Here are the errata to date. It should be distributed with the message that anyone who wants a copy of catmac.tex should write to me directly. The transmission problems you have make it silly to try to distribute it directly. On the other hand, the comments make it perfectly comprehensible without actually texxing it. Michael \documentstyle{article} \textheight9in \headsep 0in \headheight0in \topmargin0in \textwidth 6.5in \oddsidemargin0in \input catmac \long\def\ig#1{\relax} \def\pf{\par\addvspace{\medskipamount}\noindent Proof.\enskip} \mathchardef\T="0454 \def\o{\circ} \def\op{^{\rm op}} \def\mathrm#1{\expandafter\def\csname#1\endcsname{\mathop{\rm#1}\nolimits}} \def\mathbf#1{\expandafter\def\csname#1\endcsname{\mathop{\rm\bf#1}\nolimits}} \mathrm{id} \mathbf{Cat} \begin{document} \resetparms \begin{center} Corrections to {\it Toposes, Triples and Theories}\\ Michael Barr and Charles Wells\end{center} The corrections are listed by page number. The name in parentheses after the page number shows who told us of the error. \begin{trivlist} \item?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. \item?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 {\bf identity property} if $e\o f= f$ whenever $e\o f$ is defined and $g\o e= g$ whenever $g\o e$ is defined. Then we require that for any element $f$, there is an element $e$ with the identity property for which $e\o f$ is defined and an element $e'$ with the identity property for which $f\o e'$ is defined. \item?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.'' \item?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.'' \item?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 {\bf 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. \item?p. 49? Closing parenthesis missing at end of Exercise (EAPL)(a). \item?p. 75? Geometric morphisms are discussed in Chapter 6, not Chapter 5. \item ?p. 125? Third line from bottom. The word ``morphism'' is repeated. \item?p. 126? (Felipe Gago-Couso). Proposition 1 has an omitted hypothesis. We include here a complete restatement of the proposition and its proof: \item??The following proposition gives one method of constructing morphisms of triples. We are indebted to Felipe Gago-Couso for finding the gap in the statement and proof in the first edition and for finding the correct statement. \noindent{\bf Proposition 1.} {\em In the notation of the preceding paragraphs, let $\sigma:TT'\to T'$ be a natural transformation for which \begin{center} \qtriangle?T'`TT'`T';\eta T'`\id`\sigma? \end{center} \ig{\bfig \eta%T' T'----------\>TT' \ | \ | \ | \ | \ | (3) \ | \id \ |\sigma\l \ | \ | \ | \ | \ | \ | \vvb T' \efig} and \begin{center} \xext=1500 \yext=700 \adjust?`\mu T';`T\sigma;`\sigma;`\mu'? \bfig \putsquare(0,200)?TTT'`TT'`TT'`T';\mu T'`T\sigma`\sigma`\sigma? \putsquare(1000,200)?TT'T'`TT'`T'T'`T';T'\mu% `\sigma T'`\sigma`\mu'? \put(250,0){\makebox(0,0){{\rm(a)}}} \put(1250,0){\makebox(0,0){{\rm(b)}}} \efig \end{center} \ig{\bfig \mu%T' T\mu' TTT'------\>TT' TT'T'------\>TT' | | | | | | | | | | | | (4) T\sigma| \sigma| \sigma%T'| \sigma| | | | | | | | | \v \v \v \v TT'-------\>T' T'T'-------\>T' \sigma \mu' $(a)$ $(b)$ \efig } commute. Let $\alpha = \sigma\o T\eta':T\to T'$. Then $\alpha$ is a morphism of triples.} \pf That (1) commutes follows from the commutativity of \begin{center} \xext=500 \yext=1000 \adjust?`\eta;`\eta';`T\eta';T'`? \bfig \resetparms \putsquare(0,500)?\id`T`T'`TT';\eta`T'`T\eta'`\eta'? \settriparms?0`1`1;500? \putqtriangle(0,0)?