summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/barr/diaxydoc.tex
blob: 07ea061fb6e5803cbe1d30ab5fb562046cf5fec9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342

\documentclass[12pt]{article}
 \input diagxy
 \xyoption{curve}
 \textwidth 6in
 \oddsidemargin 0pt
 \makeindex
\begin{document}
\def\xypic{\hbox{\rm\Xy-pic}}

\title{A new diagram package (Version \thedate)}
\author{Michael Barr\\Dept of Math and Stats\\McGill University\\805
Sherbrooke St. W\\Montreal, QC Canada H3A 2K6\\[5pt] barr@barrs.org}
 \date{}
 \maketitle

\makeatletter
\contentsline {section}{\numberline {1}Why another diagram package?}{2}
\contentsline {subsection}{\numberline {1.1}The latest version}{3}
\contentsline {subsection}{\numberline {1.2}Main features}{3}
\contentsline {subsection}{\numberline {1.3}Compatibility}{3}
\contentsline {subsection}{\numberline {1.4}Errant spaces}{4}
\contentsline {subsection}{\numberline {1.5}Font sizes}{4}
\contentsline {subsection}{\numberline {1.6}On \LaTeX \ 2.09 and \LaTeX \ 2e}{5}
\contentsline {subsection}{\numberline {1.7}Acknowledgments}{5}
\contentsline {subsection}{\numberline {1.8}License}{6}
\contentsline {section}{\numberline {2}The basic syntax}{6}
\contentsline {subsection}{\numberline {2.1}A word about parameters}{7}
\contentsline {section}{\numberline {3}Defined diagrams}{8}
\contentsline {section}{\numberline {4}Examples}{10}
\contentsline {subsection}{\numberline {4.1}Complex diagrams}{19}
\contentsline {subsection}{\numberline {4.2}Empty placement and moving labels}{23}
\contentsline {subsection}{\numberline {4.3}Inline macros}{25}
\contentsline {subsubsection}{\numberline {4.3.1}Added:}{26}
\contentsline {subsection}{\numberline {4.4}2-arrows}{26}
\contentsline {subsection}{\numberline {4.5}Mixing \hbox {\rm \unhbox \voidb@x \hbox {\kern -.1em X\kern -.3em\lower .4ex\hbox {Y\kern -.15em}}-pic}\ code}{27}
\contentsline {subsection}{\numberline {4.6}Diagram from TTT}{29}
\makeatother

\kern 40pt
{\narrower
For the fashion of Minas Tirith was such that it was built on seven levels,
each delved into a hill, and about each was set a wall, and in each wall
was a gate.

-- J.R.R.\ Tolkien, ``The Return of the King''\footnote{These macros are
at the seventh level of nesting.  The first level is the code actually
executed directly by the processor.  The second is the microcode, burned
into the chip, that interprets the assembly language instructions into
the ones that are directly executed.  Level three is the language,
usually C, but originally Pascal, into which the Web code, the fourth
level, is compiled.  The fifth level is \TeX\ itself, which is best seen
as a high level programming language for mathematical text processing.
Level six is the \xypic\ code in which these macros, level seven, are
written.}

}


\kern 5pt
 Note:  This is the first increase in functionality since the original
general release.  The macros \index{\backslash to}\verb.\to. and
\index{\backslash two}\verb.\two. that make inline arrows and pairs of arrows now
accept an option to vary the types of the arrow, for example to get a
pair of adjoints going in opposite directions.  A macro that makes three
arrows has been added as well as a macro for a 3 x 3 diagram that can be
expanded into a map between two exact sequences.  A new optional
parameter has been added to \index{\backslash to}\verb.\to.,
\index{\backslash two}\verb.\two. and \index{\backslash three}\verb.\three. to allow the
user to override the automatic computation of the arrow size.  See the
sections on inline arrows, page~\pageref{added} and
page~\pageref{complex} for details.

\section{Why another diagram package?}

This started when a user of my old package, diagram, wrote to ask me if
dashed lines were possible.  The old package had dashed lines for
horizontal and vertical arrows, but not any other direction.  The reason
for this was that \LaTeX\ used rules for horizontal and vertical arrows,
but had its own fonts for other directions.  While rules could be made
any size, the smallest lines in other directions were too long for
decent looking dashes.  Presumably, Lamport was worried about compile
time and file size if the lines were too short, considerations that have
diminished over the years since.  Also arrows could be drawn in only 48
different directions, which is limiting.  My macros were not very well
implemented for slopes like 4 and 1/4, since I never used such lines.

There was certainly an alternative, \xypic, for those who wanted
something better.  But it was hard to learn and I was not entirely happy
with the results.  The basic interface used an
\index{\backslash halign}\verb.\halign..  This
meant that no extra space was allotted to an arrow that had a large
label.  In addition, the different slots could have different horizontal
size, which could result in a misshapen diagram.  I used it in a paper
that had a `W' shaped diagram whose nodes had different widths and the
result was quite obviously misshapen.  On the other hand, the graphics
engine underlying \xypic\ is really quite remarkable and it occurred to
me that I could try to redraft diagram as a front end.  The result is
the current package.  It has been tested mainly with version 3.7, so
there is no guarantee it would work with any earlier version (or, for
that matter, later).  A limited amount of testing with version 3.6
suggests that the only thing that does not work is the 2-arrows; these
come out vanishingly short.

So despite the desire for logical programming, it still seems that for
some purposes, it is desirable to have a system in which you specify
what goes where and where the arrows are drawn.

\subsection{The latest version}
The latest version can be downloaded at
\verb,ftp://ftp.math.mcgill.ca/pub/barr/diagxy.zip, or by anonymous ftp
from \verb,ftp.math.mcgill.ca//pub/barr/diagxy.zip,.


\subsection{Main features}

\begin{itemize}
 \item A general arrow drawing function
\index{\backslash morphism}\verb.\morphism..
 \item Various common diagram shapes such as squares, triangles, etc.
 \item A few special shapes such as cube and $3\times3$ diagrams.
 \item Small 2-arrows that can be placed anywhere in a diagram, much
like \LaTeX's
picture
 environment.
 \item A uniform syntax, while allowing access to all of \xypic's
capabilities
 \item Never expires and does not request acknowledgment.
\end{itemize}

\subsection{Compatibility}
The syntax described below is not compatible with the original diagram
package.  Every front end of this sort represents a trade-off between
simplicity and utility.  A package that simply upgraded the syntax to
allow more dashed (and dotted) lines would have been just as easy to
implement, but would have made poor use of the wonderful possibilities
of the underlying \xypic\ package.  Also there would have been too many
different arrow specifications (they would have had to go at least in
the range $[-9,9]$) for easy memory) and still would not have included
things like inclusion arrows.  To those who would have liked a simpler
syntax, I apologize.  Those who would want more flexibility, I remind
that the entire \xypic\ package is there for use.

\subsection{Errant spaces}
 There is one point that cannot be made too strongly.  {\em Watch for
errant spaces.} Unlike the old diagram package, which was carried out in
math mode so that spaces were ignored, \xypic\ is not in math mode and
spaces will mess up your diagrams.  On the other hand, it will not be
necessary to enclose duplicate nodes inside
\index{\backslash phantom}\verb.\phantom., since the
registration between different nodes is perfect.  In the old package,
for reasons that I now understand, objects did not always register
properly.  This was a flaw built in to the very heart of the package and
is not worth correcting, although it could have been done better in the
first place.  If you see an object in double vision, then the almost
certain cause is that a space has gotten in there somewhere.  I have
attempted to prevent this by liberal use of
\index{\backslash ignorespaces}\verb.\ignorespaces.  in the
definitions, but one cannot always be sure and while testing, I found a
number seemingly innocuous spaces that messed up diagrams.  When in
doubt, always terminate a specification with a
\index{%}\verb.%..  See the
examples.

