LaTeX Math Symbols 3/29/17, 10*20 AM
LaTeX Math Symbols
The following tables are extracted from The Not So Short Introduction to LaTeX2e, aka. LaTeX2e in 90 minutes, by Tobias Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl. It can be located here.
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Last modified: Wed May 31 14:04:55 CDT 2000
http://www.math.ubc.ca/~cautis/tools/latexmath.html Page 9 of 9 XY-pic User’s Guide
Kristo↵er H. Rose [email protected] ⇥ h i Version 3.7, February 16, 1999
Abstract 1 Basics 2 XY-pic is a package for typesetting graphs and diagrams 1.1 Loading ...... 2 using Knuth’s TEX typesetting system. XY-pic works with 1.2 Entries ...... 2 most of the many formats available; e.g., plain T X, E 1.3 Arrows ...... 2 LAT X, and -T X. Several styles of input for various E A S E diagram typesM are supported; they all share a mnemonic 1.4 Labels ...... 3 notation based on the logical composition of visual com- 1.5 Breaks ...... 3 ponents. This guide concentrates on how to typeset 1.6 Curving ...... 4 “matrix-like” diagrams, such as commutative diagrams, 1.7 Speeding up typesetting ...... 4 in the following style: U 2 More Arrows and Labels 4 x (x,y) 2.1 Explicit label positioning ...... 4 # % 2.2 Labeling with any object ...... 5 X Z Y / X y ⇥ p 2.3 More arrow styles ...... 5 q f 2.4 Sliding arrows sideways ...... 6 ✏ g ✏ Y / Z 2.5 More targets ...... 6 2.6 Changing the target ...... 7 was typeset by the X -pic input lines Y 2.7 Arrows passing under ...... 7 \xymatrix{ 2.8 More bending arrows ...... 8 U \ar@/_/[ddr]_y \ar@/^/[drr]^x \ar@{.>}[dr]|-{(x,y)} \\ 2.9 Defining new arrow types ...... 8 & X \times_Z Y \ar[d]^q \ar[r]_p & X \ar[d]_f \\ 3 More Entries 9 & Y \ar[r]^g & Z } 3.1 Manual entry formatting ...... 9 Such diagrams have the following characteristics: 3.2 Extra entries outside the matrix . . . 9 Specified as a matrix of entries that are automati- 3.3 Spacing and rotation ...... 9 • cally aligned in rows and columns. 3.4 Entry style ...... 10 Any entry may be connected to any other en- • 3.5 Naming for later use as targets . . . . 10 try using a variety of arrow styles all rotated and 3.6 Grouping objects ...... 10 stretched as required. Arrows may be decorated with labels that are tied • 4 Availability and Further Information 11 to a specified point along the arrow and extend in a particular direction; and arrows may be paired, 4.1 Getting XY-pic ...... 11 cross, and visit/bend around other entries “on the 4.2 Backwards compatibility ...... 11 way.” 4.3 Further reading ...... 12 Several other styles of input are supported; a short survey 4.4 Credits ...... 13 of the possibilities is included last at the end along with information on how XY-pic can be obtained. A Answers to all exercises 13
Contents References 14
Preface 2 Index 15
⇥Laboratoire de l’Informatique du Parall´elisme,Ecole Normale Sup´erieurede Lyon; 46, All´eed’Italie; F–69364 Lyon 7, France.
1 Preface where the “. . . ” should be replaced by entries to be aligned in rows and columns where This guide explains some features of XY-pic that are entries in a row are separated by &,4 and relevant to typesetting of “matrix-like diagrams” as • used in, for example, category theory; please refer to entire rows are separated by \\. the reference manual [8] for complete information on • the described constructions. The guide assumes that For example, you have some experience in using TEX for typeset- m 2 A i=n i ting mathematics, e.g., have studied [2, ch. 16–19], [3, cGG GG sec. 3.3], or [9], and that XY-pic is installed on your P GG GG TEX system as described in the INSTALL file accom- G D panying the distribution. • The first section describes what you need to get was typeset by started, in particular all that is needed to typeset \xymatrix{ the diagram in the abstract. Section 2 and 3 explain A &*+[F]{\sum_{i=n}^m {i^2}} \\ advanced use of arrows and entries, respectively. Fi- & {\bullet} & D \ar[ul] } nally, section 4 explains where and under what condi- tions XY-pic is available, gives the relation of version Notice the following: 3.7 to previous versions, and lists further sources of entries are typeset as mathematics (using “text information. • style”); entries should not start with a macro Throughout we give exercises that you should be (as illustrated by the use of {} around \bullet . able to solve as you go along; all exercises are an- all entries are centered and the separation be- swered at the end just prior to the references and • index. tween rows and columns is usually quite large in a diagram, 1 Basics empty entries at the end of rows may be omit- • ted, This section explains the X -diagram construction Y “XY-decorations” (here \ar[ul]) in entries al- concepts needed to get started with typesetting • low drawing of arrows and such relative to the matrix-like diagrams. entries without changing the overall layout, and
1.1 Loading “XY-modifiers” (here *+[F]) first in entries al- • low changing the format and shape in many The XY-pic setup used in this guide is loaded by in- ways. serting the lines
\input xy 1.3 Arrows \xyoption{all} An “arrow” in an XY-pic diagram is a generic term for the drawn decorations between the entries of the 1 in the definitions part of your document. If you wish basic matrix structure. In XY-pic all arrows must be to load only the features you use, or you wish to use specified along with the entry in which they start; this non-standard facilities like the v2 backwards compat- is called their base entry. Each particular arrow com- 2 3 ibility mode or the ps PostScript backend then mand then refers explicitly to its target entry. This this is also possible as described in the reference man- is obtained using the \ar command which accepts ual [8]. many options of which we will describe a few here and some more in section 2. In its simplest form an 1.2 Entries arrow is entered as \ar[hop] where hop is a sequence of single letters: u for up, d for down, l for left, and A diagram is created by the command r for right, e.g., the arrow \ar[ur] reads “typeset an arrow from the current entry to that one up and one \xymatrix{ ... } right.” 1 LATEX2" [3] users can use \usepackage[all]{xy}. 2If you use the version 2 loading command \input xypic (or the xypic document style option) then the v2 option described in section 4.2 will be loaded automatically. 3PostScript is a registered Trademark of Adobe, Inc. [1]. 4Thus when using XY-constructions involving & inside other tabular constructions then enclose the XY-pic construction in an extra pair of braces!
