User's guide for the Diagram Macros Contents
Contents
1. 58pt 10 8 Ternary diagram arrows The ternary diagram arrows are those oriented along the diagonals of a rect angle of sides 1 2 they have a standard horizontal extent equal to 66pt or 132pt according to the case Taking some liberty with trigonometry and geography we shall name them after the ternary directions of the compass card north north east nne Jeast north east ene east south east ese south south east sse south south west ssw west south west wsw west north west wnw north north west nnw The only type of ternary arrows provided by the Diagram Macros are the plain arrows Again you will obtain the command for a ternary arrow by pre ceeding the abbreviation ar with the prefix indicating the direction of the arrow Here are thus the eight available ternary arrows A I 2 nnwar is nnear e wnwar enear wswar esear e e sswar ssear 11 9 Naming the diagram arrows The rules for naming the diagram arrows are analogous to those for naming the in text arrows with the additional possibility of naming the arrow on either of its sides It suffice to type the first or the last letter of the command as an upper case letter and give the name of the arrow as an argument Here is an exhaustive list of the available possibilities Eql An equality cannot be named Quadriar quadriadjar quintiar quintiadjar Quadruple and quintuple ar rows cannot be named Verti2. arrow backward plain arrowNbkar backward dotted arrowNbkdotar lt backward monomorphism bkmono backward epimorphism bkepi backward bimorphism bkbimo backward isomorphism bkiso lt backward pair of parallel arrows bkbiar lt _ backward equality bkeq1 backward pair of adjoint arrows bkadjar backward triple of parallel arrows bktriar backward triple of adjoint arrowsNbktriadjar backward quadruple of parallel arrows bkquadriar backward quadruple of adjoint arrows bkquadriadjar _ backward quintuple of parallel arrows bkquintiar backward quintuple of adjoint arrowsNbkquintiadjar A 4 Naming the in text arrows In a line of text it is probably a better idea to type f A B than A B Nevertheless the Diagram Macros allow you to give an upper name to the single and triple in text arrows as well as an upper lower name to the double in text arrows It is a general rule of the Diagram Macros that to give an upper name to a single or a triple arrow or an upper lower name to a double arrow it suffices to 1 type the first letter of the corresponding command as an upper case letter 2 give the name s of the arrow s as argument s of the command On next page is a list of the available named in text arrows The name of an arrow will always be processed in Math mode at the scrip style size therefore the text characters will be printed
3. fit nicely in a dia gram their variety and their names are the same as those of in text arrows plain arrow ar dotted arrow dotar monomorphism mono epimorphism epi bimorphism bimo isomorphism iso pair of parallel arrows biar equality eq1 pair of adjoint arrows adjar triple of parallel arrows triar triple af adjoint arrows triadjar quadruple of parallel arrows quadriar quadru ple of adjoint arrows quadriadjar quintuple of parallel arrows quintiar and quintuple of adjoint arrows quintiadjar You should never use an in text arrow in a diagram nor a diagram arrow in a line of text the technical characteristics of those pictures are just incompatible While an in text arrow is always horizontal the possibility must exist to give a diagram arrow a rather arbitrary direction For that reason the name of a diagram arrow starts always with the prefix indicating its direction even in the case of an horizontal right pointing arrow The direction of an arrow is indicated by a direction of the compass card using the classical abbreviations north n south s east e west w The command for producing a primary arrow is obtained by typing the abbreviation for its direction followed by the abbreviation for the type of arrow For example the command wbimo will produce a west pointing bimorphism Here are other examples E e nar e wmono ug Veepi A sbiar The case of adjoint arrows requ