\subsection{Font sizes}
 According to the documentation of \xypic, the declarations
\begin{verbatim}
\xyoption{tips}
\SelectTips{xy}{12}
\end{verbatim}
 \index{tips} will cause 12 point tips to be used.  This does not appear
to be the case.  The only way I can seem to get larger font sizes is to
add \index{\backslash fontscale{x}}\verb.\fontscale{x}., with \index{x}\verb.x.
taking on the values \verb.0,h,1,2,3,4,5., after \xypic\ is loaded.
With \verb.11pt. you will want \verb.h. and with \verb.12pt. you will
want 1. The others are in case you use larger sizes for transparencies
or for later reduction.  While on the subject, I might mention that if
you want thicker arrow shafts without enlarging anything else you could
add the declaration
\begin{verbatim}
\font\xydashfont=xydash scaled \magstep1
\end{verbatim}
or even larger.  However, this does not thicken the arrow tips and is
not really recommended.  There is probably no way (short of creating
your own fonts) to thicken the tips without also lengthening them.


\subsection{On \LaTeX\ 2.09 and \LaTeX\ 2e}
 Since I have no other forum to do so, I use this space to indulge in a
personal rant, which you may skip since it has nothing to do with the
package (except periphally, since it explains why I have not made 14,
17, and 20 point style files for \LaTeX\ 2e).  Recent distributions of
\TeX\ have omitted latex.tex, the source file for \LaTeX\ 2.09.  If you
make a posting on comp.text.tex, admitting to using the older version,
you are certain to get one or more replies admonishing you that \LaTeX\
2.09 is obsolete and should not be used.  There are at least a couple of
good reasons for continuing to use the older version (along with some
reasons for switching, especially if you are using a language other than
English).  For one thing, the compatibility mode is useless if you have
used additional fonts, as I have (most importantly, cmssbx, where many
use bbbold, one of the ugliest fonts to see the light of day).  In fact,
the most significant change in \LaTeX\ 2e is that it uses the new font
selection system and the compatibility mode makes no attempt to be
compatible with the old one.  The second reason for not switching is
that, ``If it ain't broke, don't fix it.''  For many the old \LaTeX\
worked fine, why change?  If you want to use new fonts in \LaTeX\ 2e it
is certainly possible, but the instructions for doing so are embedded
holographically in Chapter 7 of The \LaTeX\ Companion and took me days
to decipher.  I am still not convinced that I am doing it right,
although it seems to work.  The third---and to me most important---is
that the file latex.tex is extensively documented internally and it is
relatively easy to make whatever changes one wants.  I have never
changed the source, but always put the changes in another file.  In this
way, I have changed, for example,
\index{\backslash @startsection}\verb.\@startsection.  in various ways.
It is clear that Lamport intended that his version of \LaTeX\ be
user-modifiable.  It is equally clear that the implementers of the new
version had no interest in that.  They didn't go out of their way to
make it hard, but they did nothing to encourage it and the documentation
is entirely on how to use it.

\subsection{Acknowledgments}
 First, I would like to thank Kris Rose for a superb programming job
that made this package possible.  The original versions of \xypic\ were
based on matrices, but this version includes code to place an arrow with
two nodes (more precisely, their centers) a user-definable distance
apart and that is exactly what I needed.  Second I would like to thank
Ross Moore who answered innumerable questions about the use of the
macros that the documentation did not answer or at least I could not
understand what it said.  I thank Donald Arsenau answered several of my
questions about the internals of \TeX\ that I did not understand
clearly and Steve Bloom who found a couple
of typos in the first version of these notes.  Finally, I would like to
thank all those, but especially Charles Wells, who gave me opinions on
the best syntax.  But I made the final decisions and if you have any
complaints, you can direct them to me.  Not that I am likely to change
anything at this date.

\subsection{License}
 The use of this package is unrestricted.  It may be freely distributed,
unchanged, for non-commercial or commercial use.  If changed, it must be
renamed.  Inclusion in a commercial software package is also permitted,
but I would appreciate receiving a free copy for my personal examination
and use.  There are no guarantees that this package is good for
anything.  I have tested it with LaTeX 2e, LaTeX 2.09 and Plain TeX.
Although I know of no reason it will not work with AMSTeX, I have not
tested it.


\section{The basic syntax}

The basic syntax is built around an operation
\index{\backslash morphism}\verb.\morphism.  that is
used as \index{\backslash morphism}
 \begin{verbatim}
 \morphism(x,y)|p|/{sh}/<dx,dy>[N`N;L]
 \end{verbatim}
 The last group of parameters is required.  They are the source and
target nodes of the arrow and the label.  The remaining parameters are
optional and default to commonly used values.  Currently, the
\verb.L.\index{label} is set in \index{\backslash scriptstyle}\verb.\scriptstyle.
size, but this can easily be changed by putting
\index{\backslash let\backslash labelstyle=\backslash textstyle}
\verb.\let\labelstyle=\textstyle. in
your file.

 The parameters \verb.x. and \verb.y. give the location of the source
node within a fixed local coordinate system and \verb-x+dx- and
\verb-y+dy- locate the target.  To be precise, the first coordinate is
the horizontal center of the node and the second is that of the base
line.  These distances are given in terms of \index{\backslash ul}\verb.\ul.'s,
(for unitlength), which is user assignable, but currectly is .01em.  The
placement parameter \verb.p. is one of \verb.a,b,l,r,m. that stand for
above, below, left, right, or mid and describe the positioning of the
arrow label on the arrow.  If it is given any value other than those
five, it is ignored.  We describe below why you might want it ignored.
The label \index{m}\verb.m. stands for a label positioned in the middle
of a split arrow.  When used on a non-vertical arrow, \index{a}\verb.a.
positions the label above the arrow and \index{b}\verb.b. positions it
below.  On a vertical arrow, the positioning depends on whether the
arrow points up or down.  Similarly, when used on a non-horizontal
arrow, \index{l}\verb.l. positions the label to the left and
\index{r}\verb.r. positions it to the right, while on a horizontal arrow
it depends on which way it points.

The shape parameter \verb.sh. describes the shape of the arrow.  An
arrow is thought of as consisting of three parts, the tail, the shaft
and the head.  You may specify just the head, in which case the shaft
will be an ordinary line, or all three.  However, since the tail can be
(and usually is) empty, in practice you can also describe the shaft and
tail.  In addition, it is possible to modify the arrow in various ways.
Although the parameter is shown within braces, the braces can be omitted
unless one of the modifier characters is \verb./., in which case, {\em
the entire parameter} must be put in braces.  It is important to note
that it will not work just to put the \verb./. inside the braces, since
this will interfere with the internal parsing of \xypic.  The head and
tail shapes are basically one of \index{>}\verb.>., \index{>>}\verb.>>.,
\verb.(., \verb.)., \index{>}\verb*+>+, and \index{<}\verb*+<+.  Each of
these may also be preceded by \index{^}\verb.^. or \index{_}\verb._. and
others are user definable.  For details, see the \xypic\ reference
manual.  The first of these is an ordinary head, while the second is for
an epic arrow.  The third is not much used, but the superscripted
version makes and inclusion tail, as will be illustrated below.  The
reverse ones give reversed arrowheads.  The sign \verb*+ + stands for an
obligatory space and it leaves extra space for a tailed (monic) arrow,
which otherwise runs into the source node.  Although there are many
possibilities for shafts, including alphanumeric characters, the ones
that mainly interest us are:  \index{-}\verb.-., which is an ordinary
shaft, \index{--}\verb.--., which produces a dashed arrow,
\index{=}\verb.=., which gives a double arrow (although with only one
arrowhead), and ., which makes a dotted arrow.  Thus \index{>}\verb.>.
or \index{->}\verb.->. will produce an ordinary arrow,
\index{->>}\verb.->>. an epic arrow, \index{ >->}\verb*+ >->+ makes a
monic arrow, and \index{-->>}\verb.-->>. would make a dashed epic arrow.
The descriptions \index{<-}\index{<<-}\verb.<-,<<-,.  \index{<-<
}\verb*+<-< +, and \index{<<--}\verb.<<--. give the reversed versions.
Note that \index{<}\verb.<. does not give a reversed arrow, since
\xypic\ interprets that as a reversed head, not a tail.