``T';`\id`\sigma? \efig \end{center} \ig{\bfig \eta \id------------\>T | | | | | | | | \r \eta'| \r T\eta'| | | | | \v \eta' \v T'----------\>TT' (5) \ | \ | \ | \ | \ | \id \ |\l\sigma \ | \ | \ | \ | \ | \ | \ | \vvb T' \efig } In this diagram, the square commutes because $\eta$ is a natural transformation and the triangle commutes by (3). The following diagram shows that (2) commutes. \begin{center} \xext=2100 \yext=2100 \adjust?`\mu;`T\eta'T;`\sigma;`T'T\eta'? \begin{picture}(\xext,\yext)(\xoff,\yoff) \putmorphism(0,2100)(0,-1)?``T\eta'T?{1400}1l \putmorphism(0,2100)(1,0)?TT`T`\mu?{700}1a \putmorphism(0,2100)(1,-1)?`TTT'`TT\eta'?{700}1l \putmorphism(700,2100)(1,-1)?`TT'`T\eta?{700}1r \put(700,1750){\makebox(0,0){1}} \putmorphism(700,1420)(1,0)?\phantom{TTT'}`\phantom{TT'}`\mu T'?{700}1a \putmorphism(700,1380)(1,0)?\phantom{TTT'}`% \phantom{TT'}`T\sigma?{700}1b \setsqparms?0`1`1`1;700`700? \putsquare(700,700)?TTT'`TT'`TT'TT'`TT'T';`T\eta'TT'``? \putmorphism(700,700)(1,0)?\phantom{TT'TT'}`% \phantom{TT'T'}`TT'\sigma?{700}1a \put(300,1400){\makebox(0,0){2}} \put(950,1050){\makebox(0,0){3}} \settriparms?0`1`0;700? \putbtriangle(1400,700)?``TT';T\eta'T'`id`? \putmorphism(1400,700)(1,0)?\phantom{TT'T'}`% \phantom{TT'}`T\mu'?{700}1a \put(1600,1050){\makebox(0,0){6}} \setsqparms?1`1`0`1;700`700? \putsquare(0,0)?TT'T`\phantom{TT'TT'}`T'T`T'TT';% TT'T\eta'`\sigma T``T'T\eta'? \putmorphism(700,0)(1,0)?\phantom{T'TT'}`% \phantom{T'T'}`T'\sigma?{700}1b \setsqparms?0`0`1`1;700`700? \putsquare(1400,0)?``T'T'`T';``\sigma`\mu'? \putmorphism(700,700)(0,-1)?``\sigma TT'?{700}1m \putmorphism(1400,700)(0,-1)?``\sigma T'?{700}1m \put(300,350){\makebox(0,0){4}} \put(1050,350){\makebox(0,0){5}} \put(1750,350){\makebox(0,0){7}} \end{picture} \end{center} \ig{\bfig \mu TT-------------\>T |\ \ | \ \ | \ \ | \ \ | \ \ | \ \ | \ \ | \ 1 \ | TT\eta'\ T\eta'\ | \ \ | \ \ | \ \ | \ \ | \lr \mu%T' \lr | TTT'-----------\>TT' T\eta'T| 2 | -----------\>| \ | | T\sigma | \ | | | \ | | | \ | T\eta'TT'| 3 T\eta'T'| 6 \\$id$\l (6) | | | \ | | | \ \v TT'T\eta' \v TT'\sigma \v T\mu' \lr\l TT'T---------\>TT'TT'-------\>TT'T'----\>TT' | | | | | | | | | | | | | | | | \sigma%T| 4 \sigma%TT'| 5 \sigma%T'| 7 |\sigma\l | | | | | | | | \v \v \v \v T'T---------\>T'TT'---------\>T'T'-----\>T' T'T\eta' T'\sigma \mu' \efig } In this diagram, square 1 commutes because $\mu$ is a natural transformation, squares 2 and 3 because $T\mu'$ is and squares 4 and 5 because $\sigma$ is. The commutativity of square 6 is a triple identity and square 7 is diagram 4(b). Finally, diagram 4(a) above says that $\sigma\o\mu T'=\sigma\o T\sigma$ which means that although the two arrows between $TTT'$ and $TT'$ are not the same, they are when followed by $\sigma$, which makes the whole diagram commute. Squares 1 through 5 of diagram (6) are all examples of part (a) of Exercise (GOD), Section 1.3. For example, to see how square 1 fits, is $\eta'\mu$. \noindent{\bf Corollary 2.} {\em With $\T$ and $\T'$ as in Proposition 1, suppose $\sigma:T' T \to T'$ is such that $\sigma\o T'\eta =\id$, $\sigma\o\sigma T' = \sigma\o \eta T:T\to T'$ is a morphism of triples.