2 Exercise 1: Which entry does [] refer to? $\xymatrix@1{ A\times B\times C\times D \ar[r]^-{+} &B The relative coordinates specified in this way are }$ purely logical, e.g., if the diagram contains very wide + entries then “diagonal” arrows will be nearly horizon- (it becomes A B C D /B without the -). tal. The constructed arrows are aligned along the line ⇥ ⇥ ⇥ In fact - is in just one of the may possible placings between the centers of the base and target entries; of labels described in section 2.1. they will not automatically disappear under entries that they cross (we discuss how this is achieved in section 2.7). Exercise 3: Typeset the second axiom of category The arrow style kan be changed by writing the theory as command as \ar@style[hop]. This will be described f in more detail in section 2.3; here we just list the most A / B @ @ common @styles (obvious variations also work): @ @ g;h @@ g @@ f;g @@ @@ @{=>} @{.>} @{:>} @{~>} @{-->} @{-} @{} @ ✏ @ .. . . C / D .. V . h .. V . . ... V . ✓⇢ ✓⇢ . .
Exercise 2: Typeset 1.5 Breaks
• @@ • It is also possible to “break” an arrow with a label us- @@@@ @@@@ ing the | character: $\xymatrix@1{A\ar[r]|f&B}$ @@@@ o will set A f /B . •• If you just want an empty break you should use the special \hole break: the arrow A /B 1.4 Labels was typeset by including $\xymatrix@1{ A\ar[r]|\hole & B }$ in the text. You can put labels on arrows. Labels are conceptual- A di↵erent use of breaks is to place a label some- ized as sub- and superscripts on arrows such that they where in a diagram outside the normal matrix mesh: are placed in the usual positions (as “limits”), i.e., ^ this is accomplished by “breaking” an invisible arrow reads “above” and _ “below” on an arrow pointing obtained using the @{} arrow style: the square right. Notice that the positions depend only on the direction of the arrow, the absolute notions of “up,” A / B “down,” etc. are not important. For example, $\xymatrix@1{ = ✏ ✏ X\ar[r]^a_b & Y & Z\ar[l]^A_B }$ B / C a B will set X /YZo (the @1 is a special code b A was typeset by that can be used for “one-line” diagrams to improve the placement on the line; more such spacing codes \xymatrix{\ar @{} [dr] |{=} are described in section 3.3). A \ar[d] \ar[r] & B \ar[d] \\ It is possible to use labels that are not single let- B \ar[r] & C } ters, digits, or control sequences: if a simple math There is more on breaks in section 2.7. formula in the default style (script style) is desired then simply enclose in {...}. In practice anything can be used as a label as described in section 2.2. Exercise 4: Typeset the first axiom of category Each label is placed perpendicular to the arrow at theory as the display the point halfway between the centers of the base and target objects. This is usually the most æsthethic, f however, in diagrams where the sizes of the entries A / B ~~ vary much it is sometimes nicer to place the label f i ~ g ~B at the center of the actual arrow. This behaviour is ✏ ~ ~ ✏ requested by inserting a - (minus) right after the ^ B g / C + or _: A B C D /B was typeset by ⇥ ⇥ ⇥
3 1.6 Curving matrix that can be loaded very quickly; they are au- tomatically recreated when a matrix is changed. Arrows can be made to curve, for example to avoid going through another entry, using the special style If this causes some diagrams to not work, then @/curving/. The simplest styles of curving are the such compilation can be explicitly switched o↵by \xymatrixnocompile \xymatrix following, shown applied to an arrow from A to B: using in place of . Compilation can be switched o↵completely with @/^/ A ) B \NoCompileMatrices (which respects TEX grouping as does \CompileMatrices, by the way). @/_/ A 5 B And if you are still not satisfied with the speed @/_1pc/ A B then you can add the following: = As the last example shows a dimension can be in- \OnlyOutlines serted just after ^ or _ if more or less curving is de- sired. In case it is easier to specify the in- and out-going which will omit all compiled pictures; the additional directions of the curving then that is also possible: command \ShowOutlines will add a dotted rectangle use outlining the size of the picture. @(in,out) where in and out are one of the following directions: 2 More Arrows and Labels u O In this section we explain a number of variations of ul ur _? ? the arrow commands that are useful in commutative ?? ? diagrams. l o / r ? ?? ? dl ONMLHIJK✏ dr 2.1 Explicit label positioning d The label commands explained in section 1.4 place In this case the curving is computed such that the the label text near the point along the arrow halfway curve begins at the base entry in the in direction between the centers of the base and target entries. and ends at the target entry from the out direction This, however, may be changed by inserting a place (this means that @(d1,d2) and @(d2,d1) are mirror between the ^, _, or |, and the actual label (in fact - images. See section 2.6 for more directions). is a place). In general you may insert the following:
Exercise 5: Typeset < will place the label at the point where the • f actual arrow begins, i.e., “appears from under” id , x k 1 f(x) 9 f the base, so $\xymatrix@1{A\ar[r]^<{+}&B}$ + will typeset A /B .
1.7 Speeding up typesetting Similarly, > will place the label at • the point where the actual arrow ends, One thing that you will notice is that X -pic is some- Y i.e., “disappears below” the target, times slow in typesetting diagrams (this is to be ex- so $\xymatrix@1{A\ar[r]^>{+}&B}$ will pected considering the number of drawing operations + performed as reflected by the number last in each typeset A /B . xymatrix message). If you follow the rule of start- ing all entries with a (nonexpandable) character or { << and >> will place the following la- then you can insert the declaration • bel at a point just a bit6 further from the beginning and end of the arrow, \CompileMatrices so $\xymatrix@1{A\ar[r]^>>{+}&B}$ will + in the preamble of your document: this will create typeset A /B . Using more s will temporary files5 containing compiled versions of each move the label further in. 5The temporary files are named the same as your document but .tex is replaced by -n.xyc where n is a sequence number. 6 “A bit” is in fact a TEX \jot which is usually 3pt.