If the shape parameter begins with an \index{@}\verb.@., it is
interpreted differently.  In that case, it has the form
\index{@{shape}@}\verb.@{shape}@. modifier, where the modifier is as
described in the \xypic\ reference guide.  I just mention a couple of
them.  The parameter \index{@{->}@<3pt>}\verb.@{->}@<3pt>., for example,
would give an ordinary arrow moved three points in the direction
perpendicular to that of the arrow.  If you give
\index{{@{-->}@/^5pt/}}\verb.{@{-->}@/^5pt/}., you will get an epic
arrow that is curved in the direction perpendicular to the direction of
the arrow by five points.  It is imperative that a specification such as
\index{@{>}@/5pt/}\verb.@{>}@/5pt/. be enclosed in braces because of the
\index{/}\verb./.s.

\subsection{A word about parameters}
 I have already mentioned the necessity of enclosing certain arrow shape
specifications in braces.  Because of the way \TeX\ operates, I have
used a number of different delimiters:  \index{(}\verb.(.,
\index{)}\verb.)., \index{|}\verb.|., \index{/}\verb./.,
\index{<}\verb.<., \index{>}\verb.>., \index{[}\verb.[.,
\index{]}\verb.]., \index{`}\verb.`., and \index{;}\verb.;..  Any of
these that appear inside an argument could conceivably cause problems.
They were chosen as the least likely to appear inside mathematics
(except for \index{(,)}\verb.(,). which appear in positions that are
unlikely to cause problems).  However, be warned that this is a possible
cause of mysterious error messages.  If this happens, enclosing the
offending parameter in braces should cure the problem.  The exceptions
come when the braces interfere with \xypic's somewhat arcane parsing
mechanism.  One place it imperative to use braces is if you attempt to
use a disk as a tip.  Although most tips do not have to be enclosed in
braces, you get a small solid disk tip from \index{{*}}\verb.{*}. and
not from \index{*}\verb.*.  (see the note at the bottom of the first
column of page 42 in the reference manual.  The first three of
\begin{verbatim}
\morphism(0,300)|a|/-*/<500,0>[A`B;f]
\morphism(0,0)|a|/{-*}/<500,0>[A`B;f]
\morphism(0,0)|a|/{*}/<500,0>[A`B;f]
\morphism(0,0)|a|/-{*}/<500,0>[A`B;f]
\end{verbatim}
 give error messages.  Only the fourth works:
 $$\bfig
\morphism(0,0)|a|/-{*}/<500,0>[A`B;f]
 \efig$$

In addition to the diagrams, there are macros that are intended to be
used inline to make horizontal arrows, pointing left or right, plain,
monic, epic, or user-definable shapes, and calculating their own length
to fit the label.  Finally, there is a macro for making short 2-arrows
that may be placed (actually, \index{\backslash place}\verb.\place.d) anywhere in
a diagram.


\section{Defined diagrams}

Using the basic
\index{\backslash morphism}\verb.\morphism.  macro, I have a defined a number of
diagrams:  squares (really rectangles) and variously oriented triangles
and a few compound diagrams.  The basic shapes are exactly the same as
in the old diagram package, but the options are done entirely
differently.  Here is the syntax of the
\index{\backslash square}\verb.\square.  macro:
 \begin{verbatim}
 \square(x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dx,dy>[N`N`N`N;L`L`L`L]
 \end{verbatim}
 Each of the first four sets of parameters is optional and any subset of
them can be omitted.  (Note that only the sets can be omitted, once you
give \index{dx}\verb.dx., you also have to give \index{dy}\verb.dy. and
so on.)  The first set gives the horizontal and vertical position (in a
local coordinate system) of the lower left corner of the square, the
next four give the label placements using the same five characters
previously described.  The next four give the shapes of the arrows using
the same syntax as discussed above.  The last group is horizontal and
vertical size of the square.  More precisely, the \index{x}\verb.x.
coordinate is that of the midpoint of the node, while the
\index{y}\verb.y. coordinate is that of the baseline of the node.  This
is entirely based on \xypic.

In the case of other shapes, discussed below, the positioning parameter
may be different.  The \index{x}\verb.x. coordinate is the midpoint of
the leftmost node and the \index{y}\verb.y. coordinate is the baseline
of the lowest node.  In the case of the
\index{\backslash qtriangle}\verb.\qtriangle.,
\index{\backslash Vtriangle}\verb.\Vtriangle., and
\index{\backslash Ctriangle}\verb.\Ctriangle. described below, these are different
nodes.  What this positioning means is that if you specify the
coordinates and sizes correctly the shapes will automatically fit
together.  The last example on Page~\pageref{TTTdiag} illustrates this.

Here is a listing of the shapes, together with the groups of parameters.
In all cases, the first four groups are optional and any subset of them
will work.  However, they must come in the order given.  Note that the
names of the triangles are related to the shape as the shape that best
approximates the shape of the letter.  For example, a
\index{\backslash ptriangle}\verb.\ptriangle. is a right triangle that has its
hypotenuse going from upper right to lower left.  Triangles with lower
case names have their legs horizontal and vertical and the dimension
parameters are the lengths of the legs.  Those with capitalized names
have their hypotenuse horizontal or vertical.  In those cases, one of
\index{dx}\verb.dx. or \index{dy}\verb.dy. is the length of a leg and
the other is {\em half} the length of the hypotenuse.  In all cases, the
order of the nodes and of the arrows is linguistic, first moving from
left to right and then down.  The defaults are reasonable, but with
triangles, there is not always a natural direction for arrows.  I always
made mistakes in the order with my macros and this is certainly a
defect.  But the order is the same.  In every case the braces around the
shape specification can be removed unless it includes the following
delimiter (that is, \index{`}\verb.`. or \index{/}\verb./., as the case
may be.)
 \begin{verbatim}
\square(x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dx,dy>%
 [N`N`N`N;L`L`L`L]
\ptriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\qtriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\dtriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\btriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\Atriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\Vtriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\Ctriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\Dtriangle(x,y)|ppp|/{sh}`{sh}`{sh}/<dx,dy>[N`N`N;L`L`L]
\Atrianglepair(x,y)|ppppp|/{sh}`{sh}`{sh}`{sh}`{sh}/%
<dx,dy>[N`N`N`N;L`L`L`L`L]
\Vtrianglepair(x,y)|ppppp|/{sh}`{sh}`{sh}`{sh}`{sh}/%
<dx,dy>[N`N`N`N;L`L`L`L`L]
\Ctrianglepair(x,y)|ppppp|/{sh}`{sh}`{sh}`{sh}`{sh}/%
<dx,dy>[N`N`N`N;L`L`L`L`L]
\Dtrianglepair(x,y)|ppppp|/{sh}`{sh}`{sh}`{sh}`{sh}/%
<dx,dy>[N`N`N`N;L`L`L`L`L]
\end{verbatim}
 \index{\backslash square} \index{\backslash ptriangle} \index{\backslash qtriangle} \index{\backslash dtriangle}
\index{\backslash btriangle} \index{\backslash Atriangle} \index{\backslash Vtriangle}
\index{\backslash Ctriangle} \index{\backslash Dtriangle} \index{\backslash Atrianglepair}
\index{\backslash Vtrianglepair} \index{\backslash Ctrianglepair} \index{\backslash Dtrianglepair}
Note that the
\verb.%.  signs are required if you break the
macro at such points.  See also the discussion of errant spaces above.

To make a diagram, you have to enclose it inside \index{\backslash xy ...
\backslash endxy}.\verb,\xy ...  \endxy,.  You will usually want it
displayed,
for which the simplest way is to enclose it in \index{$$\xy ...
\endxy$$}\verb,$$\xy...\endxy$$,.  For old times' sake, I have also let
\index{\backslash bfig}\verb.\bfig. and \index{\backslash efig}\verb.\efig. be synonyms for
\index{\backslash xy}\verb.\xy. and \index{\backslash endxy}\verb.\endxy., resp.  (In case
you wonder, these go all the way back to a main frame formatter running
at McGill when Charles Wells and I were first writing TTT, where
\verb,.BFIG, and \verb,.EFIG, were used to make display---at least as
far as that primitive formatter was capable of.)

