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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 $Set$.
ummm, shouldn’t
Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking an object $c$ to
$\sum_{n \geq 0} c^{\otimes n},$which thereby becomes the free monoid on $C$.
really be
Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking $C$ to
$\sum_{n \geq 0} C^{\otimes n},$which thereby becomes the free monoid on $C$.
an object $c$ of $C$ is not involved.
ummm… no. There was one typo in what I wrote: that should have been a lower-case $c$ before the period. I’ll go fix that. (Edit: done.)
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.)
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.)
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.)
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 $\epsilon_A:\mathrm{List}(A)$ and a function $\eta_A:A \to (\mathrm{List}(A) \to \mathrm{List}(A))$, cannot in general be proven to be set-truncated; i.e. see the list on the circle type $\mathrm{List}(S^1)$. Thus, I would propose splitting list out to its own article, while leaving this article explicitly for free monoids.
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.
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