4 A factor in ()s: (a) indicates that the += grow to enclosing square • label should be “tied” to the point a of - shrink the way from the center of the base en- -s. Hence Since objects specified this way start⌫ with no mar- $\xymatrix@1{A\ar[r]^<(0){+}&B}$ will ⇣⌘ ✓◆ + + gin, a single is usually included to get the default typeset A /B . spacing.
Finally, there is a simple way to denote the x Exercise 7: Typeset A /B . • place on an arrow where it intersects with a straight line between two points: the place '&%$ !"# There can only be one boxing. This can be any !{t ;t } places the label relative to the point 1 2 box generation command. The following are the most on the arrow where the line from the target t 1 useful boxing{text} combinations: to the target t2 crosses it. Bug: Only works for straight arrows at present. @variant{tip} tip (or shaft) object As usual more possibilities can be found in the refer- \txt{...} ordinary text ence manual [8, fig. 2]. \composite{...*...} combined objects \frm{} repeat last object Exercise 6: Typeset (the possibilities for variant and tip are given in the 7 1000000x following section). Finally, \hbox{...} is a quick 1 H / 1000000 HH jj4 way to ensure text-mode interpretation of a single ob- 2000x H jj 2 HH jjjj x ject. However, \txt allows the use of \\ in text to cre- H jjjj jjjjH# ate a line break, and the special form \txt<6pc>{...} 1000 j / 2000 2x will constrain the text to a centered 6pc wide column. By the way, \txt can be used outside of XY-pic con- structions. 2.2 Labeling with any object Finally, several objects can be combined using the last form; the *s serve to separate the composed ob- XY-pic supports a general format for entering any TEX ject. text as labels (as well as entries to be explained later). The character * is reserved for this: in its simplest High form *{math} will typeset the math material as an label object. This is like {math} except that the default Exercise 8: Typeset A ++++++ B . style is ignored and there is no added blank margin. ⇥⇥⇥⇥⇥⇥ However, in general the following form of *object 2.3 More arrow styles is available: The arrow styles described in section 1.3 are all ex- *modifiers{text} amples of the general arrow style constructions where modifiers can be used to change the shape @variant{tail shaft head} and size of the constructed object. The following are @variant{head} the most common, the full list of possibilities can be found in the reference manual [8, fig. 3]: that in describes arrows with the indicated tail, shaft, and head (on the first form the tail and head can be + grow omitted; the second style defaults to having no tail +
7 The plain TEX command \hbox corresponds to \mbox in LATEX and \text in the variants. AMS
5 The following possibilities exist for head and tail ' ' which we will denote tips (here shown as heads): {|-} g ^{|-} g _{|-} 'g ' '' {|=} 'g 3{|-} g' % ⇧ < s > 3 x L {*} • ^{‘} _{‘} ' E e << s s >> 3 3 + g ^{’} _{’} ' ' ' |< s >| 3 | ' ' ' |<< s s >>| 3 3 || ' % ⇧ e ( ) E o The even more general form *object can be used, / // where object refers to any of the constructions de- scribed in section 2.2. and the shaft should be one of the following: v ) Exercise 9: Typeset A 0 6 N B . •pv?H • jjjj j j - jjjj -- j j 4t 4t 4t 2.4 Sliding arrows sideways ~ 4t 4t ~~ 4t 4t It is often desirable to have several parallel arrows . between two objects. This can be done by sliding ei- ther or both arrows sideways a distance given as a The variant should be empty or one of the follow- TEX dimension enclosed in @<>s: it specifies how far ing: “sideways” the arrow should be moved, e.g.,
^ “above” variant \xymatrix{ _ “below” variant A \ar@<1ex>[dr]^a_{.} \\ 2 “double” variant & B \ar@<1ex>[ul]^b \ar@<1ex>[r]^c 3 “triple” variant & C \ar@<1ex>[l]^d_{.} } will typeset Here are some standard arrows in this notation, all from A to B as usual: A @ _@ @@ a @@ . @@ @{<->} A o / B @@ @@ b @@ c . / @^{<->} A o / B B o C d @_{<->} A o / B @2{<->} A ks +3 B A positive distance will slide the arrow in the “^- @3{<->} A jt_ _*4 B direction,” e.g., the two arrows above are slid in the direction of the labels a, b, c, and d, respectively; As a special convenience = and : are provided as ab- a negative distance in the “_-direction.” The dis- breviations for - and . with variant forced to 2. tance <@1ex> is often appropriate since it corresponds As it can be seen, the variant will a↵ect the entire roughly to the height of letters like “x,” in the type arrow. Sometimes this is not what is wanted. In that size being used. case a local variant can be used by entering any of the tail, shaft, and head, on the following form: Exercise 10: Typeset A ''B . variant{tip} variant{shaft} 2.5 More targets The target address can be given in a large number of Here are some arrows where this is required: formats called positions. The full range of possibili- ties is described in the reference manual [8, fig. 1]; @{^{(}->} A / B here is a number of useful ones in addition to the hop @{|-_{>}} A / B format described in section 1.3:
Notice that there is no distinction between shafts and [r,c], where r, c are integers, denotes the rela- tips using this form, thus it is necessary to include all • tive entry found r rows below and c columns to three of tail, shaft, and head, when using it. The ad- the right of the current entry (the current entry vantage is that it is possible then to “fill with a tip.” itself is thus [0,0]). This always corresponds Furthermore, the following additional possibilities are to a [hop], e.g., [1,2] is the same as [drr] available when using this notation: and [-2,0] is the same as [uu].
