Date: Wed, 25 Apr 90 20:00:33 EDT From: INHB000 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} Date: Tue, 15 May 90 08:46:58 EDT From: INHB000 To: Subject: Dear Bob: In addition to the new errata from Jim Otto that I got a copy of this morning, I also recd directly the following, that you may want to distribute to the net. I HAVE NOT CHECKED OUT EITHER ONE, although Charles says the two errors reported below are correct(ly identified as errors). The sales having fallen to nearly infinitesimal, it is unlikely that there will ever be a second edition to include these errata in. =============================================================== I noted your errata for Toposes, Triples, and Theories, and apparently the errors I detected below were not on your list: 1. Page 27, second line from bottom: "T:S -> T" should be t:S -> T. 2. Page 53, middle of page: using "." as the composition operator and "n" for eta, the derivation should be i(T,LA)(L(f.h)) = nA.f.h = RLf.nD.h = RLf.i(T,LD)(Lh) = i(T,LA)(Lf.Lh) . - Dwight Spencer Dept. of Computer Science and Engineering Oregon Graduate Institute =============================================================== Meantime, Category Theory for Computing Science ought to be out about now. Cheers, Mike Date: Wed, 23 May 90 20:21:38 EDT From: INHB000 Vaughn Pratt has discovered something missing from the Atrianglepair and Vtrianglepair macros. Anyone who wants the corrections should write to me. There is no point in trying to send it to you since they would get screwed up in transit. I must say that the phrases ``dependent sum'' and ``dependent product'' really meant nothing to me and Larry Paulson's note on ``dependent product'' and ``dependent function'' was a revelation to me. So I would advocate going back to those old terms, for what my opinion is worth. Michael Barr inhb@mcgillb.bitnet inhb@musicb.mcgill.ca Date: Thu, 24 May 90 15:55:30 BST From: Charles.Wells@ prg.oxford.ac.uk Subject: Dependent Products and Sums Here is a short note for the bulletin board: I agree with Mike Barr that "dependent function" and "dependent product" express exactly what is going on and should be the terminology used. It is a THEOREM that a dependent product is a kind of sum, generalizing the well known fact that in sets A\x B is a sum of A copies of B. A similar remark applies to dependent functions. However, the introduction of the names "dependent product" for dependent function and "dependent sum" for "dependent product" has expelled us permanently from Paradise. The confusion would be as bad as it has been for the notation for set inclusion, which was irretrievably messed up when in the sixties people started using the sideways horseshoe to mean "included in but not equal to" instead of merely "included in", which had been the standard meaning for fifty years. What about "indexed product" and "indexed sum"? Date: Sun, 27 May 90 19:05:38 EDT From: INHB000 Dear Bob: How do you feel about whole papers. Charles and I have a joint paper of which I attach the title and (most of) the introduction. Perhaps it is just as well to distribute this much and say that anyone who wants the whole paper, either by email or pmail should write to me. Regards, Mike =========================================== On the limitations of sketches Michael Barr and Charles Wells Introduction Sketches, as described for example in ?Barr and Wells, 1985?, can be used to describe many, but not all, kinds of mathematical structure. Recently Wells ?1990? has described an extension of the notion to allow more powerful constructors than those given by limits and colimits to be used to described structures. This raises the question of exactly what can be sketched with an ordinary sketch. A remarkable theorem of Makkai and Par\'e ?1990? (and discovered independently by Lair) says that a category is sketchable if and only if it is accessible. This means that for some cardinal $\kappa$ the category has colimits of all $\kappa$ filtered diagrams and that every object is a $\kappa$ filtered colimit of $\kappa$ presentable objects. An object $C$ of a category is $\kappa$ presentable if the functor $\Hom(C,-)$ preserves the colimits of $\kappa$ filtered diagrams. Since in practice one can usually decide quite easily if a category is accessible, this gives a usable criterion for sketchability, without, unfortunately, giving any idea how to sketch certain theories. Consider the category of categories with finite limits and functors that preserve them. We are not supposing canonical finite limits; the functors are merely required to take a finite limit diagram in the source to some finite limit diagram over the same base in the target. At first, it would seem that a theory to describe the set of all finite limit cones in a category would require a universal quantifier, and thus would not be sketchable. On the other hand, it is easy to see that the category of these categories with finite limits and functors that preserve them is $\aleph_0$ accessible and therefore from the theorem mentioned previously is sketchable. In this paper we actually exhibit a simple sketch for that category. A similar argument works for the category of categories with weak finite limits (a cone is a weak limit if every other cone has at least one arrow to it). We indicate the minor change needed in the argument for the case of weak terminal objects. On the other hand, the related category of categories with sublimits (a cone is a sublimit if every other cone has at most one arrow to it) is not accessible and hence not sketchable. In this case, a universal quantifier or some higher order construct is needed. We show that a universal quantifier suffices. Among other things this shows that there is a class of first order sketches that has more expressive power than that of ordinary sketches. Date: Additional errata (checked out): \item?p.27?, (Dwight Spencer) second line from bottom: T:S\to T should be $t:S\to T$. \item?p. 53? (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?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 {\bf universal arrow} for $R$ and $A$.)'' \item?p. 55? line 4: (Dwight Spencer) change $RLA$ to $RWA$. \item?p. 55? line 14: (Dwight Spencer) Change $Ry$ to $y$. \item ?