# 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 the paperback

The following typos (from August 2023 and later) were too late to be corrected in the paperback; below these are typos found in the first edition hardback that will be corrected in the paperback.

Chapter 23

p. 356: Just below T23.3 should say “…this time we are making a functor that sends an object *a* to the set *C(a,x)”*

p. 364: The last paragraph says “*closure of C under colimits”. *This is ambiguous and should really have the word “free” somewhere. [I was trying to avoid introducing the word “cocompletion”, but it’s really the free cocompletion.]

Chapter 24

p. 370: At the bottom of the page, the bullet point on unit laws should also say “identity 2-cells on identity 1-cells act as identities with respect to horizontal composition”

### 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, b* ∈ *S* ” 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 *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 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 “∀ *x* ∈ *m* *fs(x) = ft(x)*” not ∀ *x* ∈ *a*

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 *∈?” (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