6 "r,c", where r, c are positive integers, denotes /d dimen/ is the vector going dimen in the par- • the absolute entry found in the rth row and cth • ticular direction d which can be either the eight column of the diagram. The top left entry is simple ones in section 1.6, empty to denote that "1,1". the current direction (the last direction of an arrow) should be used, or one of the following: t0;t , where t0 is any target, changes the base • entry of the present arrow to t0 and then sets va(↵) absolute angle the target to t relative to the original base en- d:a(↵) relative angle in degrees try. For example, d:(x,y) relative vector \xymatrix{ A \\ d^ / d_ short for :a(90) / B & C \ar@<1ex>[ul] :a(-90) \ar@<1ex>[ul];[] } where the d in the last four may be empty to typesets denote the “current direction.” A @ 0 is the zero vector. _@ @@ • @@ @@ @@ @@ Many, many more possibilities are described in the @@ BC reference manual [8]. i.e., the second \ar[ul] arrow starts at the Exercise 11: What is the di↵erence between a tar- [ul] entry and ends in the current entry. get t and the target t+0? See section 3.5 for how to use a label as a target. 2.7 Arrows passing under 2.6 Changing the target Arrows can pass under (or via) any other entry: Just insert ’t, i.e., a quote (apostrophe) character followed It is possible to overwrite a target with another by by a target, for each entry that should be visited ex- appending something of the form *object to it. This cept the last, “ordinary & final” entry: has the e↵ect of typesetting the object at the current position, thus e↵ectively on top of the target, and \xymatrix{ then use what was typeset as the target. {\circ} A target may also have its position changed by \ar@{|-->} ’[dr] ^a one of the following constructions: ’[rr]+D*{\bullet} ^b [drrr] ^c +vector or -vector which changes the target to & {\circ} & {\circ} & {\circ} \\ • be a zero-sized one at the position obtained by {\circ} & {\circ} & {\circ} & {\circ} } adding or subtracting the vector to its center, typesets or @ D @ a b z• D c !vector which moves the center of the target by @ z D @ z • the vector; z D" where a vector should have the form As you see, labels are set separately on each segment.
7 2.8 More bending arrows 2.9 Defining new arrow types Finally, arrows can bend around entries: just insert Last in this treatment of arrows we will explain how ‘dt, i.e., a backquote and a direction d followed by new arrows can be defined. The crucial fact is that a target t, for each “turn” that starts out in the d- the characters used for tips and shafts are restricted direction and ends in a quarter turn towards the tar- to the following: get t. This is di↵erent from the curving described earlier in that all turns consist of a straight part end- ><|ox+/()[] tip characters ing in a turn which is a multiple of 1/8 circle segments, -.~:= shaft characters and each segment allows separate labels. The possible directions are those of section 1.6 When an arrow is interpreted by X -pic it is first split and 2.6, and the possible targets include all those Y into the three components and then each component discussed above. Actually the direction is only re- is looked up in a library of so-called “directionals.” quired for the first in a series of turns since the final It is possible to add new such directionals using the direction of one turn is the obvious choice for the fol- command lowing turn. Furthermore, turns can be changed from the default by adding either ^d for anticlockwise turn \newdir{ directional }{ composite } to d or _d for clockwise turn to d, where d is the “exit direction” of the turn. Finally, the turns will have radius 10pt by default, where directional should be a sequence either of but this can be changed to any dimension R from a tip characters or of shaft characters, and composite particular turn and onwards by inserting /R imme- should be a list of objects separated with * just like diately after the ‘‘ of the turn. Here is an example the argument to \composite described in section 2.2. involving all of these features: If arrows of a particular variant (always one of the let- ters ^_23) needs an alternate definition then another \xymatrix{ declaration can be given with the variant inserted be- {\circ} \ar ‘r[d] ^a tween \newdir and the first {. ‘[rr] ^b There is one object modifier which is very useful ‘/4pt[rr] ^c in this context, in addition to those of section 2.2: ‘[rrr] ^d ‘_dl[drrr]^e !vector shift object vector [drrr]^f & {\circ} & {\circ} & {\circ} \\ {\circ} & {\circ} & {\circ} & {\circ} } (where the possibilities for vector are described in section 2.6). Combined with the direction code this typesets is very powerful, for example, a e \newdir{|>}{% b d c !/4.5pt/@{|}*:(1,-.2)@^{>}*:(1,+.2)@_{>}} f ⌧ defines a new tip that makes ED@A ⇢ \xymatrix{ A \ar @{=|>} [r] & B } The example illustrates the following points: typeset ,2 . Notice how the “relative di- If the segment can not be made as short as re- A B rection” is used here to rotate some of the composed • quired then it will point “past” the target. This components. is useful for “going around” entries. There is one ‘ per turn thus each target ap- Exercise 14: Often tips used as “tails” have their • pears as many times as there are turns towards ink on the wrong side of the point where they are it, except the last target that appears one more placed. Fortunately space ( ) is also a tip charac- time namely both with ‘ for each turn towards ter so we can define the directional > to generate a it and once as an “ordinary” target to set the “tail-spaced” arrow. Do this such that final stretch. \xymatrix{ A \ar @{>->} @< 2pt> [r] The sizes of the intermediate targets are ig- \ar @{ >->} @<-2pt> [r] & B } • nored. typesets A / / // B Exercise 13: Typeset A B . O O
