Corrections etc. to version 2.4 of An Invitation to General Algebra and Universal Constructions.

P.6, 5th line of section 0.4: Change  A minimum of 5 minutes  to  One to five minutes

P.12, Exercise 1.2:3: I don't know the answer to the part of this exercise corresponding to part (ii) of Exercise 1.2:2.

P.19, Exercise 1.7:3:  (of various arities)  should be  (of various finite arities)

P.42, last line:  delete the sentence beginning on this line, and on p.390, the reference [103].  As far as I can tell, the website in question no longer exists.

P.65, beginning of 6th line of section 3.11:  (G, ... )  should be  (|G|, ... )

P.74, Exercise 3.13:3(i), lines 3-4:  should be  k  in both places.

P.75, Exercise 3.13:6:  The symbols  Z S  etc. denote the monoid ring construction discussed in the middle two paragraphs of p.70.

P.95, Exercise 4.1:13:  This describes the reconstruction problem for finite partially ordered sets.  Looking in Math Reviews, I've found a recent survey article on the topic (here is its MR review), in case you are interested in what is known.  To view the article, you'll need to use a campus computer, or activate Berkeley library proxy service on some other computer, or in some other way take advantage of a subscription to the journal.

P.111, first line of next-to-last paragraph of proof:  a set of ordinals  should be  a set  S  of ordinals

P.116, Exercise 4.5:11(vi):  (ii) above  should be  (iii) above

Exercise 4.6:½.  Show that in a commutative ring, every prime ideal contains a minimal prime ideal.  (Note: though the phrase “maximal ideal” by convention means a maximal element of the set of proper ideals, “minimal prime ideal” means minimal among all prime ideals, without restriction.)

Exercise 4.6:3½.  Suppose  X  is a partially ordered set which contains a cofinal chain.  Show that every cofinal subset of  X  also contains a cofinal chain.  (I found this harder than expected.  Maybe there's a simple proof I haven't noticed.)

P.121, Exercise 4.6:2:  In part (iii),  I  must be assumed finite.

P.132, start of last paragraph:  any set  X  should be  any nonempty set  X

Pp.134-140:  I see that in this section, I've gotten carried away on interesting and challenging exercises that are not important for what follows, even more than in the rest of the text.  It's hard to draw a line as to what to include in that category; but if you're short of time, you might skip or skim some or all of the following material, and perhaps revisit it later:  The last line of p.134 through the top three lines of p.136; the last paragraph of p.136 through the exercise that follows it (Exercise 5.2:13, p.137), and everything after Exercise 5.2.17(iii) (p.138), to the end of the section.

P.144, line 6:  Hausdorff  should be  T1

P.176, Exercise 6.5:4(i):  after  Show, however, that  add  , in general,.  (In other words, such choices may exist for some  K;  what you are to prove is that they do not exist for all  K.)

P.182, 2nd paragraph after “Examples”:  in the next-to-last line, interchange  V  and  W.  (But in the final line,  W  and  V  are correct as they stand!)

P.203, second line of Exercise 6.9:6(iv):  of the full subcategory  should be  of the identity functor of the full subcategory

P.218, last line:  Change small limits and colimits to small products and coproducts, pushouts, pullbacks, equalizers and coequalizers, all of which can be constructed “objectwise”.  (The general concepts of limit and colimit, of which the constructions named in the new wording are special cases, will be defined in section 7.6.)

P.220, line 4:  isomorphic to  D should be  isomorphic to  Dop

P.222, proof of Lemma 7.3.2:  As one of you pointed out, the proof that given a morphism of functors  a  as in (7.3.3), the family of maps  α(D)  that we construct from it comprises a morphism of functors (7.3.4) is not as straightforward as I thought.  Given a morphism  D1D2  in  D,  we need to show that the square with horizontal arrows  α(D1): FU(D1) → D1  and  α(D2):  FU(D2) → D2,  and  vertical arrows  FU()  and  f,  commutes.  To do this, consider the following three instances of  a

C(U(D1), U(D1)) → D(FU(D1), D1)

C(U(D1), U(D2)) → D(FU(D1), D2)

C(U(D2), U(D2)) → D(FU(D2), D2)

and connect these by the commuting diagrams coming from the fact that  a  is a bifunctor. This means putting in a pair of arrows joining the first of the above arrows to the middle one, given by left composition with  U(f )  and  f   respectively, and a pair going upward from the bottom arrow to the middle one, given by right composition with  U(f )  and  FU(f )  respectively.  Now the top-left hom-set contains the identity element of  U(D1),  and the bottom-left hom-set contains the identity element of  U(D2).  Under the maps described, these both go to  U()  in the middle left hom-set, and hence have the same image in the middle right hom-set.  If we follow the images of these identity morphisms around the two commuting squares, we find that the equality of their images in the middle right hom-set (together with the definition of  α)  says that  α(D1FU() = α(D2),  as required.

P.230, beginning of paragraph before Exercise 7.4:2: In the symbol  Zp  (sorry, I don't know how to do blackboard-bold in html, so I'm substituting ordinary boldface here), the  Z  should have a ^ over it, and the  p  should be  (p),  as shown four lines earlier.

P.235, last line of Lemma 7.5.3:  (j > ...)  should be  (j ≥ ...)