\section{Examples}

Many people---including me---learn mainly by example and
I will give a number of examples here.  The formal
syntax that is not given here can be learned in the \xypic\ reference
manual. More samples from an actual paper can be found in
\verb,ftp.math.mcgill.ca/pub/barr/derfun.tex,.  If you want to compile
that paper, you will need \verb,tac.cls,, available from \verb,tac.ca,.
We begin with \index{\backslash morphism}
\begin{verbatim}
 $$\bfig
 \morphism[A`B;f]
 \morphism(0,300)[A`B;f]
 \morphism(0,600)|m|[A`B;f]
 \morphism(0,900)/<-/[A`B;f]
 \morphism(900,500)<0,-500>[A`B;f]
 \morphism(1200,0)<0,500>[A`B;f]
 \efig$$
\end{verbatim}
which gives the diagram
 $$\bfig
 \morphism[A`B;f]
 \morphism(0,300)[A`B;f]
 \morphism(0,600)|m|[A`B;f]
 \morphism(0,900)/<-/[A`B;f]
 \morphism(900,500)<0,-500>[A`B;f]
 \morphism(1200,0)<0,500>[A`B;f]
 \efig$$
\begin{verbatim}
 $$\bfig
\square[A`B`C`D;e`f`g`m]
 \efig$$
\end{verbatim}
 produces
 $$\bfig
\square[A`B`C`D;e`f`g`m]
 \efig$$
 This can be modified, for example
\begin{verbatim}
 $$\bfig
\square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
 \efig$$
\end{verbatim}
 produces
 $$\bfig
\square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
 \efig$$
 This can be put together with a morphism as follows:
\begin{verbatim}
 $$\bfig
\square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
 \morphism(500,500)|m|/.>/<-500,-500>[B`C;h]
 \efig$$
\end{verbatim}
 \index{\backslash square}
 which makes a familiar diagram:
 $$\bfig
\square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
 \morphism(500,500)|m|/.>/<-500,-500>[B`C;h]
 \efig$$
 The same diagram could have been made by
\begin{verbatim}
 $$\bfig
 \ptriangle|alm|/>>`>`.>/[A`B`C;e`f`h]
 \dtriangle/`>` >->/[B`C`D;`g`m]
 \efig$$
\end{verbatim}
\index{\backslash ptriangle} \index{\backslash dtriangle}

There are four macros for making pairs of triangles put together:
 $$\bfig
 \Vtrianglepair[A`B`C`D;f`g`h`k`l]
 \efig$$
 comes from
\begin{verbatim}
 $$\bfig
 \Vtrianglepair[A`B`C`D;f`g`h`k`l]
 \efig$$
\end{verbatim}
 \index{\backslash Vtrianglepair}
 \index{\backslash Atrianglepair}
 \index{\backslash Ctrianglepair}
 \index{\backslash Dtrianglepair}

 The other three are called \verb.\Atrianglepair.,
\verb.\Ctrianglepair., and \verb.\Dtrianglepair..

 You can fit two squares together, horizontally:
\begin{verbatim}
 $$\bfig
\square|almb|[A`B`C`D;f`g`h`k]
\square(500,0)/>``>`>/[B`E`D`F;l``m`n]
 \efig$$
\end{verbatim}
 $$\bfig
\square|almb|[A`B`C`D;f`g`h`k]
\square(500,0)/>``>`>/[B`E`D`F;l``m`n]
 \efig$$
 or vertically
\begin{verbatim}
 $$\bfig
\square(0,500)|alrm|[A`B`C`D;f`g`h`k]
\square/`>`>`>/[C`D`E`F;`l`m`n]
 \efig$$
\end{verbatim}
 $$\bfig
\square(0,500)|alrm|[A`B`C`D;f`g`h`k]
\square/`>`>`>/[C`D`E`F;`l`m`n]
 \efig$$
or a square and a triangle
\begin{verbatim}
 $$\bfig
\Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0]
 \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm
   rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t]
 \efig$$
\end{verbatim}
 gives the diagram
 $$\bfig
\Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0]
 \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm
   rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t]
 \efig$$
 This diagram is on page 361 of the third edition of Category Theory for
Computing Science to describe recursion.
 Here is an example using the procedure for sliding an arrow sideways.
This one could even be used in a text,
 \index{sliding arrows}
$\xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B;d]
\morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B;e]\endxy$
 which was made using
\begin{verbatim}
$\xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B;d]
\morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B;e]\endxy$
\end{verbatim}
 Indidentally, if you don't put this in math mode, the diagram will come
out too low, for reasons I do not understand but must be buried within
the \xypic\ code.
Later we will introduce a number of inline procedures.

Something a bit different that illustrates the use of another shaft

\index{=}\verb.=.  that gives a 2-arrow, as well as curved arrows:
\begin{verbatim}
 $$\bfig
\morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B;f]
\morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B;g]
\morphism(250,50)|a|/=>/<0,-100>[``]
 \efig$$
\end{verbatim}
 $$\bfig
\morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B;f]
\morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B;g]
\morphism(250,50)|a|/=>/<0,-100>[`;]
 \efig$$
 In order to use curved arrows, you must insert
\index{\backslash xyoption{curve}}\verb.\xyoption{curve}.
into your file. Here are two ways of doing three arrows between two
objects, depending on what you like:
\begin{verbatim}
 $$\bfig
 \morphism(0,0)|a|/@{>}@<5pt>/<500,0>[A`B;f]
 \morphism(0,0)|m|/@{>}/<500,0>[A`B;g]
 \morphism(0,0)|b|/@{>}@<-5pt>/<500,0>[A`B;h]
 \efig$$
\end{verbatim}
 which gives
 $$\bfig
 \morphism(0,0)|a|/@{>}@<5pt>/<500,0>[A`B;f]
 \morphism(0,0)|m|/@{>}/<500,0>[A`B;g]
 \morphism(0,0)|b|/@{>}@<-5pt>/<500,0>[A`B;h]
 \efig$$
and
\begin{verbatim}
 $$\bfig
 \morphism(0,0)|a|/{@{>}@/^5pt/}/<500,0>[A`B;f]
 \morphism(0,0)|m|/@{>}/<500,0>[A`B;g]
 \morphism(0,0)|b|/{@{>}@/^-5pt/}/<500,0>[A`B;h]
 \efig$$
\end{verbatim}
 which gives
 $$\bfig
 \morphism(0,0)|a|/{@{>}@/^5pt/}/<500,0>[A`B;f]
 \morphism(0,0)|m|/@{>}/<500,0>[A`B;g]
 \morphism(0,0)|b|/{@{>}@/^-5pt/}/<500,0>[A`B;h]
 \efig$$
 Either of these could also be used inline.

 There is a macro
\index{\backslash place}\verb.\place.
that places that object anywhere.  I have changed the name from
\verb.\put. in order to avoid clashing with the \LaTeX\ picture mode's
\index{\backslash put}\verb.\put..
Here is an example that uses a construction that is undocumented here,
but uses a documented \Xy\ construction:
\begin{verbatim}
\newbox\anglebox
\setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy}
 \def\angle{\copy\anglebox}
 $$\bfig
 \square[A`B`C`D;f`g`h`k]
 \place(100,400)[\angle]
 \efig$$
\end{verbatim}
\newbox\anglebox
\setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy}
 \def\angle{\copy\anglebox}
 $$\bfig
 \square[A`B`C`D;f`g`h`k]
 \place(100,400)[\angle]
 \efig$$
 Notice that you get a headless arrow by using
\index{\backslash ar@{-}}\verb.\ar@{-}..

