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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeMay 3rd 2016

    New entry archetypal example. I am very glad to thank Prof. Joel W. Robbin, my thesis advisor, for making me aware of this useful concept in late 1990s.

    • CommentRowNumber2.
    • CommentAuthorfastlane69
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    Archetype is being used as a synonym for “universal”?

    As in a coset space is a universal construction on homogeneous spaces for Hausdorff paracompact topological groups?

    Or is universal reserved for another context or meaning or I’m using it incorrectly?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 3rd 2016

    I don’t understand the concept either. The example given is what I would call a skeleton of the category of homogeneous spaces. However, the previous paragraph seems to indicate that not every “archetypal example” is a skeleton, but I don’t know what “typical in mathematician practice and generic in the sense of some measure” means. Can you give some more examples?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 3rd 2016

    I took it that “archetypal example” was meant (only) in a very informal sense. For example, when one is trying to explain to a mathematician how Lawvere theories work by means of an example, a very typical example to reach for is the case of groups. So (in my understanding) that would be an archetypal example.

    Of course, in a very different area of investigation, the Persona, the Shadow, the Anima, and the Wise Old Man might be “archetypal examples”. ;-)

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    Maybe the wording “measure” etc. is indeed not very good.

    One wants to say informally that the archetypal examples are those which exhibit the essence of all examples. The case of skeleton is basic but there are other cases. Of course, the latter “essence” is not well defined, unless we are able to say which process can produce the rest and which problem (of description or alike) we are trying to solve.

    For example, the archetypal example of calculation of an exponential of a matrix is the calculation for Jordan form matrices (and if we restrict to diagonalizable matrices then just calculations for the diagonal matrices – in the latter case just the exponentiating of entries on the diagonal). Not every matrix is diagonal, and for many purposes the similarity of matrices is not a desirable notion of equivalence (as in the context where one might care on unitarity) but for the purpose of exponentiating a matrix (where exponential is defined by the exponential series) one can exponentiate any similar matrix and transform back. So we may be in a context where we generally do care on the difference between representatives of a similarity class, but still for the purposes of subtask of studying the exponential the case of Jordan forms suffices, provided one knows how to do the similarity transformation. But even if we take that the similarity is sort fo equivalence, we can in fact take two special Jordan forms: single Jordan cell and diagonalizable matrix. In some sense these two special cases are archetypal because to do the general Jordan cell you need to be able to do the case of a single Jordan cell plus using the rule for exponentiating the block diagonalized matrix. Surely the block diagonalized matrix is more general than diagonal matrix but the rule for exponentiating is the same (just over more general ring). This kind of thinking is useful in doing research in mathematics (and even in education, by starting with archetypal cases we get concrete handle of the general case which is maybe too abstract in the first encounter).

    Another example would be to do some algorithm or proof about Riemann surfaces and the nontrivial part is just to solve the problem for each connected component. Then the archetypal example is the compact case.

    Overall, sometimes it is possible and useful and sometimes not to describe the category in which the archetypal examples will be either representatives of all isomorphism classes, or – more weakly – generators of some sort, but the purpose in bringing up them to assert that the general case of the problem is essentially described once they are covered.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    I agree with Todd, that it should be taken informally, but still I do not think that “typical” traditional example is to be called archetypal. If it is showing the essential features of the general case then yes, otherwise it is just a traditional or easy or concrete choice of an example, nothing archetypal. Archetypal structure should really reflect the general case in a satisfactory manner. This satisfaction is often not seen at the level when one encounters the example but realized much later when one realizes that the general case (or its treatment, if that is in the focus) reduces somehow to the archetypal example.

    For example, with what I know on algebra I would definitely not call vector spaces the archetypal example of a module, even for educational purpose, as the case of modules has so many strikingly new features. Like there can be many isomorphic modules of the same rank, there are syzigies and so on. Truly, the vector spaces are the example which helps remembering the axiom and having some mental image. On the other hand, if one asks for an archetypal example of a non-split exact sequence the standard example with integers and integers modulo p shows the essence of the phenomenon. However if one wants to build various complex counterexamples with short exact sequences, then one sees that the case is far from exhaustive in phenomena and would not call this example archetypal.

    • CommentRowNumber7.
    • CommentAuthorNoam_Zeilberger
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    [deleted, sorry wrong thread]

    • CommentRowNumber8.
    • CommentAuthorfastlane69
    • CommentTimeMay 3rd 2016

    Is there a Jungian context to the use of Archtype perhaps?

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 3rd 2016

    Zoran’s #5 and #6 make the general idea reasonably clear to me. I would agree with his vector space example being a non-example.

    Some of this discussion should be incorporated into the article.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeMay 4th 2016

    How about the present version: archetypal example ? Unfortunately I teach tomorrow and will have to concentrate on students and geometry the rest of the day.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2016

    Better, thanks!

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeMay 4th 2016

    Somehow the first paragraph was not visible in the version above (the first line software glitch).

    An archetypal example of a notion in mathematics is a class of examples such that every instance of the notion in some sense reduces to it: by being isomorphic to it, equivalent in some other sense, or at least that the archetypal example has all essential features (for the problem at hand) found in general case.