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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 7th 2021

    Added two references.

    diff, v16, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeApr 15th 2021

    In universal algebra the standard name is associative Mal’cev algebra (in various spellings).

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 15th 2021

    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.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 16th 2021

    Redirects herd, associative Malcev algebra.

    diff, v18, current

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeApr 16th 2021
    • (edited Apr 16th 2021)

    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.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 16th 2021

    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.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 16th 2021


    • Peter T. Johnstone, The ‘closed subgroup theorem’ for localic herds and pregroupoids, Journal of Pure and Applied Algebra 70 (1991) 97-106. doi.

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2021

    cross-linked with pseudo-torsor

    diff, v24, current

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 13th 2022

    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 AbHeapAbHeap 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.

    diff, v29, current

  1. The term “heap” is used concurrently as a name of a certain algebraic-combinatorial structure introduced by X.Viennot in the 80s.


    diff, v31, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2023

    I have added some formatting and touched wording and hyperlinking of the idea-section.

    I couldn’t make sense of this passage:

    shifting aa via the (right) translation in the group which translates bb into cc.

    (apparently trying to narrate the expression ab 1ca \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?)

    diff, v32, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2023

    also touched wording and formatting of the section “Automorphism groups of heaps” (here) for better flow.

    diff, v33, current

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeSep 4th 2023

    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 aa to another fixed torsor element bb makes the translation translating aa to bb a well defined automorphism. In geometry we say translation by a “translation vector” from aa to bb (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 PP is a free right GG-space where GG is a topological group then the map which to a pair (a,b)(a,b) of points in the same GG-orbit assigns the unique element of GG which sends aa to bb is called translation map τ:P *G\tau: P^*\to G (where P *P×PP^*\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 PP is a principal GG-bundle if τ\tau is continuous. This is automatic if the bundle PP/GP\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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2023
    • (edited Sep 4th 2023)

    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(,,)t(-,-,-) via one of the three constructions in this Prop.. But it needs to be said better in order to be informative.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2023

    I have added a wording (here) along the lines that I suppose was intended. But this may be a case where the (really simple) formulas say more than many words.

    diff, v34, current

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeSep 4th 2023
    • (edited Sep 4th 2023)

    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 aa by b 1cb^{-1}c is translating “point” aa by “vector” from bb to cc. 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 ab 1ca 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 1cb^{-1} c is the element which sends bb to cc, hence one applies to aa the same right multiplication which sends bb to cc; this as a measure of the action is more pertinent to the intuition than saying what if bb is identity.

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 5th 2024


    Examples in homotopy theory

    The map [0,1][0,1][0,1]\to[0,1] that sends 000\mapsto 0, 1/311/3\mapsto 1, 2/302/3\mapsto 0, 111\mapsto 1, and interpolates linearly between these points yields an associative Malcev operation on [ΣX,Y][\Sigma X,Y], where XX and YY are (unpointed) spaces, Σ\Sigma is the suspension functor, and [,][-,-] denotes the set of morphisms in the homotopy category.

    Thus, [ΣX,Y][\Sigma X,Y] is a (nonabelian) heap. Likewise, the full mapping space Map(ΣX,Y)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.

    diff, v35, current