Here is a special code installed at the request of Jonathon Funk:
\begin{verbatim}
 $$\bfig
\pullback|brrb|<800,800>[P`X`Y`Z;t`u`v`w]%
|amb|/>`-->`>/<500,500>[A;f`g`h]
 \efig$$
\end{verbatim}
 \index{\backslash pullback}
 $$\bfig
\pullback|brrb|<800,800>[P`X`Y`Z;t`u`v`w]%
|amb|/>`-->`>/<500,500>[A;f`g`h]
 \efig$$
The full syntax for this is
\begin{verbatim}
\pullback(x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dx`dy>[N`N`N`N;L`L`L]%
    |ppp|/{sh}`{sh}`{sh}/<dx,dy>[N;L`L`L]
\end{verbatim}
 Of these only the nodes placed inside brackets are obligatory.  The
first sets of parameters are exactly as for
\index{\backslash square}\verb.\square.  and the
remaining parameters are for the nodes and labels of the outer arrows.
There is no positioning parameters for them; rather you set the
horizontal and vertical separations of the outer node from the square.

Here are some more special constructions.  In general, if you are doing
a square, you should use
\index{\backslash Square}\verb.\Square.  instead of
\index{\backslash square}\verb.\square.
because if figures its own width.  The syntax is almost the same, except
that
\index{dx}\verb.dx.  is omitted.  For example,
\begin{verbatim}
 $$\bfig
 \Square/^{ (}->`>`>`^{ (}->/<350>[{\rm Hom}(A,2^B)`{\rm Sub}(A\times B)`
 {\rm Hom}(A',2^{B'})`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')]
 \efig$$
\end{verbatim}
 will produce the square
 $$\bfig
 \Square/^{ (}->`>`>`^{ (}->/<350>[{\rm Hom}(A,2^B)`{\rm Sub}(A\times B)`
 {\rm Hom}(A',2^{B'})`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')]
 \efig$$
 There are a couple of points to note here.  Note the use of the
argument
\index{^{ (}->}\verb.^{ (}->.  to get the inclusion arrow.  The
complication is
created by the necessity of adding a bit of extra space before the hook.
You get pretty much the same effect by putting a bit of extra space
after the node:
\begin{verbatim}
 $$\bfig
 \Square/^(->`>`>`^(->/<350>[{\rm Hom}(A,2^B)\,`{\rm Sub}(A\times B)`
 {\rm Hom}(A',2^{B'})\,`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')]
 \efig$$
\end{verbatim}
 The full syntax is
\begin{verbatim}
 \Square(x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dy>[N`N`N`N;L`L`L`L]
\end{verbatim}

There are also macros for placing two
\index{\backslash Square}\verb.\Square.s together
horizontally or vertically.  The first is
\index{\backslash hSquares}\verb.\hSquares.  with the
syntax
\begin{verbatim}
\hSquares(x,y)|ppppppp|/{sh}`{sh}`{sh}`{sh}`{sh}`{sh}`{sh}/%
<dy>[N`N`N`N`N`N;L`L`L`L`L`L`L]
\end{verbatim}
The second is
\index{\backslash vSquares}\verb.\vSquares.  with a similar syntex except that there
are two
\index{dy}\verb.dy.  parameters, one for each square:
\begin{verbatim}
\hSquares(x,y)|ppppppp|/{sh}`{sh}`{sh}`{sh}`{sh}`{sh}`{sh}/%
<dy,dy>[N`N`N`N`N`N;L`L`L`L`L`L`L]
\end{verbatim}
 Similarly, there are four macros for making pairs of triangles put
together.  For example,
 $$\bfig
 \Vtrianglepair[A`B`C`D;f`g`h`k`l]
 \efig$$
 comes from
\begin{verbatim}
 $$\bfig
 \Vtrianglepair[A`B`C`D;f`g`h`k`l]
 \efig$$
\end{verbatim}

There is a macro for making cubes\index{\backslash cube}.  The syntax is
\begin{verbatim}
 \cube(x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dx,dy>[N`N`N`N;L`L`L`L]%
      (x,y)|pppp|/{sh}`{sh}`{sh}`{sh}/<dx,dy>[N`N`N`N;L`L`L`L]%
      |pppp|/{sh}`{sh}`{sh}`{sh}/[L`L`L`L]
\end{verbatim}
 The first line of parameters is for the outer square and the second for
the inner square, while the remaining parameters are for the arrows
between the squares.  Only the parameters in square brackets are
required; there are defaults for the others.  Here is an example:
\begin{verbatim}
 $$\bfig
\cube(0,0)|arlb|/ >->` >->`>`>/<1500,1500>[A`B`C`D;f`g`h`k]%
(300,300)|arlb|/>`>`>`>/<400,400>[A'`B'`C'`D';f'`g'`h'`k']%
 |mmmm|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$
\end{verbatim}
 gives the somewhat misshapen diagram
 $$\bfig
\cube(0,0)|arlb|/ >->` >->`>`>/<1500,1500>[A`B`C`D;f`g`h`k]%
(300,300)|arlb|/>`>`>`>/<400,400>[A'`B'`C'`D';f'`g'`h'`k']%
 |mmmm|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$
 because the parameters were oddly chosen.  The defaults center the
squares.  I discovered accidently, while debugging the cube that what I
thought was an out-of-range choice of parameters would produce an offset
cube:
\begin{verbatim}
 $$\bfig
\cube|arlb|/ >->` >->`>`>/<1000,1000>[A`B`C`D;f`g`h`k]%
(400,400)|arlb|/>`>`>`>/<900,900>[A'`B'`C'`D';f'`g'`h'`k']%
 |rrrr|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$
\end{verbatim}
 gives
 $$\bfig
\cube|arlb|/ >->` >->`>`>/<1000,1000>[A`B`C`D;f`g`h`k]%
(400,400)|arlb|/>`>`>`>/<900,900>[A'`B'`C'`D';f'`g'`h'`k']%
 |r`r`r`r|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$


\subsection{Complex diagrams}\label{complex}
In homological algebra one often has a $3\times3$ diagram, with or
without 0's on the margins.  There is a macro to do that:
 \index{3 by 3}\index{\backslash iiixiii}
\begin{verbatim}
 $$\bfig
\iiixiii(0,0)|aammbblmrlmr|/>`>`>`>`>`>`>`>`>`>`>`>/<500,500>{'5436}%
 <400,400>[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
\end{verbatim}
 that gives
 $$\bfig
\iiixiii(0,0)|aammbblmrlmr|/>`>`>`>`>`>`>`>`>`>`>`>/<500,500>{'5436}%
 <400,400>[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
 Here is the explanation.  The arrow parameters are not given in the
usual order, but rather first all the horizontal arrows and then all the
vertical ones, each in their usual order.  The nodes are given in the
usual order.  The fifth parameter is the octal number
\verb.`5436.   (In
\TeX, the
\verb.'.  introduces octal numbers and
\verb.".  introduces
hexadecimal numbers) which corresponds to the binary number
101,100,011,110 that describes the distribution of the 0's around the
margins.  The same results would have been obtained if the number had
been the hexadecimal number
\verb."B1E.  or the decimal number

\verb.2846..  The sixth parameter gives the horizontal and vertical
offset of the 0's, which you often want smaller than the others.  You
must not give the sixth parameter unless you have given a value (which
could be 0) to the fifth or an error condition will result.  Note that
the positioning parameters ignore the 0's so that it is the lower left
node that appears at the position
\verb.(x,y)..  As usual, there are
defaults.
 \begin{verbatim}
 $$\bfig
\iiixiii[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
 $$\bfig
\iiixiii%
 {'5436}[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
 $$\bfig
\iiixiii{'5436}<250,450>[A'`B'`C'`A`B`C`A''`B''`C'';%
f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
\end{verbatim}
 $$\bfig
\iiixiii[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
 $$\bfig
\iiixiii%
 {'5436}[A'`B'`C'`A`B`C`A''`B''`C'';f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$
 $$\bfig
\iiixiii{'5436}<250,450>[A'`B'`C'`A`B`C`A''`B''`C'';%
f'`g'`f`g`f''`g''`u`v`w`u'`v'`w']
 \efig$$