HI89 JK:; 8 Finally, when XY-pic diagrams are used in con- Exercise 16: Typeset junction with Knuth’s computer modern fonts then the declaration • 2 3 1 \SelectTips {cm}{} x' x! x x x will change the tips to some that look similar, e.g., x $\SelectTips{cm}{} \xymatrix@1{A\ar@{->>|}[r]&B}$ typesets A / / B . The second argument (here {}) 3.2 Extra entries outside the matrix can be used to specify a point size {10}, {11}, or It is possible to put extra entries in your diagrams {12}, if desired (the default is {10}8); {cm} can be that are not part of any “entry” of the matrix cre- replaced by {eu} and {xy} to get tips in “Euler” and ated by & and \\. This is done with the excursion the default technical style, respectively. The declara- command tion respects TEX grouping. \save t \restore / Exercise 15: Typeset A / / /B . Hint: With the where t should be a target in one of the formats de- construction \objectobject one can typeset an XY-pic scribed in sections 2.5-2.6. t can do any kind of type- object (anything that could follow a * in section 2.2) setting desired, for example, in any context. \xymatrix{ A \ar@{-}[dr] 3 More Entries &{}\save[]+<3cm,0cm>*\txt<8pc>{% This is a very big commentary This section explains what can go in an entry and but it does not otherwise affect how the general form of the entries is changed. the diagram.} \ar[l] \ar[d] \restore \\ & B \ar@{-}[r] & C \ar@{-}[r] & D 3.1 Manual entry formatting } All the entries we have seen thus far have been simple will typeset math objects. However, it is possible to change the format of an individual entry by using the form: This is a very big commentary but it A o @ does not otherwise *object arrows @@ @@ a↵ect the diagram. @@ mmm mmm This allows complete control over what object is B vm C D placed in the entry, overriding any spacing and other conventions for the entry. This was how the frame It illustrates how a “down” arrow does not necessar- was obtained in the figure in section 1.2. ily have to point particularly straight down – in this A simple use of this is to insert text in entries us- case because it is based in the displaced pseudo entry. ing \txt objects just like labels as described above in section 2.2: 3.3 Spacing and rotation \xymatrix{ The *object form described above can be used to *\txt{A very long and stupid\\program} space individual objects di↵erently, however, it is also \ar[rr]^-{\txt{weird\\arrow}} possible to change the overall spacing of a matrix by &&*\txt<2pc>{Com\-pli\-cated\\Code}} inserting the following codes between \xymatrix and will typeset the following {:
weird Com- @=dimen set spacing A very long and stupid arrow pli- @R=dimen set row spacing / program cated @C=dimen set column spacing Code @M=dimen set entry default margin @W=dimen set entry default width
8 Except when XY-pic is loaded as a LATEX2" package where the default size is used.
9 @H=dimen set entry default height Similarly, all entries are rectangular by default, @L=dimen set label margin but it is possible to change this to round by declaring the default modifiers to be applied to all entries that = can be replaced by any of +, +=, -, and -= of sec- do not override them (cf. section 2.2): tion 2.2 with the same meaning, i.e., replace “set” \entrymodifiers={++[o][F-]} with “increase,” “increase to at most,” “decrease,” \SelectTips{cm}{} and “decrease to at least,” respectively. For exam- \xymatrix @-1pc { ple, $\xymatrix@1@=0pt@M=0pt{A&B\\C&D}$ in the *\txt{start} \ar[r] text typesetsAB. CD & 0 \ar@(r,u)[]^b \ar[r]_a In case uniform spacing is desired, several alter- & 1 \ar[r]^b \ar@(r,d)[]_a native forms exist. & 2 \ar[r]^b \ar ‘dr_l[l] ‘_ur[l] _a [l] @! force all spaces equal &*++[o][F=]{3} @!0 — and ignore entry sizes \ar ‘ur^l[lll]‘^dr[lll]^b [lll] @!R force equal row spacing \ar ‘dr_l[ll] ‘_ur[ll] [ll] } @!C force equal column spacing will typeset Finally a special notation allows rotation of an entire matrix: b ⌦ b / / b / b / start 0 a 1 2 3 @ d rotate towards d ? T a a ECD@GF Only the matrix grid will rotate, however, not the ?>=<89:; ?>=<89:; ?>=<89:; ?>=<89:;/.-,()*+ actual contents. Notice how we obtain the doubleBCD@GA ringBCD@GA using the *object form which then has to include all the desired Exercise 17: Typeset the “strip lemma” modifiers (and how the use of computer modern tips is nice for diagrams as these). a ? ?? a0 3.5 Naming for later use as targets b If you build an entry with a long and complicated b0 excursion then you might wish to be able to refer to it later. XY-pic provides a mechanism for this: there is a special target form which we haven’t discussed 3.4 Entry style yet: As mentioned above, the entries of a diagram are set t ="name" in math mode in text style. You may change this by redefinining the macro \objectstyle, and the la- bel style by redefining \labelstyle. We can combine This will introduce the new target "name" which will this with the above to get “small diagrams,” e.g., typ- refer to the target just before the =. This is particu- ing larly useful inside excursions, of course, and can also be used after labels. $\left( \def\objectstyle{\scriptstyle} a ( b \def\labelstyle{\scriptstyle} Exercise 18: Typeset A / B / C . \vcenter{\xymatrix @-1.2pc @ur { A \ar[r]^{a} & B \ar[d]^{b} \\ A’\ar[u]^{a’} & B’\ar[l]^{b’} }} 3.6 Grouping objects \right)$ Sometimes you wish to frame or otherwise treat a B a b rectangle of objects as a single object. This is pos- ? ?? sible with the last two target position forms that we in a paragraph will typeset “0 A B0 1.” _?? will mention: a0 b0 B A0 C B C You can even abandon the@ use of mathA mode en- t.s merge t with simple s tirely: the command \def \objectstyle {\hbox} {t} make t simple will change the format of entries to plain text.