} \pf This is Proposition 1 stated in $\Cat\op$ (which means: reverse the functors but not the natural transformations). \item?p. 134? (Colin McLarty). In the second through fourth paragraphs of the proof of (a), every occurrence of ``$L$'' should be ``$W$''. General comment about chapters 4 and 8 (C. Lair). In many places we state that some extension of a functor is unique, when in fact it is only unique up to isomorphism of functors in the functor category. These occur on p. 153 (Theorem 4), p. 156 (Theorem 2), and implicitly in p. 293, Theorem 2 and p. 294, Theorem 1. \item?p. 146.? C. Lair has told us that Ehresmann proved a more general form of Kennison's Theorem in Ehresmann ?1967a?, ?1967b?. \item?p. 162? (C. Lair). The sketch for LE categories constructed here has LE categories with specified limits of finite diagrams as models, and morphisms of models are functors which preserve the specified limits. A similar remark should be made about the sketch for toposes on p. 165. \item?p. 214? The reference, third line from bottom, to section 6.4 should be to section 7.3. \item?p. 233? ``Epimorphic family'' should be boldface and indexed. \item?p. 242? ``Cocontinuous'', in Theorem 12, was not defined. A cocontinuous functor is one which preserves all colimits. \item?p. 250? Fourth line is broken. \item?p. 250? Theorem 7 is referred to several times elsewhere as Freyd's embedding theorem, and should be named as such here. \item?p. 261? second line from bottom: Freyd's Theorem is Theorem 7 of 7.1, not Theorem 5 of 7.2. \item?p. 293? (Peter Johnstone). In Exercise (TOTO), the maps should be strictly increasing rather than nondecreasing. \item?p. 294? We should point out the connections between Theorem 1 here and Theorem 12, p. 242 and Theorem 2, p. 263. \item?p. 295? second line before exercises. ``Function'' misspelled. \item?p. 295? (Peter Johnstone). The description of the realizability topos is completely incorrect; in particular, the realizability topos is not a classifying topos, so the reference does not belong here at all. The reference which {\it does} belong here is to Mulry, \item?p. 296? Same change for Exercise (DLO) as for Exercise (TOTO) above. \item?p. 297? (Peter Johnstone and many others). Theorem 1 omits the very important fact that models of geometric theories have filtered colimits. \item?p. 300? The statement on line 6 that filtered colimits of regular functors are regular deserves some discussion, or at least should be made an exercise! \item?p. 301? In connection with the first sentence beginning on this page, we now know that the category of orthodox semigroups and their morphisms is the category of models of an LE-sketch and is regular, but is not the category of models of an FP-sketch. (An orthodox semigroup is one in which the product of idempotents is an idempotent.) Details in a forthcoming paper by Wells. \item?p. 302? (Peter Johnstone). Because models of geometric theories preserve filtered colimits (see correction to p. 297), the answer to Exercise (CYCGRP)(c) is easily seen to be: No. \item?p. 307? diagram (5). The two rightmost arrows lack labels. The one from $UB$ to $C$ is $c$ and the one from $C$ to $UB$ is $s$. \item?p. 318? (Colin McLarty). Exercise (DL) should say that all composites {\it of length three} are the identity. \item?p. 325? (Peter Johnstone). In line 15, $(R:C)$ is not a full subcategory of the comma category $(R,C)$.\end{trivlist} INDEX, pp. 342ff? Some omissions: {\obeylines Beck's Tripleability Theorem, 112. Butler's Theorem, 135. epimorphic family, 233. Freyd's Representation Theorem, 246ff. Freyd's characterization of natural number objects, 273.}\vskip1ex \section*{SUPPLEMENTAL BIBLIOGRAPHY} \noindent C. Ehresmann, Probl\`emes universels relatifs aux cat\'egories n-aires. C.R.A.S. 264 (273-276), 1967a.\vskip1ex \vskip1ex\noindent C. Ehresmann, Sur les structures alg\'ebriques. C.R.A.S. 264 (840-843), 1967b. \end{document}