A similar macro\index{3 by 2}%
\index{\backslash iiixii}\verb.\iiixii.  has been added for a map between exact
sequences, with parameters similar to the above.  An actual example is
\begin{verbatim}
 $$\bfig
 \iiixii|aaaalmr|<1000,800>[H`G`F`H\oplus H_0`G\oplus H_0\oplus F_0`
F\oplus F_0; f`g`\pmatrix{f&0\cr0&1\cr0&0}`\pmatrix{g&0&0\cr0&0&1}`
 \pmatrix{1\cr0}`\pmatrix{1\cr0\cr0}`\pmatrix{1\cr0}]
\efig$$
\end{verbatim}
                which gives
 $$\bfig
 \iiixii|aaaalmr|<1000,800>[H`G`F`H\oplus H_0`G\oplus H_0\oplus F_0`
F\oplus F_0; f`g`\pmatrix{f&0\cr0&1\cr0&0}`\pmatrix{g&0&0\cr0&0&1}`
 \pmatrix{1\cr0}`\pmatrix{1\cr0\cr0}`\pmatrix{1\cr0}]
\efig$$
 The general syntax is
\begin{verbatim}
 \iiixii(x,y)|ppppppp|/1`2`3`4`5`6`7/<dx,dy>{n}<dx'>[...]
\end{verbatim}
 with the usual meaning.  The number n is a number between 0 and 15
(default 15) that specifies whether and where 0's appear (think binary,
with the high bit at the upper left) and dx' specifies the separation of
the zeroes.  You get two squares side by side if both n and dx' are 0.

\subsection{Empty placement and moving labels}
 The label placements within \verb.|p|. is valid only for
\verb.x=a,b,r,l,m..  If you use any other value (or leave it empty) the
label entry is ignored, but you can use any valid \xypic\ label, as
described in Figure 13 of the reference manual.  One place you might
want to use this is for the placement of the labels along an arrow.  In
\xypic\ the default placement of the label is midway between the
midpoints of the nodes.  If the two nodes are of widely different sizes,
this can result in strange placements; therefore I always place them
midway along the arrow.  However, as the following illustrates, this can
be changed.
 \begin{verbatim}
 $$\bfig
\morphism(0,900)||/@{->}^<>(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,600)||/@{->}^<(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,300)||/@{->}^>(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,0)||/@{->}^(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
 \efig$$
\end{verbatim}
 which produces
 $$\bfig
\morphism(0,900)||/@{->}^<>(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,600)||/@{->}^<(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,300)||/@{->}^>(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,0)||/@{->}^(0.7){f}/<800,0>[A^B\times B^C\times C`C;]
 \efig$$
 Here is the explanation.  The label placement argument is empty (it
cannot be omitted) and the arrow entry is empty.  However, placing
\verb.^(0.7){f}. inside the arrow shape places the label $f$ 7/10 of the
way between the nodes.  Unmodified, this places it 7/10 of the way
between the centers of the nodes.  This may be modified by
\index{<}\verb.<., which moves the first (here the left) reference point
to the beginning of the arrow, \index{>}\verb.>. which moves the second
reference point to the end of the arrow, or by both, which moves both
reference points.  In most cases, you will want both.  Incidentally,
\index{-}\verb.-. is a synonym for the sequence
\index{<>(}5).\verb.<>(.5). and that is the default placement in my
package.

Here are some more examples that illustrates the special sequence
\index{\backslash hole}\verb.\hole. used in conjunction with \index{|}\verb.|.
that implements \index{m}\verb.m. as well as the fact that these things
can be stacked.  For more details, I must refer you to the \xypic\
reference manual.
 \begin{verbatim}
 $$\bfig
\morphism(0,600)||/@{->}|-\hole/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,300)||/@{->}|-\hole^<>(.7)f/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,0)||/@{->}|-\hole^<>(.7)f_<>(.3)g/<800,0>[A^B\times
B^C\times C`C;]
 \efig$$
\end{verbatim}
 produces
 $$\bfig
\morphism(0,600)||/@{->}|-\hole/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,300)||/@{->}|-\hole^<>(.7)f/<800,0>[A^B\times B^C\times C`C;]
\morphism(0,0)||/@{->}|-\hole^<>(.7)f_<>(.3)g/<800,0>[A^B\times
B^C\times C`C;]
 \efig$$

Here is another version of the cube we looked at above, using these
special placements and\index{\backslash cube}
\index{\backslash hole}\verb.\hole.'s to break some lines and make it
neater.
\begin{verbatim}
 $$\bfig
\cube|arlb|/@{ >->}^<>(.6){f}` >->`@{>}_<>(.4){h}`>/%
<1000,1000>[A`B`C`D;`g``k]%
(400,400)|axxb|/>`@{>}|!{(300,1000);(500,1000)}\hole^<>(.6){g'}`>`@{>}%
|!{(1000,500);(1000,300)}\hole_<>(.4){k'}/<900,900>[A'`B'`C'`D';f'``h'`]%
 |rrrr|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$
\end{verbatim}
 $$\bfig
\cube|arlb|/@{ >->}^<>(.6){f}` >->`@{>}_<>(.4){h}`>/%
<1000,1000>[A`B`C`D;`g``k]%
(400,400)|axxb|/>`@{>}|!{(300,1000);(500,1000)}\hole^<>(.6){g'}`>`@{>}%
|!{(1000,500);(1000,300)}\hole_<>(.4){k'}/<900,900>[A'`B'`C'`D';f'``h'`]%
 |rrrr|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta]
 \efig$$
 This one is probably worth saving as a template.  Later I will explain
the meaning of the strings \verb.!{(300,1000);(500,1000)}\hole. and
 \verb.!{(1000,500);(1000,300)}\hole. along with a caveat on their use.
If the nodes are unusually large, the cube may be magnified using
\index{\backslash scalefactor}\verb.\scalefactor.  .

\subsection{Inline macros}
 Here we illustrate a few of the macros for inline---or
displayed---equations the package contains.  In each case, the macro may
have a superscript or subscript or both (in which case the superscript
must come first) and the arrow(s) grow long enough to hold the super- or
subscript.  If you type\par\noindent
\verb.$A\to B\to^f C\to_g D\to^h_{{\rm Hom}(X,Y)} E$., you get
 $A\to B\to^f C\to_g D\to^h_{{\rm Hom}(X,Y)} E$.  Similarly, the macro
\index{\backslash toleft}\verb.\toleft. reverses the arrows.  The remaining macros
of this sort are \index{\backslash mon}\verb.\mon. which gives a monic arrow,
\index{\backslash epi}\verb.\epi. which gives an epic arrow,
\index{\backslash two}\verb.\two. that gives a pair of arrows, as well as
leftwards pointing versions, \index{\backslash monleft}\verb.\monleft.,
\index{\backslash epileft}\verb.\epileft., and \index{\backslash twoleft}\verb.\twoleft. of
each of them.  Here is one more example:
\begin{verbatim}
 $A\twoleft B\twoleft^f C\twoleft_g D\twoleft^h_{{\rm Hom}(X,Y)} E$
\end{verbatim}
 gives  $A\twoleft B\twoleft^f C\twoleft_g D\twoleft^h_{{\rm Hom}(X,Y)} E$.
There is an almost unlimited variety of such procedures possible.  The
ones that are provided can be used as templates to define new ones with,
say, curved arrows or three arrows or whatever a user might have need
of.

