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nLab > Latest Changes: free monoid

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- CommentRowNumber1.
- CommentAuthorTodd_Trimble
- CommentTimeSep 9th 2012
- (edited Sep 9th 2012)
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Author: Todd_Trimble
Format: MarkdownItexI 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 $Set$.

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>$ .
- CommentRowNumber2.
- CommentAuthorRodMcGuire
- CommentTimeSep 9th 2012
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Author: RodMcGuire
Format: MarkdownItexummm, 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, shouldn’t

Then a left adjoint to the forgetful functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Mon(C) \to C</annotation></semantics></math>$ exists, taking an object $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>$ to
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></munder><msup><mi>c</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo></mrow><annotation encoding="application/x-tex">\sum_{n \geq 0} c^{\otimes n},</annotation></semantics></math>$
which thereby becomes the free monoid on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ .

really be

Then a left adjoint to the forgetful functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Mon(C) \to C</annotation></semantics></math>$ exists, taking $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ to
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></munder><msup><mi>C</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo></mrow><annotation encoding="application/x-tex">\sum_{n \geq 0} C^{\otimes n},</annotation></semantics></math>$
which thereby becomes the free monoid on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ .

an object $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ is not involved.
- CommentRowNumber3.
- CommentAuthorTodd_Trimble
- CommentTimeSep 9th 2012
- (edited Sep 9th 2012)
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Author: Todd_Trimble
Format: MarkdownItexummm... 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.)

ummm… no. There was one typo in what I wrote: that should have been a lower-case $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>$ before the period. I’ll go fix that. (Edit: done.)
- CommentRowNumber4.
- CommentAuthorTobyBartels
- CommentTimeSep 9th 2012
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Author: TobyBartels
Format: MarkdownItexThe 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.)

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
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Author: Todd_Trimble
Format: MarkdownItexShould 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.)

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
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Author: TobyBartels
Format: MarkdownItexI 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.)

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.)
- CommentRowNumber7.
- CommentAuthorMadeleine Birchfield
- CommentTimeJan 24th 2023
- (edited Jan 24th 2023)
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Author: Madeleine Birchfield
Format: MarkdownItex[[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.

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi> <mi>A</mi></msub><mo>:</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon_A:\mathrm{List}(A)</annotation></semantics></math>$ and a function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mo stretchy="false">(</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_A:A \to (\mathrm{List}(A) \to \mathrm{List}(A))</annotation></semantics></math>$ , cannot in general be proven to be set-truncated; i.e. see the list on the circle type $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">List</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{List}(S^1)</annotation></semantics></math>$ . Thus, I would propose splitting list out to its own article, while leaving this article explicitly for free monoids.
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJan 24th 2023
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Author: Urs
Format: MarkdownItexIt 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.

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.
- CommentRowNumber9.
- CommentAuthornLab edit announcer
- CommentTimeJan 25th 2023
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Author: nLab edit announcer
Format: MarkdownItexadded section about free monoids in dependent type theory Anonymous <a href="https://ncatlab.org/nlab/revision/diff/free+monoid/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+monoid/30">v30</a>, <a href="https://ncatlab.org/nlab/show/free+monoid">current</a>

added section about free monoids in dependent type theory

Anonymous

diff, v30, current
- CommentRowNumber10.
- CommentAuthorvarkor
- CommentTimeSep 15th 2024
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Author: varkor
Format: MarkdownItexAdded some references. <a href="https://ncatlab.org/nlab/revision/diff/free+monoid/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/free+monoid/35">v35</a>, <a href="https://ncatlab.org/nlab/show/free+monoid">current</a>

Added some references.

diff, v35, current

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nLab > Latest Changes: free monoid