P.240, Exercise 7.5:19:  on the 4th line, change  Zpi+1  to  pi+1Z,  and at the beginning of the next line, change  Zpi  to  piZ

P.244, statement of Proposition 7.6.3:  In the first line, after be categories, add with  D  small.  Then add after the proposition:

(Above, we assumed  D  small so that  CD  would be a legitimate category, to be sure that  CD(Δ(-), F)  would take values in the category  Set  of small sets.  If we are interested in the case where  D  is merely assumed legitimate, we can apply the above result in any universe  U'  larger than  U,  replacing  Set  in the statement with  Set(U').)

Pp.249-250: Replace Observation 7.8.2 with the following (hopefully clearer) version:

Observation 7.8.2.  Suppose that for each universe  U  we are given a condition  P(U)  on functors between U-legitimate categories, such that

(i)  for every universe  U,  all functors  F  between U-legitimate categories that satisfy  P(U)  respect U-small limits (respectively, U-small colimits; or U-small limits or colimits of a particular sort, such as products or coproducts), and

(ii)  functors  F  satisfying  P(U)  also satisfy  P(U')  for all universes  U' ⊇ U(The commonest case will be where  P(U)  does not refer to  U;  e.g., where it is a condition such as “F  is a right adjoint functor”.)
Then all functors  F  satisfying  P
(U)  in fact respect U-large limits (respectively, U-large colimits, products, coproducts, etc.)

Hence, in discussing properties  P  which are preserved, in the sense of (ii) above, under enlarging the (usually unnamed) universe, if we make assertions that all functors  F  satisfying  P  “respect limits” etc., we need not specify “small” or “large”.

Proof.  Regard any U-large limit as a U'-small limit for a universe  U' ⊇ U,  and apply (i) with  U'  in the role of  U.    ☐

Pp.249-250:  Shorten to large the eight occurrences of the phrase possibly large (namely, the three in the 2nd paragraph of Definition 7.8.1, the two in the 3rd paragraph thereof, the one in the sentence before Observation 7.8.2, and the one in the sentence after that Observation.  I don't mention the three in the original version of that Observation, since they were implicitly eliminated by the preceding change.)  The reason is that “large” means “not necessarily small”; so “possibly” is not needed.

P.256, end of first sentence:  need not turn small colimits into limits should be need not turn small limits into colimits

P.265, Exercise 7.10:4, near end of 2nd line:  Z∈Ob(D)  should be  Z∈Ob(C)

P.273:  If you want to see a more detailed discussion of how binary symmetries lead to three sorts of adjunctions, sketched in the two middle paragraphs on this page, I have a write-up here.  (It assumes familiarity with the concept of the 2-sided orbit-space  \ G / K  of a group  G  by two subgroups  H  and  K.)

P.278, Lemma 8.1.9:  Change the words after  may be constructed  to  as  (B/E, q),  where  E  is the congruence on  B  generated by  {(f(x), g(x)) | x∈|A|},  and  q: B → B/E  is the canonical map.

Exercise 8.1:7 .  Let  Ω  be a type with finitely many operations, all finitary.

If  A  and  B  are finitely generated Ω-algebras, must the Ω-algebra  A × B  be finitely generated?

(If the answer is yes, you might examine whether this result needs the full assumptions on  Ω  made above, while if it is no, you might look for conditions on finitely generated  A  and  B  that are sufficient for their direct product to be finitely generated.)

P.281, 4th line above Exercise 8.2:1:  (x1, ... , xn, ...)  should be  (x0x1, ... , xn, ...)

P.287, last sentence of Exercise 8.3:8:  after  every finitely related Ω-algebra  add  (i.e., every Ω-algebra that has a presentation  < X | Y >  with  Y  finite)

P.295:  change condition (iii) of Exercise 8.4:6 to:  Are the implications from some point on in the above chain all reversible?

P.299, Exercise 8.5:3(ii):  delete the words  in finitely many variables

P.305, Exercise 8.6:8:  The display at the end of the exercise should end with  |Frather than  F.

P.308, 2nd line of Exercise 8.7:2:  B(L)  should be  B(A)

P.311, 5th line below display:  2.4:2(i) should be 2.4:2(ii).

P.314, 3rd and 4th lines above Definition 8.9.1:  < γ0  should be  ≤ γ0.  Also make the same correction on p.317, last two lines of Definition 8.9.4.

P.328, beginning of paragraph before the Exercise:  Lemma 8.10.6(ii)  should simply be  Lemma 8.10.6

P.341, second line above diagram:  replace the words  and let  with

and  s  and  t  are evaluated on this Z-tuple using the co-V structure of  R  assumed in (ii).  Let

P.346, second line of third paragraph of Exercise 9.5:5:  for clarity, change  define  V(A)  = ...  to  define  V(A)  to be the hom-set ...

P.351, line after (9.4.1):  finite coproducts should be  <γ-fold coproducts.

P.361, end of first line of Lemma 9.8.21:  for clarity, change  object of  K-Mod  R  to  object  R  of  K-Mod

P.365, third line of Exercise 9.9:3:  K-Mod → L-Mod  should be  L-Mod → K-Mod

P.367, 2nd line of Exercise 9.10:2(v):  no nontrivial functors  should be  no nontrivial representable functors

P.417, description of Exercise 4.2:9: Delete the words ; & 2 open q’s.  (I don't know what open questions I once had in mind; but none are stated in this exercise.)