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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 9th 2012
    • (edited Sep 9th 2012)

    I added to the “abstract nonsense” section in free monoid a helpful general observation on how to construct free monoids. “Adjoint functor theorem” is overkill for free monoids over SetSet.

    • CommentRowNumber2.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 9th 2012

    ummm, shouldn’t

    Then a left adjoint to the forgetful functor Mon(C)CMon(C) \to C exists, taking an object cc to

    n0c n,\sum_{n \geq 0} c^{\otimes n},

    which thereby becomes the free monoid on CC.

    really be

    Then a left adjoint to the forgetful functor Mon(C)CMon(C) \to C exists, taking CC to

    n0C n,\sum_{n \geq 0} C^{\otimes n},

    which thereby becomes the free monoid on CC.

    an object cc of CC is not involved.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 9th 2012
    • (edited Sep 9th 2012)

    ummm… no. There was one typo in what I wrote: that should have been a lower-case cc before the period. I’ll go fix that. (Edit: done.)

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeSep 9th 2012

    The adjoint functor theorem is useful to do the proofs. (None of the constructions currently come with proofs that they are what we claim they are.)

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 10th 2012

    Should I include a proof of the theorem I quoted? (Hm, not sure I really want to put myself out there, but I’ll ask anyway.)

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 10th 2012

    I don’t think that it’s necessary now. It might be better to leave the proofs for somebody who doesn’t find the result obvious and wants to write down what they think of. (That’s usually what I do … not that I always find proofs obvious when I leave them out if I’m quoting the results from elsewhere.)

  1. list currently redirects to free monoid. However, in homotopy type theory, lists as usually defined in the literature such as in section 5.1 of the HoTT book, as an inductive type generated by a term ε A:List(A)\epsilon_A:\mathrm{List}(A) and a function η A:A(List(A)List(A))\eta_A:A \to (\mathrm{List}(A) \to \mathrm{List}(A)), cannot in general be proven to be set-truncated; i.e. see the list on the circle type List(S 1)\mathrm{List}(S^1). Thus, I would propose splitting list out to its own article, while leaving this article explicitly for free monoids.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2023

    It would make general sense to distinguish fine-print between the notions of free monoids and lists, and be it just to highlight that different terminology may depend on difference of applications. So as soon as there is material and editorial energy for a split, we should do it, and you are welcome to go ahead with it.

  2. added section about free monoids in dependent type theory


    diff, v30, current