# Joy of Abstraction: Errata

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

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

### Corrections to first edition hardback

Note that this list (up to July 30, 2023) will all be corrected in the paperback. Corrections found after this will be in a new list.

Chapter 5

p. 59: Para 3 is ill worded, and to be clearer should say something like “For example a rhombus is required to have all sides the same, and opposite angles the same, whereas a kite only has to have adjacent pairs of edges the same, and one pair of opposite angles the same.”

p. 59: T 5.6 should say “A trapezoid is a generalization of a rhombus and of a parallelogram” (not “special case”)

Chapter 7

p.87: Definition 7.3 has an extra comma. It should say “∀ a, bS ” not “∀ a, b, S

Chapter 12

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

Chapter 13

p.149: bottom of the page, in the expression g(f(1)) the leftmost parenthesis is a bigger than the rightmost; they should match

p.154 I should have said that we will call the category of posets either Poset or Pst, because later on in Chapter 20 I use the latter.

Chapter 14

p. 169: at the top of the page that first function should be labelled with a square root sign to make it clearer which function I am modifing here

p. 176: After T14.10 the isomorphism diagram should have A and B instead of a and b

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

“By definition, for any bB, g(b) is the unique element aA such that f(a) = b. In this case we’re looking for g(f(a)), so we’re looking for the unique element x ∈ A such that f(x) = 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 “∀ xm   fs(x) = ft(x)” not ∀ xa

p. 200: In definition 15.12 f and g are referred to the wrong way round. To correct this, you can leave the diagram the same but change the text to:

“Let f : a → b be a morphism in a category. If there is a morphism g as shown here with fg = 1_b then f is certainly epic. Then f is called a split epic and g is called a splitting, a right inverse, or a section for f.”

However, I suspect it will be clearer (when I do the corrections) to change the diagram so that it is the same as the one on p.196, and say

“Let g : b → a be a morphism in a category. If there is a morphism f as shown here with gf = 1_a then g is certainly epic. Then g is called a split epic and f is called a splitting, a right inverse, or a section for g.”

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 18

p.244: “The fact that the bijection is produced via composition with a factorization” should say “with the projections”.

Chapter 19

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

Chapter 20

p. 291: In the “More general definition of functor” the 2nd bullet point’s 2nd arrow should go Fx -> Fy.

p. 291: In the footnote Fc should be Fx.

p. 298: At the top I used Poset and then later I used Pst. Those should be the same, so I will change that so they’re all Pst.

p. 299: second paragraph has one instance of “homomomorphism”! This should be “homomorphism”.

Chapter 21

p.323: In Definition 21.3 it should say “for all a, b ∈𝒞” (not x, y); this is in both the bullet points for faithful and for full.

p.326: In Definition 21.5 it should say “such that F(c) ≅ d”” not f(c)

Chapter 22

p. 335: In the top diagrams it should be Δ_v(0) and Δ_v(1) rather than Δ_v (i) and Δ_v (i)

pp.336 – 337: In the last diagram at the bottom of p.336 and the subsequent iterations on p.337, the bottom curved arrow should be the identity on Fy, not on Gx.

Chapter 23

p.356 T23.3 Note that “aaall” is not a typo. I really mean “aaall” or even “aaaaaaaaaaaall”. 🙂

Index

homset also appears on p.290