10 The first will enlarge t to also “cover” the “simple” 4.1 Getting XY-pic s (simple means that it cannot have changes etc. at- tached unless encapsulated in {}s). Here is an exam- The easiest way to retrieve the latest version of ple where we merge and frame: XY-pic is through the XY-pic home page [5] on the World Wide Web; it is also available by Inter- \xymatrix @R=1pc { net ftp from url: ftp://ftp.diku.dk/diku/users/kris/ 1,1 & 1,2 & 1,3 & 1,4 & 1,5 \\ TeX/ as wellh as url: ftp://ftp.mpce.mq.edu.au/ 2,1 & 2,2 & 2,3 & 2,4 & 2,5 pub/maths/TeX/i inh files starting with xy. Finally, i \save "1,2"."2,4"*[F.]\frm{} XY-pic has been contributed to the CTAN archives \ar"1,1" \ar"2,1" \ar"1,5" \ar"2,5" where it is located in the directory macros/generic/ \restore } diagrams/xypic. will typeset License: XY-pic is free software in the sense that it 1, 11o , 21, 31, 41/ , 5 is available under the following license conditions: r rrr yrr YY X -pic: Graphs and Diagrams with T X 2, 12, 22, 32, 42YYY, , 5 Y E c 1991–1997 Kristo↵er H. Rose c 1994–1997 Ross Moore As you can see, the center of the merged object is the same as the one of the target before the “..” The XY-pic package is free software; you can redis- Finally a more advanced example where we create tribute it and/or modify it under the terms of the two merged objects with center in their “real” center, GNU General Public License as published by the Free name them and then connect to them. It also shows Software Foundation; either version 2 of the License, how macros can be used inside diagrams: they should or (at your option) any later version. always expand to “commands” like \ar. . . , etc.: The XY-pic package is distributed in the hope that it will be useful, but without any warranty; without A / B A / B even the implied warranty of merchantability or fit- O 0 2 0 O ness for a particular purpose. See the GNU General Public License for more details. e e e2 e e e✏ ✏ C oe D C0 o D0 You should have received a copy of the GNU Gen- eral Public License along with this package; if not, can be typeset by write to the Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. \def\g#1{\save In practice this means that you are free to use X -pic [].[dr]!C="g#1"*[F]\frm{}\restore}% Y for your documents but if you distribute any part of \xymatrix{ X -pic (including modified versions) to someone then \g1 A\ar[r]&B\ar[d]&\g2 A’\ar[r]&B’\ar[d]\\ Y you are obliged to ensure that the full source text of C\ar[u]&D\ar[l]& C’\ar[u]&D’\ar[l] X -pic is available to them (the full text of the license \ar @{.>} "g1" ;"1,4" Y in the file COPYING explains this in somewhat more \ar @{-->} "2,1";"g2" } detail ^¨ ). Then we can make arrows from/to the two frames by using the two new targets "g1" and "g2" as shown. 4.2 Backwards compatibility
The first widely distributed version of XY-pic was ver- Exercise 19: Change the lambda cube of exer- sion 2 (from release 1.40 to release 2.6). A special cise 12 such it is enclosed in a transparent (i.e., dot- compatibility mode is used automatically if the old ted) cube. style of loading is used (using files named xypic.tex and xypic.sty). You can also mix old and new di- agrams in a document if you load as described in 4 Availability and Further In- section 1.1 and add the declaration \xyoption{v2}. This provides almost full backwards compatibility: formation the following are the only known exceptions:
Below we describe how to and the conditions for ob- Automatic “shortening” of arrow tails by |<< • taining XY-pic version 3.7, the compatibility with pre- break was a bug and has been “fixed” so it does vious versions, and we conclude with a few appetisers not work any more. Fix: Put a |<\hole break to lure the reader into reading more about XY-pic. before it as described in section 2.3.
11 The release 2.6 * position operator is not avail- categorical twocells like • able. Fix: Use the : and :: operators (de- ⇣ scribed in detail in the reference manual [8]). 1 BC ( ↵◆ 6 RRR ↵ RR ( Using t ;t :(x,y) as the target of an arrow JJJ ⇠ • 1 2 !) v~ ttt command does not work. Fix: Enclose it in ✏ braces, i.e., write {t1;t2:(x,y)}. A D 8 ↵◆ The old \pit, \apit, and \bpit commands are • not defined. Fix: Use *@{>} (or \tip) with variants and rotation. The “poly” feature allows polygon-like struc- • tures: The even older notation where an argument in /'$ < • ⌅⌅ ✓$'/ < braces to \rto and the others was automati- ⌅ ✓ $''// << ⌅ ✓ $ ' / < ⌅⌅ ✓✓ $ ' / ⇧99 << cally taken to be a “tail” is not supported. Fix: ⌅ \ ✓\\ $ ' / ⇧ 9 Hm⌅⌅mm ✓ \\$ HH// ⇧⇧ 99 Use the supported |<. . . notation. HH✓ $$ H ⇧ 9 \✓ \\\\mmm ⇧⇧ 9
Finally note that sometimes the spacing with ver- The “web” feature allows composition in lat- sion 3.7 is “improved” relative to earlier versions ^¨ . • tices such as Please report all other things that do not work the same in version 2.6 and 3.7 to the author. (2) O ...... • . . . . . 4.3 Further reading ...... The reference manual [8] describes several more in- • . . / (1) put modes that are useful when the diagram is not . • . • . • . . organised as a matrix. We’ll give some examples of . • ...... such diagrams but refer to the reference manual for . . . . . the details. • . The “arc” feature is suitable for e↵ects like The “graph” feature allows input of data struc- • • tured as directed graphs to make it easy to pro- p duce such pictures as •?? ?? ?? w ?? ?? ? ? •c x A s gg 1 O ggggg ✏ O R z?WgWW ?? WW+ WWW ?? x2 ?? ? y In each case the figure is produced with relatively few lines using techniques documented in the respective The “knot” feature allows drawing of mathe- sections of the reference manual together with the de- • matical knots and links like tails of the XY-pic kernel and extensions, “arrow,” and “matrix” features as used in this guide; further exten- ⌫ sions provide support for use of colour, PostScript 2 e↵ects, and much more. Finally, the following documents might prove use- J ful: How to Typeset Pretty Diagram Arrows with M TEX [6] presents some of the design decisions be- g hind XY-pic, XY-pic and Notation for Categorical Di- agrams [7] explains how the modularity of X -pic can simple link figure-8 knot cinquefoil Y be used to obtain complex e↵ects with commuta- tive diagrams, and Typesetting Neural Nets using XY- The “2cell” feature provides special support for pic [4] explains how neural networks can be typeset •
12 using the package. Finally, a collection of complex Note that both arrows are curved “above” relative to examples is maintained on the XY-pic home page and their direction. the ftp servers discussed above. Answer to exercise 6 (p.5): The author used the 4.4 Credits display \xymatrix{ X -pic version 2 was a small package created by the Y 1 \ar[rr] ^-{1000000x} author. The present version 3.7 is the result of more \ar[dr] _(.2){2000x} |!{[d];[rr]}\hole than three years of continued collaboration with Ross && 1000000 \\ Moore,9 and would never have been developed with- 1000 \ar[r] _{2x} out his support and T X-nical expertise.10 This in- E \ar[urr] _>>>>{x^2} cludes many of the modes mentioned above which & 2000 } were created for use within specialised branches of mathematics, however, the notations are designed to In particular notice how the break was specified to fit quite naturally within the framework of XY-pic and happen exactly where the two arrows cross. For an are useful more generally. Finally, we are grateful to easier but not so general method see exercise 12 last Y&Y Inc. for making very nicely done PostScript in section 2.7. versions of the XY-pic fonts available. Answer to exercise 7 (p.5): The author typed A Answers to all exercises \xymatrix@1{ A \ar[r]^*+[o][F-]{x} & B }
Answer to exercise 1 (p.2): The target [] is the Answer to exercise 8 (p.5): The author did current entry itself. \xymatrix@1{ A \ar @{/*\composite{{+}*{\times}}/} [rr] Answer to exercise 2 (p.3): The author did ^*+\txt{High\\label} \xymatrix{ && B} {\bullet} \ar@{--}[d]\ar@{=}[dr]\ar@{-}[r] & {\bullet} \ar@{.}[d] \\ {\bullet} & {\bullet} \ar[l] } Answer to exercise 9 (p.6): The author entered \xymatrix{ Notice how has been enclosed in {} since it is an A \ar @/^/ @{^{<}-_{>}} [r] “expandable”• entity, i.e., a defined macro: this is rec- \ar @/_/ @{{*}{x}{*}} [r] & B } ommended.
