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# Mistakes in "Seven Sketches in Compositionality"

edited March 2018

If you catch typos or other mistakes, please add them to David Spivak's list:

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1.

Equation 1.4, the P and Q subscripts of A should be p and q. And maybe the big-U notation deserves a footnote explanation.

Comment Source:Equation 1.4, the P and Q subscripts of A should be p and q. And maybe the big-U notation deserves a footnote explanation.
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2.

Chapter 1, page 4:

For example, the two constituent systems could be the views of two local authorities on possibly contagion between an infected person • and a vulnerable person ∗.

Language seems off here, "possibly contagion between".

Comment Source:Chapter 1, page 4: For example, the two constituent systems could be the views of two local authorities on possibly contagion between an infected person • and a vulnerable person ∗. Language seems off here, "possibly contagion between".
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3.
edited March 2018

Chapter 1, page 2:

One might call these surprised {\it generative effects}.

Should be:

One might call these surprises {\it generative effects}.

page 5:

A ≤ B in the poset if

seems to be a grammatical whoopsie.

Also, poset'' hasn't been defined at this point, which I suspect is not what the authors wanted, with this being a math book and all, although personally I rather like words being used before they're defined.

The existence of the generative effect, however, is captured in the inequality Φ(A) ∨ Φ(B) ≤ Φ(A ∨ B). (1.3) This can be a strict inequality ...

If I understand correctly, the formula only captures a generative effect if it IS a strict inequality (not just can be one).

page 5 again:

the map Φ preserves some structure but not others

should be:

the map Φ preserves some structures but not others

Comment Source:Chapter 1, page 2: One might call these surprised {\it generative effects}. Should be: One might call these surprises {\it generative effects}. ---- page 5: A ≤ B in the poset if seems to be a grammatical whoopsie. Also, poset'' hasn't been defined at this point, which I suspect is not what the authors wanted, with this being a math book and all, although personally I rather like words being used before they're defined. ---- The existence of the generative effect, however, is captured in the inequality Φ(A) ∨ Φ(B) ≤ Φ(A ∨ B). (1.3) This can be a strict inequality ... If I understand correctly, the formula only captures a generative effect if it IS a strict inequality (not just can be one). ---- page 5 again: the map Φ preserves some structure but not others should be: the map Φ preserves some structures but not others
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4.
edited March 2018

Chapter 1, page 7: In Definition 1.8, right before the exercise, the notation used is $$a_1, a_2$$ but becomes $$a, b$$ by the end of the sentence

Chapter 1, page 9: Remark 1.16, the surjective function is showing a double headed arrow for the first time, maybe it's worth adding a notation footnote

Chapter 1, page 9: Definition 1.19, missing period in "for any $$x \in X$$ It is often"

Comment Source:Chapter 1, page 7: In Definition 1.8, right before the exercise, the notation used is \$$a_1, a_2\$$ but becomes \$$a, b\$$ by the end of the sentence - - - - - - - - - - - Chapter 1, page 9: Remark 1.16, the surjective function is showing a double headed arrow for the first time, maybe it's worth adding a notation footnote - - - - - - - - - - - Chapter 1, page 9: Definition 1.19, missing period in "for any \$$x \\in X\$$ It is often" 
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5.

"Example 1.61. Consider the two-element set P = {p, q, r} with ..."

Looks like a 3-element set to me!

Comment Source:"Example 1.61. Consider the two-element set P = {p, q, r} with ..." Looks like a 3-element set to me!
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6.

Chapter 1 page 7 Definition 1.8: $$A_P ∩ A_Q$$ should be $$A_p ∩ A_q$$ ?

Comment Source:Chapter 1 page 7 Definition 1.8: \$$A_P ∩ A_Q\$$ should be \$$A_p ∩ A_q\$$ ?
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7.

Chapter 1 page 8 Exercise 1.12 $$(A_p)_{p \in P}$$ should be curly braces for "partition"?

Comment Source:Chapter 1 page 8 Exercise 1.12 \$$(A_p)_{p \\in P}\$$ should be curly braces for "partition"?
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8.

Chapter 1 page 9 Exercise 1.15 question 2: second relation is the same as the first, is that deliberate, or are arrows pointing in wrong direction?

Comment Source:Chapter 1 page 9 Exercise 1.15 question 2: second relation is the same as the first, is that deliberate, or are arrows pointing in wrong direction?
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9.
edited March 2018

In the beginning of 1.2: "a poset - a preordered set" with emphasis on letters "p" and "o" of preordered. I find it a little strange. Maybe the authors meant a "partially ordered set" instead. This mnemonic makes it easier to forget about antisymmetry requirement for posets. Calling a partially ordered set a "poset" and a preordered set a "preset" makes more sense and it's more obvious where the names come from.

Update: I read further and now I understand it is not a mistake. The author made a deliberate decision to use the name "poset" for presets "skeletal posets" for posets, "partial order" for a preorder and "preorder" for a partial order.

