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1. • CommentRowNumber1.
   • CommentAuthorltrujello
   • CommentTimeOct 28th 2020
   • PermaLink
   Author: ltrujello
   Format: MarkdownItexI'm interested in editing _[[Mac Lane's proof of the coherence theorem for monoidal categories]]_, as I recently went through all the gory details myself and wrote it up. I was wondering if anybody has any thoughts on what should be left alone with regard to any future changes. Many people clearly put in a lot of work into the page, but it looks like people got busy and it hasn't been updated in a while. I think the first few paragraphs are fine, but I think the rest is a bit wordy, it could be more formal, and notation could be changed (very slightly) to be less clunky. I specifically want to make the current document more formal (e.g., saying "Definition: blah blah"), include some nice diagrams, change the notation (e.g., to avoid using double primes, to avoid denoting a monoidal category as B since I think the letter M pedagogically makes more sense), and complete the incomplete entries at the bottom. I'm not really sure if anyone would be against such changes, hence my inquiry.

   I’m interested in editing Mac Lane’s proof of the coherence theorem for monoidal categories, as I recently went through all the gory details myself and wrote it up. I was wondering if anybody has any thoughts on what should be left alone with regard to any future changes. Many people clearly put in a lot of work into the page, but it looks like people got busy and it hasn’t been updated in a while.

   I think the first few paragraphs are fine, but I think the rest is a bit wordy, it could be more formal, and notation could be changed (very slightly) to be less clunky. I specifically want to make the current document more formal (e.g., saying “Definition: blah blah”), include some nice diagrams, change the notation (e.g., to avoid using double primes, to avoid denoting a monoidal category as B since I think the letter M pedagogically makes more sense), and complete the incomplete entries at the bottom. I’m not really sure if anyone would be against such changes, hence my inquiry.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 28th 2020
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   Author: DavidRoberts
   Format: MarkdownItexNot at all, it is a good idea! If you are very unsure, you can post sections here for comments, but there is no problem with messing up the article, it can always be massaged back into shape if it gets a bit wobbly.

   Not at all, it is a good idea! If you are very unsure, you can post sections here for comments, but there is no problem with messing up the article, it can always be massaged back into shape if it gets a bit wobbly.

3. • CommentRowNumber3.
   • CommentAuthorJ-B Vienney
   • CommentTimeJun 23rd 2024
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   Author: J-B Vienney
   Format: MarkdownItexI don't understand the beginning of the introduction. It is written: "Let $\langle B, \otimes_B, \alpha^{(B)}, e_B, \lambda^{(B)}, \varrho^{(B)}, \rangle$ be some [[monoidal category]]. At first, we might like the _[[coherence theorem for monoidal categories]]_ to state that _every_ [[diagram]] in $B$ built up from instances of $\alpha^{(B)}$, $\lambda^{(B)}$, $\varrho^{(B)}$ and identity arrows by multiplications $\otimes_B$ (such as the pentagon diagram) commutes. This, however, is not possible, because some formally different vertices of such a diagram might turn out to be the same in our particular monoidal category $B$, in a way that makes the diagram non-commutative." Then it says: "As a concrete example of the problem in a particular monoidal category..." but this example only shows that we don't necessarily obtain a strict monoidal category by taking the skeleton of a monoidal category. Isn't it true that by the coherence theorem for monoidal categories, every diagram built up from associators, unitors, identities and tensor product commutes? If it is false, what is a counterexample?

   I don’t understand the beginning of the introduction. It is written:

   “Let $\langle B, \otimes_B, \alpha^{(B)}, e_B, \lambda^{(B)}, \varrho^{(B)}, \rangle$ be some monoidal category. At first, we might like the coherence theorem for monoidal categories to state that every diagram in $B$ built up from instances of $\alpha^{(B)}$, $\lambda^{(B)}$, $\varrho^{(B)}$ and identity arrows by multiplications $\otimes_B$ (such as the pentagon diagram) commutes. This, however, is not possible, because some formally different vertices of such a diagram might turn out to be the same in our particular monoidal category $B$, in a way that makes the diagram non-commutative.”

   Then it says:

   “As a concrete example of the problem in a particular monoidal category…” but this example only shows that we don’t necessarily obtain a strict monoidal category by taking the skeleton of a monoidal category.

   Isn’t it true that by the coherence theorem for monoidal categories, every diagram built up from associators, unitors, identities and tensor product commutes? If it is false, what is a counterexample?

4. • CommentRowNumber4.
   • CommentAuthorAlexanderCampbell
   • CommentTimeJun 24th 2024
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   Author: AlexanderCampbell
   Format: MarkdownItexThe point of that example is that the diagram consisting of the parallel pair of arrows $\alpha_{D,D,D}$ and $1_{D}$ does not commute (i.e. the two morphisms are not equal), even though it is "built up" from associators and identities in a naive sense.

   The point of that example is that the diagram consisting of the parallel pair of arrows $\alpha_{D,D,D}$ and $1_{D}$ does not commute (i.e. the two morphisms are not equal), even though it is “built up” from associators and identities in a naive sense.

5. • CommentRowNumber5.
   • CommentAuthorJ-B Vienney
   • CommentTimeJun 24th 2024
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   Author: J-B Vienney
   Format: MarkdownItexOk, thank you, now I understand!

   Ok, thank you, now I understand!