Dear Bob: After your last note, I decided to do the following. I have changed all left square brackets to `\lb ' (note the space), all right square brackets to `\rb ', all carets to `\sp ' and all underlines to `\sb '. There were no tildes, but I would have changed them to `\spc '. I suggest you send it out just as you get it, since there are some people who don't get it properly. For the time being, it would appear that you might solicit all TeX things in that form. The \sp and \sb are actually TeX macros. The other two have to be changed back before TeXxing because they are used to delimit macro arguments and no subbing is allowed in that context. I have also removed the \input ct and replaced it by the definitions needed, so that this will compile for anyone who has catmac.sty. Regards, Mike \documentstyle\lb catmac\rb {article} \textheight8in \headsep 0in \headheight0in \topmargin0in \textwidth 6.5in \oddsidemargin0in \long\def\ig#1{\relax} \def\pf{\par\addvspace{\medskipamount}\noindent Proof.\enskip} \mathchardef\Bsc="0242 \let\imp=\Rightarrow \let\meet=\wedge \mathchardef\T="0454 \def\lnm#1{Lecture Notes in Mathematics {\bf#1}} \def\o{\circ} \def\op{\sp {\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} \mathrm{Hom} \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\lb GENERAL COMMENT\rb 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\lb p. 9\rb (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\lb p.27\rb (Dwight Spencer), second line from bottom: $T:S\to T$ should be $t:S\to T$. \item\lb pp. 39-40\rb (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\lb p. 43\rb (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\lb p. 49\rb (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\lb p. 49\rb Closing parenthesis missing at end of Exercise (EAPL)(a). \item\lb p. 53\rb (Dwight Spencer) The display in the middle of the page should be: $i(T,LA)(L(f\o h)) = \eta A\o f\o h = RLf\o \eta D\o h = RLf\o i(T,LD)(Lh) = i(T,LA)(Lf\o Lh)$ In diagram (7), just below, the vertical arrows should be pointing upward. \item\lb p. 54\rb 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 {\bf universal arrow} for $R$ and $A$.)'' \item\lb p. 55\rb line 4: (Dwight Spencer) change $RLA$ to $RWA$. \item\lb p. 55\rb line 14: (Dwight Spencer) Change $Ry$ to $y$. \item \lb p. 64\rb (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).) \item\lb p. 75\rb Geometric morphisms are discussed in Chapter 6, not Chapter 5. \item \lb p. 125\rb Third line from bottom. The word ``morphism'' is repeated. \item\lb p. 126\rb (Felipe Gago-Couso). Proposition 1 has an omitted hypothesis. We include here a complete restatement of the proposition and its proof: \item\lb \rb 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\lb T'`TT'`T';\eta T'`\id`\sigma\rb \end{center} \ig{\bfig \eta%T' T'----------\>TT' \ | \ | \ | \ | \ | (3) \ | \id \ |\sigma\l \ | \ | \ | \ | \ | \ | \vvb T' \efig} and \begin{center} \xext=1500 \yext=700 \adjust\lb `\mu T';`T\sigma;`\sigma;`\mu'\rb \bfig \putsquare(0,200)\lb TTT'`TT'`TT'`T';\mu T'`T\sigma`\sigma`\sigma\rb \putsquare(1000,200)\lb TT'T'`TT'`T'T'`T';T'\mu% `\sigma T'`\sigma`\mu'\rb \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\lb `\eta;`\eta';`T\eta';T'`\rb \bfig \resetparms \putsquare(0,500)\lb \id`T`T'`TT';\eta`T'`T\eta'`\eta'\rb \settriparms\lb 0`1`1;500\rb \putqtriangle(0,0)\lb ``T';`\id`\sigma\rb \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\lb `\mu;`T\eta'T;`\sigma;`T'T\eta'\rb \begin{picture}(\xext,\yext)(\xoff,\yoff) \putmorphism(0,2100)(0,-1)\lb ``T\eta'T\rb {1400}1l \putmorphism(0,2100)(1,0)\lb TT`T`\mu\rb {700}1a \putmorphism(0,2100)(1,-1)\lb `TTT'`TT\eta'\rb {700}1l \putmorphism(700,2100)(1,-1)\lb `TT'`T\eta\rb {700}1r \put(700,1750){\makebox(0,0){1}} \putmorphism(700,1420)(1,0)\lb \phantom{TTT'}`\phantom{TT'}`\mu T'\rb {700}1a \putmorphism(700,1380)(1,0)\lb \phantom{TTT'}`% \phantom{TT'}`T\sigma\rb {700}1b \setsqparms\lb 0`1`1`1;700`700\rb \putsquare(700,700)\lb TTT'`TT'`TT'TT'`TT'T';`T\eta'TT'``\rb \putmorphism(700,700)(1,0)\lb \phantom{TT'TT'}`% \phantom{TT'T'}`TT'\sigma\rb {700}1a \put(300,1400){\makebox(0,0){2}} \put(950,1050){\makebox(0,0){3}} \settriparms\lb 0`1`0;700\rb \putbtriangle(1400,700)\lb ``TT';T\eta'T'`id`\rb \putmorphism(1400,700)(1,0)\lb \phantom{TT'T'}`% \phantom{TT'}`T\mu'\rb {700}1a \put(1600,1050){\makebox(0,0){6}} \setsqparms\lb 1`1`0`1;700`700\rb \putsquare(0,0)\lb TT'T`\phantom{TT'TT'}`T'T`T'TT';% TT'T\eta'`\sigma T``T'T\eta'\rb \putmorphism(700,0)(1,0)\lb \phantom{T'TT'}`% \phantom{T'T'}`T'\sigma\rb {700}1b \setsqparms\lb 0`0`1`1;700`700\rb \putsquare(1400,0)\lb ``T'T'`T';``\sigma`\mu'\rb \putmorphism(700,700)(0,-1)\lb ``\sigma TT'\rb {700}1m \putmorphism(1400,700)(0,-1)\lb ``\sigma T'\rb {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\lb p. 134\rb (Colin McLarty). In the second through fourth paragraphs of the proof of (a), every occurrence of ``$L$'' should be ``$W$''. \item\lb p. 137\rb (Jim Otto). Theorem 5 has too few hypotheses and Proposition 4 too many to apply the latter in the proof of the former. There are various possibilities of getting it right, including adding the finite limits and colimits to the hypotheses of Theorem 5. Nevertheless, Theorem 5 is true as it stands. The problem occurs with the last sentence of Theorem 5. Probably the best approach is to point out first that only equalizers and coequalizers are needed and then only the $U$-contractible ones. Then, if we suppose only the $U$-contractible equalizers exist (any that exist will be preserved by a functor that has a left adjoint), that is enough to do Theorem 5 (Exercise, using the fact that the underlying functor of a tripleable functor creates limits) and from that will follow that the $U$-contractible equalizers exist. As for Proposition 4, if we suppose that $\Bsc$ has $U$-contractible equalizers, the dual of Theorem 5 would imply that $U$-contractible equalizers exist. Hence sufficient for Proposition 4 is that {\em either\/} $U$-contractible equalizers or $U$-contractible coequalizers exist. Does anyone want to find an example to show that any completeness hypothesis is necessary? \item\lb p. 139\rb (Samuel Eilenberg) The story told at the end of the second paragraph is not exact. In fact the word was chosen as the manuscript was in final preparation. Nonetheless, the main the point of the story, the care that went into choosing the name of an important concept, remains. 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\lb p. 146.\rb C. Lair has told us that Ehresmann proved a more general form of Kennison's Theorem in Ehresmann \lb 1967a\rb , \lb 1967b\rb . \item\lb p. 162\rb (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\lb p. 172\rb Second sentence: $\P$ having a left adjoint does not imply $T$ does. The sentence and following one should be combined as follows: ``Since $\T$ is the composite of a functor and its right adjoint, it is the functor part of a triple $(T,\eta:1\to T,\mu:T\sp 2\to T)$.'' \item\lb p. 197\rb Lemma 3 (Francisco Marmolejo). Dropping the subscript on the arrows leaves it too ambiguous. Part (a) should read, ``$D\meet(B\imp\sb AC)=D\meet B\imp\sb DD\meet C$.'' \item\lb p. 200\rb , line $-4$ (Francisco Marmolejo). The sentence should be expanded to, ``Observe that any sheaf is $\j$-closed inside any separated presheaf. ({\bf Separated} isn't defined until page 233, but it simply means that in the definition of sheaf in the moddle of page 69, the unique existence is replaced by the assumption that there is at most one element $x$ such that $X|U\sb i=x\sb i$.)'' \item\lb p. 205\rb (Francisco Marmolejo). In the last line of Exercise TPPB, change ``induce'' to ``constitutes''. \item\lb p. 213\rb , last line (Francisco Marmolejo). Should read, ``products commute with reflexive coequalizers.'' In fact, a product of a fixed object with any coequalizer is a coequalizer. This fact follows from Lemma 6 on page 158-159. \item\lb p. 214\rb The reference, third line from bottom, to section 6.4 should be to section 7.3. \item\lb p. 233\rb ``Epimorphic family'' should be boldface and indexed. \item\lb p. 234\rb second paragraph (J\"urgen Kozlowski): Change Corollary 5 of Section 5.4 to Corollary 8 of Section 5.3. Also, Exercise CANON doesn't exist; delete the reference. \item\lb p. 242\rb ``Cocontinuous'', in Theorem 12, was not defined. A cocontinuous functor is one which preserves all colimits. \item\lb p. 250\rb Fourth line is broken. \item\lb p. 250\rb Theorem 7 is referred to several times elsewhere as Freyd's embedding theorem, and should be named as such here. \item\lb p. 261\rb second line from bottom: Freyd's Theorem is Theorem 7 of 7.1, not Theorem 5 of 7.2. \item\lb p. 293\rb (Peter Johnstone). In Exercise (TOTO), the maps should be strictly increasing rather than nondecreasing. \item\lb p. 294\rb We should point out the connections between Theorem 1 here and Theorem 12, p. 242 and Theorem 2, p. 263. \item\lb p. 295\rb second line before exercises. ``Function'' misspelled. \item\lb p. 295\rb (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\lb p. 296\rb Same change for Exercise (DLO) as for Exercise (TOTO) above. \item\lb p. 297\rb (Peter Johnstone and many others). Theorem 1 omits the very important fact that models of geometric theories have filtered colimits. \item\lb p. 300\rb 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\lb p. 301\rb 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\lb p. 302\rb (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\lb p. 307\rb 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\lb p. 318\rb (Colin McLarty). Exercise (DL) should say that all composites {\it of length three} are the identity. \item\lb p. 325\rb (Peter Johnstone). In line 15, $(R:C)$ is not a full subcategory of the comma category $(R,C)$.\end{trivlist} INDEX, pp. 342ff\rb 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. \vskip1ex\noindent (Two missing references noted by J\"urgen Kozlowski): \vskip1ex\noindent G. Osius, Categorial set theory: a characterization of the category of sets. J. Pure Applied Algebra, {\bf 4} (1974), 79--119. \vskip1ex\noindent G. Osius, Logical and set theoretical tools in elementary topoi. Model Theory and Topoi, \lnm{445} (1975), 297--346. \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle\lb 12pt,catmac\rb {article} \newcommand{\afour}{\oddsidemargin 18pt\evensidemargin 18pt \textwidth 420pt\textheight 39\baselineskip\advance\textheight by \topskip} %% Kludge to make article.sty width correct for A4 %% This only works for article.sty using the standard margin given \textheight8in \headsep 0in \headheight0in \topmargin0in \textwidth 6.5in \oddsidemargin0in \mathchardef\Csc="0243 \def\lncs#1{Lecture Notes in Computer Science {\bf#1}} \def\springer{Sprin\-ger-Ver\-lag} \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}} \mathbf{Set} \mathbf{Rel} \mathrm{Hom} \def\op{{}\sp {\rm op}} \let\x=\times \begin{document} %\afour \title{Category Theory for Computing Science: \\ Update} \author{Michael Barr and Charles Wells} \maketitle Our text, {\bf Category Theory for Computing Science}, Prentice Hall, 1990 (ISBN 0-13-120486-6) has just been published. We intend to maintain the text in the sense that programmers maintain their programs, by keeping a list of known errors and also of additions to the text that we think might be useful. (The latter will probably come in spurts as we go to meetings and find out about wonderful new applications of category theory to computing