# Joy of Abstraction: Errata

**Errata, typos and clarifications in “The Joy of Abstraction”**

Many thanks to readers who have submitted items in this list.

Chapter 12

p. 138: “moroever” should be “moreover”

Chapter 14

p. 176: argument in the last bullet point on the page is perhaps better worded like this –

“By definition, for any *b* ∈ *B*, *g(b)* is the unique element *a* ∈ *A* such that *f(a) = b*. In this case we’re looking for *g(f(a))*, so we’re looking for the unique element *a ∈ A* such that *f(a) = f(a)*, which must be *a* itself.”

pp.181 – 182: This subsection should be about tosets, not posets. The argument I give depends on the negation of “x is less than or equal to y” being “x is greater than y”. This is true in a toset (because of trichotomy) but not in a poset, because in a poset the negation would be “x is greater than y or x and y are incomparable”. A counter-example for posets would be if A consists of 2 incomparable elements (say, x and y) and B consists of two comparable elements (say x < y ). Then the bijection from A to B sending each element to itself is order-preserving (trivially), but the inverse is not.

Chapter 15

p.195: Just below the first grey box, the first like should read “∀ *x* ∈ *m* *fs(x) = ft(x)*” not ∀ *x* ∈ *a*

Chapter 16

p. 216: Second paragraph “unncessary” should be “unnecessary”

Chapter 17

p. 236: In the first box the C_1 and C_0 should be the other way round. That is, the “id” function goes from C_0 to C_1.

Chapter 19

pp. 281 – 285: Running heads on right hand pages should say “Pushouts in **Set”**