Answer to exercise 3 (p.3): The author used Answer to exercise 10 (p.6): The author typed \xymatrix{ \xymatrix@1{ A \ar[r]^f \ar[dr]_{f;g} A \ar@/^/[r] \ar@/^/@<-1ex>[r] & B } & B \ar[d]^g \ar[dr]^{g;h} \\ & C \ar[r]_h & D } Answer to exercise 11 (p.7): The size: t+0 al- ways has zero size. Answer to exercise 4 (p.3): The author entered \xymatrix{ Answer to exercise 12 (p.7): The author con- A \ar[d]_f \ar[r]^f structed & B \ar[dl]|{i_B} \ar[d]^g \\ \xymatrix@!0{ B \ar[r]_g & C } & \lambda\omega \ar@{-}[rr]\ar@{-}’[d][dd] & & \lambda C \ar@{-}[dd] \\ Answer to exercise 5 (p.4): The author did \lambda2 \ar@{-}[ur]\ar@{-}[rr]\ar@{-}[dd] \xymatrix{ & & \lambda P2 \ar@{-}[ur]\ar@{-}[dd] x \ar@(ul,dl)[]|{id} \ar@/^/[rr]|f \\ && f(x) \ar@/^/[ll]|{f^{-1}} } & \lambda\underline\omega \ar@{-}’[r][rr] 9Mathematics Department, Macquarie University, Sydney, Australia 2109; e-mail: [email protected]. 10Partial funding for this collaboration has been provided by a Macquarie University Research Grant (MURG), by the Australian Research Council (ARC), and through a research agreement with the Digital Equipment Corporation (DEC).
13 & & \lambda P\underline\omega Answer to exercise 19 (p.11): The author added \\ the XY-code \lambda{\to} \ar@{-}[rr]\ar@{-}[ur] \save [].[dd].[ddrr].[rr]*[F.]\frm{}="back" & & \lambda P \ar@{-}[ur] \restore } in the entry with !, and A special thing is added: @! forces rows and columns to be equally spaced as discussed in sec- \save [].[dd].[ddrr].[rr]*[F.]\frm{} tion 3.3; @!0 furthermore makes the spacing ignore \ar@{.} +UL;"back"+UL \ar@{.} +UR;"back"+UR the entry sizes, giving a completely fixed grid. The \ar@{.} +DL;"back"+DL \ar@{.} +DR;"back"+DR gaps could also be made with the !. . . crossing nota- \restore tion of exercise 6 last in section 2.1 but the above is in the entry with 2 to produce shorter. ! C Answer to exercise 13 (p.8): The author typed 2 P 2 \xymatrix@1{ A \ar@<-2pt> ‘d[r] ‘[r] [r] ! P! \ar@<+2pt> ‘d[r] ‘[r] [r] & B } P ! Answer to exercise 14 (p.8): The author used \newdir{ >}{{}*!/-5pt/@{>}} References
Answer to exercise 15 (p.9): The author typed [1] Adobe Systems Incorporated. PostScript Lan- \xymatrix@1{ guage Reference Manual, second edition, 1990. A \ar[r] |-{\SelectTips{cm}{}\object@{>>}} [2] Donald E. Knuth. The T Xbook. Addison-Wesley, |>{\SelectTips{eu}{}\object@{>}} E 1984. &B} [3] Leslie Lamport. LATEX—A Document Preparation Answer to exercise 16 (p.9): The author typed System. Addison-Wesley, 2nd edition, 1994. \xymatrix{ [4] Ross R. Moore. Typesetting Neural Nets using *=0{\bullet} XY-pic, 1994. Available through [5]. \ar@/^/[dr]!U|1 [5] Kristo↵er H. Rose. The X -pic home page. url: \ar@/^/[drr]!U|2 Y http://www.ens-lyon.fr/˜krisrose/Xy-pic.htmlh . \ar@/^/[drrr]!U|3 i \\ [6] Kristo↵er H. Rose. How to typeset pretty dia- &*+[F]\txt{x} gram arrows with TEX—design decisions used in &*+[F]\txt{x\\x} XY-pic. In Jiˇr´ıZlatuˇska, editor, EuroTEX ’92— &*+[F]\txt{x\\x\\x} Proceedings of the 7th European TEX Conference, } pages 183–190, Prague, Czechoslovakia, Septem- ber 1992. Czechoslovak TEX Users Group. Answer to exercise 17 (p.10): The author en- [7] Kristo↵er H. Rose. XY-pic and notation for cate- tered the display gorical diagrams. Invited talk at ECCT-94, July \xymatrix@dr@C=1pc{ 1994. Available through [5]. a \ar[r]\ar@{->>}[d] & a’ \ar@{.>>}[d] \\ b \ar@{.>>}[r] & b’ } [8] Kristo↵er H. Rose and Ross R. Moore. XY-pic Ref- erence Manual. DIKU, University of Copenhagen, Universitetsparken 1, DK–2100 København Ø, Answer to exercise 18 (p.10): The author typed 3.0 edition, June 1995. \xymatrix{ [9] Michael D. Spivak. The Joy of T X—A Gourmet A \ar[r] ^a="a" & B \ar[r] ^b="b" & C E Guide to Typesetting with the -T X Macro \ar @/^/ "a";"b" } E Package. American MathematicalAM Society,S second Notice the use of both explicit base and target in the edition, 1990. arrow between the labels.