Comment Source:In the beginning of 1.2: "a poset - a preordered set" with emphasis on letters "p" and "o" of preordered. I find it a little strange. Maybe the authors meant a "partially ordered set" instead. This mnemonic makes it easier to forget about antisymmetry requirement for posets. Calling a partially ordered set a "poset" and a preordered set a "preset" makes more sense and it's more obvious where the names come from. Update: I read further and now I understand it is not a mistake. The author made a deliberate decision to use the name "poset" for presets "skeletal posets" for posets, "partial order" for a preorder and "preorder" for a partial order.
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10.

Chapter 1 page 3: the definition of 'join' could state more clearly that the qualifier "in at least one of A or B" applies separately to each of the connections mentioned.

Comment Source:Chapter 1 page 3: the definition of 'join' could state more clearly that the qualifier "in at least one of A or B" applies separately to each of the connections mentioned.
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11.

Chapter 1 page 4: Since you suppress redundant arrows in Hesse diagrams, you might replace "we draw an arrow from system A to system B if A <= B" with "an arrow from system A to system B means that A <= B".

Comment Source:Chapter 1 page 4: Since you suppress redundant arrows in Hesse diagrams, you might replace "we draw an arrow from system A to system B if A <= B" with "an arrow from system A to system B means that A <= B".
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12.
edited April 2018

Change of plan! If you catch typos or other mistakes, please add them to David Spivak's list:

David will copy the mistakes you've already caught to that list, and he thanks you for catching them.

Comment Source:Change of plan! If you catch typos or other mistakes, please add them to David Spivak's list: * [Seven Sketches book: Typos, comments, questions, and suggestions.](https://docs.google.com/document/d/160G9OFcP5DWT8Stn7TxdVx83DJnnf7d5GML0_FOD5Wg/edit) David will copy the mistakes you've already caught to that list, and he thanks you for catching them. 
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13.

Here's something I wrote in an email to Brendan Fong. He just replied saying that he and David are fixing the mistakes you folks listed above, and the mistakes listed below - and they're switching to the standard definition of "poset". An updated version of the book will be available soon.

Someone in the course pointed out something that's more than a typo. If you're going to use "poset" to mean "preorder" (bad, bad, bad) then you can't talk about "the" meet or join of two elements in a poset, because even when it exists it's not unique.

Of course you can use "the" in the sophisticated way, meaning "unique up to canonical isomorphism"... but that seems a bit fancy for your intended audience, and it at least would need to be explained.

You guys just say things like:

Let P be a poset, and let A be a subset. We say that an element is the meet of A if ...

You could fix this by changing "the" to "a", but every equation you write down involving meets and joins is wrong unless you restrict to the "skeletal poset" case. For example, Example 1.62:

In any poset P, we have $$p \vee p = p \wedge p = p$$.

More importantly, Prop. 1.84 - right adjoints preserve meets. The equations here are really just isomorphisms!

This then makes your statement of the adjoint functor theorem for posets incorrect.

I think this is the best solution:

1. Call preorders "preorders" and call posets "posets". Do not breed a crew of students who use these words in nonstandard ways! You won't breed enough of them to take over the world, so all you will accomplish is making them less able to communicate with other people. And for what: just because you don't like the sound of the word "preorder"?

2. Define meets and joins for preorders, but point out that they're unique for posets, and say this makes things a bit less messy.

3. State the adjoint functor theorem for posets... actual posets!

Comment Source:Here's something I wrote in an email to Brendan Fong. He just replied saying that he and David are fixing the mistakes you folks listed above, and the mistakes listed below - and they're switching to the standard definition of "poset". An updated version of the book will be available soon. <hr/> Someone in the course pointed out something that's more than a typo. If you're going to use "poset" to mean "preorder" (bad, bad, bad) then you can't talk about "the" meet or join of two elements in a poset, because even when it exists it's not unique. Of course you can use "the" in the sophisticated way, meaning "unique up to canonical isomorphism"... but that seems a bit fancy for your intended audience, and it at least would need to be explained. You guys just say things like: > Let P be a poset, and let A be a subset. We say that an element is the meet of A if ... You could fix this by changing "the" to "a", but every equation you write down involving meets and joins is wrong unless you restrict to the "skeletal poset" case. For example, Example 1.62: > In any poset P, we have \$$p \vee p = p \wedge p = p\$$. More importantly, Prop. 1.84 - right adjoints preserve meets. The equations here are really just isomorphisms! This then makes your statement of the adjoint functor theorem for posets incorrect. I think this is the best solution: 1. Call preorders "preorders" and call posets "posets". Do not breed a crew of students who use these words in nonstandard ways! You won't breed enough of them to take over the world, so all you will accomplish is making them less able to communicate with other people. And for what: just because you don't like the sound of the word "preorder"? 2. Define meets and joins for preorders, but point out that they're unique for posets, and say this makes things a bit less messy. 3. State the adjoint functor theorem for posets... actual posets!
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14.
edited March 2018

John #13: I was just going to comment in the Lecture 3 thread about how "the" meet and "the" join of a subset of a preordered set are only defined up to equivalence, but I thought I would check in this thread first to see if anyone else had caught that! Glad we are switching to the more standard terminology.

Comment Source:John #13: I was just going to comment in the Lecture 3 thread about how "the" meet and "the" join of a subset of a preordered set are only defined up to equivalence, but I thought I would check in this thread first to see if anyone else had caught that! Glad we are switching to the more standard terminology.
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15.

Me too!

Comment Source:Me too!