group $G$ and a $G$-torsor $X$ we can do the following: given elements $h,h',h'' \in X$ we can define $t(h,h',h'') = g h''$ where $g$ is the unique $g \in G$ with $g h' = h$.
Is it okay if I change “automorphism group” to “underlying group”? Or does someone have a better name for it?
The established term in the corresponding situation of torsors is structure group.
I have made the change. In the course of this I touched wording, typesetting and formatting here and there, for streamlining.
I’m fixing the proof of Prop. 3.1, which had been optimized for thinking of Str(H) as a group that acts on the right on the underlying set H of the heap. The rest of the article says that Str(H) acts on the left on H, making H into a left Str(H)-torsor. (I see the nLab article torsor introduces left torsors so that seems to be the preferred convention.)
I added a proposition due to Todd Trimble:
The category of groups is equivalent to the slice category $1 \downarrow \mathrm{Heap}$ where $1$ is the terminal heap and $\mathrm{Heap}$ is the category of heaps.
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