14 Index
!, 5, 7, 8 [u],5 COPYING, 11 !{;},5 \\, 2, 5 copyright, 11 &,2 ^, 3, 4, 6–8 cover, 11 ’,7 ^{’},6 crossing arrows, 5, 7 (,6 ^{‘},6 cube, 7 (),5 ^{|-},6 current direction, 7 (0),5 _, 3, 4, 6–8 current entry, 6 (1),5 _{’},6 curve, 4, 8 ),6 _{‘},6 d,2 *, 5, 7–9, 12 _{|-},6 dashed arrow, 6 +, 5–7, 10 ‘,8 dashed frame, 5 +=, 5, 10 {*},6 default modifier, 10 -, 3, 5–7, 10 {|-},6 default spacing, 5 --,6 {|=},6 defining arrows, 8 -=, 5, 10 |, 3, 4, 6 degrees, 7 .,6 |<,6 directed graphs, 12 /, 6–8 |<<, 6, 11 direction, 4, 7, 8 //,6 ||,6 directionals, 8 :(),7 ~,6 dotted arrow, 6 :a(),7 ~~,6 dotted frame, 5 ;,7 2cell, 12 double arrow, 6 <, 4, 6 3{|-},6 double frame, 5 <<, 4, 6 0,7 =, 10 elliptical, 10 absolute angle, 7 >, 4, 6 entries outside matrix, 9 absolute entry, 7 >>, 4, 6 entries with text, 9 adjust, 5 >>|,6 entry, 2, 9 adjusting position, 7 >|,6 entry format, 9 \apit , 12 @, 3, 5, 10 entry outside matrix, 3 \ar , 2, 4 @!, 10 entry style, 10 arc, 12 @(,4 excursion, 9 arrow, 2, 4 @/^/,4 explicit positioning, 4 arrow head, 5 @/_/,4 arrow passing under, 7 extra entries, 9 @1,3 arrow shaft, 5 format, 9 @<>,6 arrow style, 3, 5 frame, 5 @=,9 arrow tail, 5 free software, 11 @C=,9 \frm ,5 @H=, 10 base entry, 2, 7 ftp, 11 @L=, 10 bend, 8 @M=,9 \bpit , 12 GNU General Public License, @R=,9 break, 3 11 @W=,9 centered, 5 going around, 8 @{},3 changing position, 7 graph, 12 [F-,],5 changing turn radius, 8 grouping, 10 [F--],5 circle segments, 8 grow, 5 [F-:<3pt>],5 circular, 10 half arrow, 6 [F.],5 columns, 2, 6 \hbox ,5 [F=],5 combined objects, 5, 8 head, 5 [F],5 commutative diagrams, 1, 4 \hole ,3 [], 3, 6 compatibility, 11 hop, 2, 6 [d],5 \CompileMatrices ,4 [l],5 \composite , 5, 8 intersects, 5 [o],5 computer modern fonts, 9 invisible arrow, 3 [r],5 coordinates, 3 \jot ,4
15 knot, 12 polygon-like, 12 target entry, 2 l,2 position, 6, 7 text, 5 label, 3, 4, 8 ps,2 text in entries, 9 label centered on arrow, 3 quarter turn, 8 text label, 5 text style, 2 label style, 10 r,2 tip, 5, 8 label with any object, 5 radius, 8 tip characters, 8 labels as targets, 10 relative angle, 7 tip in shaft, 6 labels with text, 9 relative entry, 6 triple arrow, 6 \labelstyle , 10 relative vector, 7 turn, 8 LATEX2",2 repeat last object, 5 twocells, 12 license, 11 retrieving XY-pic, 11 \txt line break, 5 rotation, 8, 10 , 5, 9 links, 12 round, 10 u,2 loading, 2 round shape, 5 uniform spacing, 10 merge, 11 rounded frame, 5 \usepackage ,2 moving target, 7 rows, 2, 6 v2,2 name, 11 segment, 7 va(),7 new arrows, 8 \SelectTips {cm}{},9 variant, 5, 8 \newdir ,8 shaded frame, 5 vector, 7 \NoCompileMatrices ,4 shaft, 5, 8 vector coordinates, 7 shaft as tip, 6 vector in direction, 7 o,6 shaft characters, 8 vector to corner, 7 object, 5 shape, 5 version 2, 11 \object ,9 \ShowOutlines ,4 object modifier, 5, 8, 10 shrink, 5 warranty, 11 \objectstyle , 10 sideways, 6 web, 12 old style of loading, 11 size, 5 World Wide Web, 11 \OnlyOutlines ,4 sliding, 6 x,6 overwrite, 7 spacing, 9 \xymatrix ,2 parallel, 6 square, 5 \xymatrixnocompile ,4 \pit , 12 squiggly arrow, 6 \xyoption , 2, 11 plain text entries, 10 tail, 5 xypic.sty, 11 poly, 12 target, 6, 7, 10 xypic.tex, 11
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