science.) We will periodically announce new errata and addenda on the category theory bulletin board. The latest TeX version of the complete list is available by e-mail or p-mail from either author. The update consists of three parts: \ref{errors}: A list of errors discovered so far. \ref{newtext}: Additional examples, problems and pointers to the literature. \ref{refs}: Additions to the list of references, pp. 417ff. of the text. Page references refer to the text. Any further corrections or suggestions for additional text will be welcome. {\footnotesize \begin{center}\begin{tabular}{ll} Michael Barr & Charles Wells \\ Department of Mathematics & Department of Mathematics \\ Burnside Hall & Case Western Reserve University \\ McGill University & University Circle \\ 805 Sherbrooke St. West & Cleveland, OH 44106-1712, USA \\ Montr\'eal, P. Q., Canada H3A 2K6 & WELLS@CWRU.BITNET \\ INHB@MUSICB.MCGILL.CA & \end{tabular}\end{center} } \newpage \section{Errors}\label{errors} \begin{description} \item\lb p. xv\rb In the Chapter Dependency Chart, there should be a diagonal line from Chapter 5 to Chapter 7. \item\lb p. 92\rb In Proposition 4.3.12 and its proof, the letter $C$ (not the script $\Csc$) is used with two different meanings. This can be corrected by changing $C$ to $S$ in the first line of the proposition, third line (first occurrence only), fourth line (last occurrence only) and in the first line of the proof (second occurrence only). \item\lb p. 96\rb The reference to Seely should be \lb 1987\rb (the entry in the list of references, p. 425, was incomplete and is updated in the list of references below.) \item\lb p. 102\rb ``The'' not ``the'' in line 7. \item\lb p. 302\rb Line $-4ff$ should say, `` In particular, ${\rm\bf Cat}$ is enriched over ${\rm\bf Cat}$ itself, since its hom sets are themselves categories (with the arrows as objects and the natural transformations as arrows) and the hom functors preserve the extra structure.'' \item\lb p.335\rb In the fourth paragraph of the discussion of the internal category of modest sets, the sentence, ``We want to describe this subset as consisting of those relations that are symmetric and transitive'', should have the following phrase added: ``and double negation closed''. \item\lb p. 345\rb Line 2 of the answer to exercise 12.a: the last letter should be ``B'', not ``b''. \item\lb p. 365\rb In the answer to problem 8, $\beta C:\Hom(C,C)\to F(C)$. \item\lb p. 431\rb The word ``relation'' should also be indexed on p. 21. \end{description} \section{Additional text}\label{newtext} \begin{description} \item\lb p. xiii\rb (Second paragraph) Another collection of papers is \lb Pitt, Rydeheard, Dybjer, Pitts and Poign\'e, 1989\rb . \item\lb p. 17\rb \lb Additional example of category.\rb Let $\alpha$ be a relation from a set $A$ to a set $B$ and $\beta$ a relation\index{relation} from $B$ to $C$ (see 1.3.4). The {\bf composite} $\beta\o\alpha$ is the relation from $A$ to $C$ defined as follows: If $x\in A$ and $z\in C$, $(x,z)\in\beta\o\alpha$ if and only if there is an element $y\in B$ for which $(x,y)\in\alpha$ and $(y,z)\in\beta$. With this definition of composition, the {\bf category of sets and relations} has sets as objects and relations as arrows. \item\lb p. 71\rb \lb New exercise\rb Show that the category of sets and relations is equivalent, in fact isomorphic, to its own dual (see 2.6.7). Answer: Let $\Rel$ denote the category of sets and relations. For a relation $\alpha$ from $A$ to $B$, that is, a subset of $A\x B$, let $\alpha\op$ denote the subset $\{(b,a)\mid (a,b)\in A\x B\}$ of $B\x A$. Let $F:\Rel\to\Rel\op$ be the identity on objects and for a relation $\alpha:A\to B$, let $F(\alpha)=\alpha\op$. $F(\alpha)$ is a relation from $B$ to $A$ in $\Rel$, hence a relation from $F(A)=A$ to $F(B)=B$ in $\Rel\op$. It is easy to