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).) \item?p. 89} (Jim Otto) Refs Osius ?74, 75? are not in the bibliography (p 337). \item?p. 137? (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. But that theorem is true as it stands. Probably the best is to first point out first that only equalizers and coequalizers are needed and then only the $U$-contractible ones. Finally, if we suppose only the $U$-contractible coequalizers exist (any that exist will be preserved by a functor that has a right 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. 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 5 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?p. 172? 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 $(T,\eta:1\to T,\mu:T^2\to T)$.'' \item?p. 234? 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. \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. Subject: TTT errata Date: Sun, 13-May-90 19:27:40 PDT X-Possible-Reply-Path: sun!portal!cup.portal.com!James_Jim_Otto Thanks for the TTT errata. I asked for it as TTT has been a big influence on me (for studying 1. initial le models of Horn theories, 2. free lcc's over generic objects). Additionally I have found: p 89: Refs Osius ?74, 75? are not in the bibliography (p 337). p 137: Thm 5 seems to lack enough hypotheses to apply prop 4 in its pf. Why is cat C^T exact? p 172: P having a left adjoint does not imply T does. (T = P o op(P). E(A, PP B) nat iso E(A x P B, P 1) nat iso E(P B, P A).) But T having a left adjoint is irrelevant to T being a triple. Also, a few months ago, a Penn grad student claimed to me, without details, that section 7.5 on Freyd's n.n.o. thm contains a subtle bug. Perhaps I will met you at lics in Phil. (1st week) or ctrs in Montreal (2nd week) this june (1st 2 weeks). Thanks Jim Otto ====================================================== Date: Tue, 15 May 90 11:36:22 -0700 From: Dwight Spencer To: barr@linc.cis.upenn.edu Subject: More TTT Errors I noted your indirect response via cat-net. Here are some more for your checking that I noticed since my first message: Errors in TTT Book - 1. Page 53, diagram (7): the vertical arrows should be pointing upward. 2. Page 54, lines 5-6 from the bottom: "... definition of universal arrow ..." - well, this definition does not appear explicitly in Section 1.5 as suggested. It certainly arises from Lemma 3 on page 28, but no hint is given there as to that lemma's translation into universal arrows. 3. Page 55, line 4: the typing of the psi-A weak universal arrow should be "A -> RWA." in order to be consistent with the discussion at that point. 4. Page 55, line 14: "The arrows Ry ..." should be "The arrows y ...". 5. Page 64: the diagram at the bottom should be labeled "(1)". - Dwight Spencer Dept. of Computer Science and Engineering Oregon Graduate Institute Date: Tue, 15 May 90 14:21 CST From: Subject: more errors in TTT To: inhb@musicb.mcgill.ca Dear Michael, Here are two other errors not listed so far (I don't know whether Jim Otto found them): On page 234, neither "Corollary 5 of Section 5.4" nor "Exercise (CANON)" exist. Well, the exercise is not listed on page 339, so I could not find it. Are you going to post Jim's list? (If it was distributed over the network, I didn't get a copy.) Also, if a TeX source exists for the book, you could distribute a second edition in that form, bypassing Springer. I ran into some strange problem when running your error list through LaTeX: all square brackets, left ones as well as right ones, had been replaced by question marks, even though the character check list was ok. After the appropriate modification it ran fine. Sincerely yours, J"urgen Date: Cc: Joseph.Goguen@PRG.OXFORD.AC.UK Subject: commutative diagrams I am writing a book on equational logic and theorem proving for final year undergraduates, and want to include a formal treatment of commutative diagrams. Actually, I have never seen a formal treatment for pasting commutative diagrams, though I would be surprised if none existed. The following is my attempt to find an approach which is both elementary and sufficiently general for the usual applications. (A diagram is of course a consistent labelling of a graph by sets on nodes and functions on edges; set theoretically it is a function, so it makes sense to take unions and intersections of diagrams, although I admit this is excessively concrete.) \bdfn A ?\bf bipath? is a diagram $P$ with given nodes $s,t$ called its ?\bf source? and ?\bf target?, and given paths $p,q$ from $s$ to $t$, such that $P$ is the union of $p,q$ and $p,q$ intersect only at $s$ and $t$. \edfn \blemma Let $P_0$ and $P_1$ be commutative bipaths that intersect in just one path (possibly empty or one point). Then $P = P_0 \cup P_1$ is a commutative diagram. \elemma We can use this in proving the following more general result, by induction on the number of commutative bipaths which intersect $p$ and $q$: \bprop Given a collection $P_i$ for $i \in I$ of commutative bipaths such that each pair intersects in one path (possibly empty or one point), then any two paths $p,q$ in $P = \bigcup_?i \in I? P_i$ which start at the same source and end at the same target form a commutative bipath. \eprop Given the lemma, the proposition has a reasonably nice proof. Unfortunately, I have no way to prove the lemma except by an exhausting case analysis. Thus I would be grateful to hear about an alternative approach; a ``reduction'' to some advanced topic (such as 2-categories) would be interesting to me, but not actually useful for the purpose at hand. With many thanks, Joseph &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Joseph A. Goguen, Professor of Computing Science, Programming Research Group, University of Oxford, 11 Keble Road, Oxford OX1 3QD, United Kingdom. email: goguen@prg.ox.ac.uk ?on the internet -- should also work in the U.K. on janet, but if not, try goguen@uk.ac.ox.prg -- Joseph.Goguen@... also works? phone: 272567 ?my office?; 272568 ?secy?; 273838 ?PRG office?; 272839 ?FAX? from USA, dial 011-44-865-...; from UK, dial (0865)-...