\subsubsection{Added:}\label{added} The macros \index{\backslash to}\verb.\to. and
\index{\backslash two}\verb.\two. can each have optional parameters of the form
\index{/{sh}/<dx>}\verb./{sh}/<dx>. and
\index{/{sh}`{sh}/<dx>}\verb./{sh}`{sh}/<dx>., resp. that allow you to
specify the shapes of the arrows and to override the automatic
computation of the lengths of the arrows.  For example,
 \begin{verbatim}
$$A\to/<-/ B\to^f C \to/ >->/<500>_g D\to/<-< /^f_g E$$
$$A\two/<-`->/<100> B\two^f C \two/ >->` >->/_g D\two/<-< `<-< /^f_g E$$
\end{verbatim}
 gives
$$A\to/<-/ B\to^f C \to/ >->/<500>_g D\to/<-< /^f_g E$$
$$A\two/<-`->/<100> B\two^f C \two/ >->` >->/_g D\two/<-< `<-< /^f_g E$$
 This renders \index{\backslash mon}\verb.\mon., \index{\backslash epi}\verb.\epi.,
\index{\backslash toleft}\verb.\toleft., \index{\backslash monleft}\verb.\monleft.,
\index{\backslash epileft}\verb.\epileft., and \index{\backslash twoleft}\verb.\twoleft.
obsolete, but they have been retained for back compatibility and
convenience.  A three arrow macro that works similarly has been added.
For example
 \begin{verbatim}
 $$A\threepppp/>`<-`>/<400>^{d^0}|{s^0}_{d^1}B\three<100>
C\three/->>`<-< `->>/ D$$
\end{verbatim}
 gives
 $$A\threepppp/>`<-`>/<400>^{d^0}|{s^0}_{d^1}B\three<100>
C\three/->>`<-< `->>/ D$$

\subsection{2-arrows}
 There is a macro for making 2-arrows of a fixed size, but varying
orientation.  They should be put at the appropriate position in a
diagram.  The two parameters are two integers \index{dx}\verb.dx. and
\index{dy}\verb.dy. whose ratio is the slope of the arrow.  They need
not be relatively prime, but arithmetic overflow could occur if they are
too large.  Note that although \index{(dx,dy)}\verb.(dx,dy). and
\index{(-dx,-dy)}\verb.(-dx,-dy). describe the same slope, the arrows
point in opposite directions.  Here is a sampler
 \begin{verbatim}
 $$\bfig
 \place(0,0)[\twoar(1,0)]
 \place(200,0)[\twoar(0,1)]
 \place(400,0)[\twoar(1,1)]
 \place(600,0)[\twoar(0,-1)]
 \place(800,0)[\twoar(1,2)]
 \place(1000,0)[\twoar(1,3)]
 \place(1200,0)[\twoar(1,-3)]
 \place(1400,0)[\twoar(-3,1)]
 \place(1600,0)[\twoar(-1,-3)]
 \place(1800,0)[\twoar(255,77)]
 \efig$$
\end{verbatim}
 $$\bfig
 \place(0,0)[\twoar(1,0)]
 \place(200,0)[\twoar(0,1)]
 \place(400,0)[\twoar(1,1)]
 \place(600,0)[\twoar(0,-1)]
 \place(800,0)[\twoar(1,2)]
 \place(1000,0)[\twoar(1,3)]
 \place(1200,0)[\twoar(1,-3)]
 \place(1400,0)[\twoar(-3,1)]
 \place(1600,0)[\twoar(-1,-3)]
 \place(1800,0)[\twoar(255,77)]
 \efig$$

Here is little amusement.
\begin{verbatim}
 $$\bfig
 \square/@3{->}`~)`=o`--x/[A`B`C`D;```]
 \place(400,100)[\twoar(-1,-1)]
 \place(100,400)[\twoar(1,1)]
 \morphism(500,500)||/{*}.{*}/<-500,-500>[B`C;]
 \efig$$
\end{verbatim}
 $$\bfig
 \square/@3{->}`~)`=o`--x/[A`B`C`D;```]
 \place(400,100)[\twoar(-1,-1)]
 \place(100,400)[\twoar(1,1)]
 \morphism(500,500)||/{*}.{*}/<-500,-500>[B`C;]
 \efig$$

\subsection{Mixing \xypic\ code}
 Here is a sample in which I have mixed code from \xypic\ with my own.
\let\tilde\widetilde
\let\hat\widehat
\begin{verbatim}
 $$\bfig
 \square(1500,500)/>`>`>`@{>}^<>(.2){r_{n-1}}/[T_{n-1}`T_{n-2}`R_{n-1}
`R_{n-2}; t_{n-1}``\sigma_{n-2}`]
 \square(1500,0)/`>`>`>/[R_{n-1}`R_{n-2}`S_{n-1}`S_{n-2};
`f_{n-1}`f_{n-2}`s_{n-1}]
 \morphism|b|[\tilde S_{n+1}`S_n;\tilde s_{n+1}]
 \square(1000,500)/>`>`>`>/[\tilde T_n`T_{n-1}`\tilde
R_n`{R_{n-1}};\tilde t_n`\tilde \sigma_n`\sigma_{n-1}`]
 \square(1000,0)/>`>``>/[\tilde R_n`R_{n-1}`\tilde S_n`{S_{n-1}};\tilde
r_n\quad ```\tilde s_n]
 \square(500,0)/>`>``>/[R_n`{R_n}`{S_n}`{S_{n}};
\hat r_n`f_n`\tilde f_n`\hat s_n]
 \POS(1500,1000)*+!!<0ex,.75ex>{T_{n-1}}
 \ar@{-}|!{(1000,500);(1500,500)}\hole(1167,334)%
 \POS(1167,334)\ar|!{(1000,500);(1000,0)}\hole_<>(.6){h_{n-1}}
  (500,0)*+!!<0ex,.75ex>{S_n}
 \morphism(2000,1000)/@{>}|\hole^<>(.8){h_{n-2}}/%
 <-500,-1000>[T_{n-2}`S_{n-1};]
 \efig$$
\end{verbatim}
 $$\bfig
 \square(1500,500)/>`>`>`@{>}^<>(.2){r_{n-1}}/[T_{n-1}`T_{n-2}`R_{n-1}
`R_{n-2}; t_{n-1}``\sigma_{n-2}`]
 \square(1500,0)/`>`>`>/[R_{n-1}`R_{n-2}`S_{n-1}`S_{n-2};
`f_{n-1}`f_{n-2}`s_{n-1}]
 \morphism|b|[\tilde S_{n+1}`S_n;\tilde s_{n+1}]
 \square(1000,500)/>`>`>`>/[\tilde T_n`T_{n-1}`\tilde
R_n`{R_{n-1}};\tilde t_n`\tilde \sigma_n`\sigma_{n-1}`]
 \square(1000,0)/>`>``>/[\tilde R_n`R_{n-1}`\tilde S_n`{S_{n-1}};\tilde
r_n\quad ```\tilde s_n]
 \square(500,0)/>`>``>/[R_n`{R_n}`{S_n}`{S_{n}};
\hat r_n`f_n`\tilde f_n`\hat s_n]
 \POS(1500,1000)*+!!<0ex,.75ex>{T_{n-1}}
 \ar@{-}|!{(1000,500);(1500,500)}\hole(1167,334)%
 \POS(1167,334)\ar|!{(1000,500);(1000,0)}\hole_<>(.6){h_{n-1}}
  (500,0)*+!!<0ex,.75ex>{S_n}
 \morphism(2000,1000)/@{>}|\hole^<>(.8){h_{n-2}}/%
 <-500,-1000>[T_{n-2}`S_{n-1};]
 \efig$$
 There are three points to note here in connection with the two lines
that begin with \index{\backslash POS}\verb.\POS.  First the objects that are the
source of the first and the target of the second are preceded by
\verb;!!<0ex,.75ex>;\verb;!!<0ex,.75ex>;.  The effect is to reset the
baseline to the baseline of the object (rather than the vertical center)
and then to lower that by 3/4 of xheight so that the arrow goes in the
right place.  This string precedes all objects.  Without that, an object
like $\tilde R $ would be set lower that $R$.  Second, the first arrow
has no target and the second no source.  This does not give the same
result as empty source and target since in the latter cases, there would
be spaces allowed around them and then the two lines would not meet.  It
would be possible to add code that tests for empty nodes, but it comes
up so seldom that I have refrained.  In the meantime, the only recourse
is to revert to the underlying \xypic\ code.  Thirdly the string
\verb.|!{(1000,500);(1500,500)}\hole. specifies that the line should be
broken at the place where the current arrow intersects the line between
the nodes located at (1000,500) and (1500,500).  One must be careful
using this construction, however, as it does not seem to work correctly
if the line segment fails to intersect the current line, or if it does
intersect, but happens to be too long.  I have not worked out how long
is too long, but you can get odd results.  I assume that this bug will
be fixed eventually.  (Ross Moore says that it works correctly in the
version he has, which, however, has not been released.)  There is a
similar string, with similar effect, in the following line.  The last
line uses simply \index{|\hole}\verb.|\hole. which positions the gap in
the middle of the arrow.