check that if $\beta:B\to C$, then $F(\beta\o\alpha)= \alpha\op\o\beta\op$ in $\Rel$, so $(\beta\o\alpha)\op=\beta\op\o\alpha\op$ in $\Rel\op$. This says $F(\beta\o\alpha)=F(\beta)\o F(\alpha)$, so $F$ preserves composition. The identity relation on $A$ is $\Delta\sb A= \{(a,a)\mid a\in A\}$, so $\Delta=\Delta\op$ and $F$ preserves identities. Since for any relation $\alpha$, $(\alpha\op)\op=\alpha$, we have $F\o F$ is the identity functor on $\Rel$, so is its own inverse. Hence $F$ is an isomorphism. By Exercise 1, it is therefore an equivalence of categories. \item\lb p. 96\rb The applications of $2$-categories have mushroomed and include \lb Hoare, Ji Feng and Martin, 1990\rb , \lb Moggi, 1989\rb , and \lb Power, 1989\rb in addition to the papers already listed. %% Goguen has told me about some other papers which I will look up. \item\lb p. 97\rb Second paragraph of Section 4.5: In addition to generalizing the Cayley Theorem, the Yoneda Lemma also has as a special case the embedding of a poset into its down-closed subsets. Also: set-valued functors are studied further in Sections 11.2 and 14.4. \item\lb p. 257\rb Besides \lb Coquand, 1988\rb , Moggi \lb 1989\rb also uses the Grothendieck construction in modeling polymorphism. \item\lb p. 287\rb Just before the exercise: See also \lb Ehrhard, 1989\rb . \item\lb p. 318\rb Add comment: We considered presheaves as actions in Section 3.2. They occur in other guises in the categorical and computer science literature, too. For example, a functor $F:A\to\Set$, where $A$ is a set treated as a discrete category, is a ``bag'' of elements of $A$. If $a\in A$, the set $F(a)$ denotes the multiplicity to which $a$ occurs in $A$. See \lb Taylor, 1989\rb for an application in computing science. \item\lb p. 325\rb Goguen \lb 1990\rb has developed a sheaf semantics for object oriented programs. \end{description} \section{Additional references}\label{refs} \fontdimen2\twlrm=4.3pt% space instead of 3.91663 \fontdimen3\twlrm=4.2pt%stretch instead of 1.95831 \fontdimen4\twlrm=1.7pt%shrink instead of 1.30554 \begin{list}{}{\leftmargin 8mm \itemindent -8mm \parsep 0pt \itemsep 0pt plus 1pt} \def\itm{\item\lb \rb } \itm \mbox{T. Ehrhard}, {\it Dictoses}. In {\bf Category theory and computer science}, D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts and A. Poign\'e, editors. \lncs{389}, \springer, 1989. \itm \mbox{J.\ Goguen}, {\it Semantics of concurrent interacting objects}. Preprint, Programming Research Group, Computing Laboratory, Oxford University, Oxford OX1 3QD, England. \itm \mbox{C.\ A.\ R.\ Hoare}, He Jifeng and C. Martin, {\it Pre-adjunctions in order-enriched categories}. Preprint, Programming Research Group, Computing Laboratory, Oxford University, Oxford OX1 3QD, England. \itm \mbox{E. Moggi}, {\it A category-theoretic account of program modules}. In {\bf Category theory and computer science}, D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts and A. Poign\'e, editors. \lncs{389}, \springer, 1989. \itm \mbox{D.\ Pitt}, D. Rydeheard, P. Dybjer, A. Pitts and A. Poign\'e, eds. {\bf Category theory and computer science}. \lncs{389}, \springer, 1989. \itm \mbox{A.\ J.\ Power}, {\it An abstract formulation for rewrite systems}. In {\bf Category theory and computer science}, D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts and A. Poign\'e, editors. \lncs{389}, \springer, 1989. \itm \mbox{R.\ Seely}, {\it Modelling computations: a $2$-categorical framework.} In {\bf Proceedings of the Conference on Logic in Computer Science}, IEEE, 1987. \lb Corrected reference\rb . \itm \mbox{P.\ Taylor}, {\it Quantitative domains, groupoids and linear logic}. In {\bf Category theory and computer science}, D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts and A. Poign\'e, editors. \lncs{389}, \springer, 1989. \end{list} \end{document}