Corrections to Toposes, Triples and Theories Michael Barr and Charles Wells February 28, 1994 The corrections are listed by page number. The name in parentheses after the page number shows who told us of the error. GENERAL COMMENT Our text is intended primarily as an exposition of the mathematics, not a historical treatment of it. In particular, if we state a theorem without attribution we do not in any way intend to claim that it is original with this book. We note specifically that most of the material in Chapters 4 and 8 is an extensive reformulation of ideas and theorems due to C. Ehresmann, J. B´ enabou, C. Lair and their students, to Y. Diers, and to A. Grothendieck and his students. We learned most of this material second hand or recreated it, and so generally do not know who did it first. We will happily correct mistaken attributions when they come to our attention. p. 9 (Peter Johnstone). Exercise (SGRPOID) is incorrect as it stands; a semilattice without identity satisfies (i) through (iii) but is not a category. Condition (iii) must be strengthened to read: Say an element e has the identity property if e ◦ f = f whenever e ◦ f is defined and g ◦ e = g whenever g ◦ e is defined. Then we require that for any element f , there is an element e with the identity property for which e ◦ f is defined and an element e ′ with the identity property for which f ◦ e ′ is defined. p. 26 (D. ˇ Cubri´ c). Property (ii) of Exercise (SUBF) should read “If f : A −→ B, then F (f ) restricted to G(A) is equal to G(f ).” p. 27 (Dwight Spencer), second line from bottom: T : S −→ T should be t : S −→ T . pp. 39-40 (Peter Johnstone). It should be noted that the product of an empty collection of objects in a category must be a terminal object. Then the phrase after the comma on line 4 of p. 40 should read, “which by an obvious inductive argument is equivalent to requiring that the category have a terminal object and that any two objects have a product.” p. 43 (Peter Johnstone). Exercise (PROD)(b) should read: “Show that if a category has a terminal object and all products of pairs of objects, then it has all finite products.” p. 49 (Peter Johnstone). Exercise (FCR) uses the concept of small category without defining it. It is used in the main body of the text on page 66 and later, and “small sketch” occurs on p. 146. A graph or a category is small if its arrows constitute a set. A sketch is small if its graph is small and its cones and cocones constitute a set. In connection with the discussion of foundations on page ix of the Preface, no matter what set theory is used, one is going to have to deal with categories and graphs whose arrows do not constitute sets. p. 49 Closing parenthesis missing at end of Exercise (EAPL)(a). p. 53 (Dwight Spencer) The display in the middle of the page should be: i(T,LA)(L(f ◦ h)) = ηA ◦ f ◦ h = RLf ◦ ηD ◦ h = RLf ◦ i(T,LD)(Lh)= i(T,LA)(Lf ◦ Lh) In diagram (7), just below, the vertical arrows should be pointing upward. p. 54 lines 5 from the bottom: (Dwight Spencer) change “arrow for” to “element of the functor”. Add to the last sentence “(A universal element of Hom(A, R(−)) is called a universal arrow for R and A.)” p. 55 line 4: (Dwight Spencer) change RLA to RWA. p. 55 line 14: (Dwight Spencer) Change Ry to y. p. 64 (Dwight Spencer) The diagram at the bottom should be labeled (1). (I don’t think it is actually referred to, but the diagram numbering in this section begins with (2).) p. 68 (D. ˇ Cubri´ c). The end of line 5 should be Sub(F × E). p. 69 (V. Pratt) The reference to Section 1.4 (line 13) should be to Subsection, “Global elements” on page 24, more precisely the second paragraph. p. 72 (D. ˇ Cubri´ c). In the next to last line, p : LF −→ X . p. 75 Geometric morphisms are discussed in Chapter 6, not Chapter 5. 1