\subsection{Diagram from TTT}
 The last example is a complicated diagram from TTT.  If you have the
documentation from the old diagram macros (or the errata from TTT), you
can see how much easier it is to describe this diagram with these
macros.  Note the use of \index{\backslash scalefactor}\verb.\scalefactor. to
change the default length from 500 to 700 that made it unnecessary to
specify the scales on the squares and triangles.
 \begin{verbatim}
 $$\bfig
 \scalefactor{1.4}%
 \qtriangle(0,1000)/>`>`/[TT`T`TTT';\mu`TT\eta'`]%
 \btriangle(500,1000)/`>`@<-14\ul>/[T`TTT'`TT';`T\eta`T\sigma]%
 \morphism(0,1500)|l|/>/<0,-1000>[TT`TT'T;T\eta'T]%
 \square(500,500)|ammx|/@<14\ul>`>`>`/[TTT'`TT'`TT'TT'`TT'T';%
    \mu T'`T\eta'TT'`T\eta'T'`]%
 \morphism(1000,1000)|r|/>/<500,-500>[TT'`TT';\hbox{\rm id}]%
 \square/>`>``>/[TT'T`TT'TT'`T'T`T'TT';TT'T\eta'`\sigma T``T'T\eta']%
 \square(500,0)|ammb|[TT'TT'`TT'T'`T'TT'`T'T';%
    TT'\sigma`\sigma TT'`\sigma T'`T'\sigma T']%
 \square(1000,0)/>``>`>/[TT'T'`TT'`T'T'`T';T\mu'``\sigma`\mu']%
\place(500,1250)[1]\place(215,1000)[2]\place(750,750)[3]%
\place(215,250)[4]\place(750,250)[5]\place(1140,750)[6]%
\place(1250,250)[7]%
 \efig$$
\end{verbatim}
 $$\bfig\label{TTTdiag}
 \scalefactor{1.4}%
 \qtriangle(0,1000)/>`>`/[TT`T`TTT';\mu`TT\eta'`]%
 \btriangle(500,1000)/`>`@<-14\ul>/[T`TTT'`TT';`T\eta`T\sigma]%
 \morphism(0,1500)|l|/>/<0,-1000>[TT`TT'T;T\eta'T]%
 \square(500,500)|ammx|/@<14\ul>`>`>`/[TTT'`TT'`TT'TT'`TT'T';\mu T'`%
    T\eta'TT'`T\eta'T'`]%
 \morphism(1000,1000)|r|/>/<500,-500>[TT'`TT';\hbox{\rm id}]%
 \square/>`>``>/[TT'T`TT'TT'`T'T`T'TT';TT'T\eta'`\sigma T``T'T\eta']%
 \square(500,0)|ammb|[TT'TT'`TT'T'`T'TT'`T'T';TT'\sigma`\sigma TT'%
    `\sigma T'`T'\sigma T']%
 \square(1000,0)/>``>`>/[TT'T'`TT'`T'T'`T';T\mu'``\sigma`\mu']%
\place(500,1250)[1]\place(215,1000)[2]\place(750,750)[3]%
\place(215,250)[4]\place(750,250)[5]\place(1140,750)[6]%
\place(1250,250)[7]%
 \efig$$


\begin{theindex}

  \item \verb@%@, 3
  \item \verb@(@, 7
  \item \verb@(@,), 7
  \item \verb@(-dx@,-dy), 25
  \item \verb@(dx@,dy), 25
  \item \verb@)@, 7
  \item \verb@*@, 7
  \item \verb@-@, 6, 23
  \item \verb@--@, 6
  \item \verb@-->>@, 7
  \item \verb@->@, 7
  \item \verb@->>@, 7
  \item \verb@/@, 7, 9
  \item \verb@/{sh}/<dx>@, 25
  \item \verb@/{sh}`{sh}/<dx>@, 25
  \item \verb@;@, 7
  \item \verb@<@, 6, 7, 22
  \item \verb@<-@, 7
  \item \verb@<-< @, 7
  \item \verb@<<-@, 7
  \item \verb@<<--@, 7
  \item \verb@<>(@, 23
  \item \verb@=@, 6, 13
  \item \verb@>@, 6, 7, 22
  \item \verb@>>@, 6
  \item \verb@[@, 7
  \item \verb@\startsection@, 5
  \item \verb@\Atriangle@, 9
  \item \verb@\Atrianglepair@, 9, 12
  \item \verb@\Ctriangle@, 8, 9
  \item \verb@\Ctrianglepair@, 9, 12
  \item \verb@\Dtriangle@, 9
  \item \verb@\Dtrianglepair@, 9, 12
  \item \verb@\POS@, 27
  \item \verb@\Square@, 15, 16
  \item \verb@\Vtriangle@, 8, 9
  \item \verb@\Vtrianglepair@, 9, 12
  \item \verb@{-}@, 14
  \item \verb@\bfig@, 9
  \item \verb@\btriangle@, 9
  \item \verb@\cube@, 16, 23
  \item \verb@\dtriangle@, 9, 11
  \item \verb@\efig@, 9
  \item \verb@\endxy@, 9
  \item \verb@\epi@, 24, 25
  \item \verb@\epileft@, 24, 25
  \item \verb@\fontscale{x}@, 4
  \item \verb@\hSquares@, 16
  \item \verb@\halign@, 2
  \item \verb@\hole@, 23
  \item \verb@\ignorespaces@, 3
  \item \verb@\iiixii@, 21
  \item \verb@\iiixiii@, 18
  \item \verb@\let\labelstyle=\textstyle@, 6
  \item \verb@\mon@, 24, 25
  \item \verb@\monleft@, 24, 25
  \item \verb@\morphism@, 2, 5, 8, 10
  \item \verb@\phantom@, 3
  \item \verb@\place@, 8, 14
  \item \verb@\ptriangle@, 8, 9, 11
  \item \verb@\pullback@, 15
  \item \verb@\put@, 14
  \item \verb@\qtriangle@, 8, 9
  \item \verb@\scalefactor@, 24, 28
  \item \verb@\scriptstyle@, 6
  \item \verb@\square@, 8, 9, 11, 15
  \item \verb@\three@, 1
  \item \verb@\to@, 1, 25
  \item \verb@\toleft@, 24, 25
  \item \verb@\two@, 1, 24, 25
  \item \verb@\twoleft@, 24, 25
  \item \verb@\ul@, 6
  \item \verb@\vSquares@, 16
  \item \verb@\xy@, 9
  \item \verb@\xy ... \endxy@, 9
  \item \verb@\xyoption{curve}@, 13
  \item \verb@]@, 7
  \item \verb@^@, 6
  \item \verb@^{ (}->@, 16
  \item \verb@_@, 6
  \item \verb@`@, 7, 9
  \item \verb@{*}@, 7
  \item \verb@3 by 2@, 21
  \item \verb@3 by 3@, 18

  \indexspace

  \item \verb@ >->@, 7

  \indexspace

  \item \verb@a@, 6

  \indexspace

  \item \verb@b@, 6

  \indexspace

  \item \verb@dx@, 8, 15, 25
  \item \verb@dy@, 8, 16, 25

  \indexspace

  \item \verb@l@, 6
  \item \verb@label@, 6

  \indexspace

  \item \verb@m@, 6, 23

  \indexspace

  \item \verb@r@, 6

  \indexspace

  \item \verb@sliding arrows@, 13

  \indexspace

  \item \verb@tips@, 4

  \indexspace

  \item \verb@x@, 4, 8

  \indexspace

  \item \verb@y@, 8

\end{theindex}


\end{document}