4. User s guide for the Diagram Macros Francis Borceux UCL Louvain la Neuve Belgium Contents OONDPAN Inserting in text arrows liess enn Backward in text AaYFfOWS 2er de ub tacet der se sea Naming the in text arrows eeseeseeeeee cence eee Emphasizing An arroW c 4k hike kk 4gemk habit are da RETE PR Re a on Primary diagram arrows sseseeseeeeee een ene nes Secondary diagram arrows cee e e Ternary diagram arrowS 0 cece eect e hn Naming the diagram arrows sseseeseeseee eee eee S Corcelving a diagram z oso dettes ti venue xa ek e eus Lyping a diagrami carea e rau rtr oh E II ER RR IER ER ILE eT The variable length option cc cece cece eee eee ee eens Superposing items in a diagram lesse Drawing curved AroWS mi isrener seur Re E RLe RR Pe aad ea be aad eid The free slope arrows cc cece eee I Bhe sealing Options sche zc etr oet exe pete Mee eed eae A Binal adjustments evene Re Week CE EORR een Some special characters lesse eee bitnet FBORCEUX BUCLLIN11 reti 17 19 wo 22 24 1 What are the Diagram Macros Diagram is a series of Macros intended to produce easily rather sophisti cated diagrams which can appear when typing category theory The range of arrows which can be produced includes plain morphisms dotted morphisms monomorphisms epimorphisms bimorphisms isomorphisms equali
5. at the new horizontal extent of a secondary arrow will be 58 n x 40 where n is the number of horizontal intervals the arrow crosses 19 13 Superposing items in a diagram Sometimes you will like to superpose two items at the same point of the pattern underlying your diagram like in the following example A B C D The command for superposing two items is cross 1 2 where 1 and 2 stand for the two items the two arguments 1 and 2 are automatically processed in Math mode The previous diagram can thus be typed as DIAG A n ear n B nn nar n cross near sear n sar nn C n ear n D diag When superposing two arrows the cross command can have the undesired effect that one of the arrows will bump into the name of the other arrow A special command has been designed to correct such a defect and is explained in section 17 For that reason the cross option could be considered as a special feature of the Diagram Macros one which requires your help 20 14 Drawing curved arrows Drawing curved arrows is another special feature of the Diagram Macros thus a process where the Diagram Macros need you help In fact you will have to decide the length of the curved arrow and possibly you will have to adjust the spacing around the diagram an easy procedure for that will be described in section 17 Just plain curved arrows in the primary directions are provided by the Dia gram Macro
6. b where e t is the vertical space expressed in points to be added at the top of the diagram e 1 is the horizontal space expressed in points to be added left of the diagram e b is the vertical space expressed in points to be added at the bottom of the diagram This command will not work properly if your diagram is wider than the tex twidth in that case include the diagram in a mini page environment see IATEXbook page 98 wide enough to contain it and position this mini page in your document using for example the commands hspace and vspace Finally some unexpected troubles can occur if you introduce in a diagram items which have not been designed in a way which is compatible with the internal structure of the Diagram Macros You will most often get rid of the problem by including the new item in a box of formal dimensions 0 0 27 18 Some special characters This last section contains some text symbols which you can want to use in connection with the Diagram Macros The commutative diagram symbol can be produced by giving the com mand com and the adjoint symbol is obtained via the command adj Finally here are some commands for designing natural transformations with or without name wat J Nat F a 6 Gaz binat oe F Vinat FHoHGH8H8 E35 In the case of named natural transformations the names of the functors and the natural transformations are processed in Math mode thus do
7. cal ar mono epi bimo dotar A first upper case letter produces a left name and a last upper case letter produces a right name thus type Smono f to draw a south monomorphism with left name f and nepI g to draw a north epimorphism with right name g Non vertical ar mono epi bimo dotar A first upper case letter produces an upper name and a last upper case letter produces a lower name thus type War f to draw a west arrow with upper name f and nebim0 g to draw a north east bimorphism with lower name g Horizontal iso A first upper case letter produces an upper name and a last upper case letter produces a lower name thus type Wiso f to draw a west isomorphism with upper name f and eisO g to draw an east isomorphism with lower name g Non horizontal iso Type a first upper case letter to name it the name will appear on one side of the arrow and the isomorphism symbol on the other side thus Nwiso f draws a north west isomorphism with name f Vertical biar adjar Type a first upper case letter to name them and give successively the name of the left arrow and that of the right arrow thus Sbiar f g draws a pair of south arrows with left name f and right name g Non vertical biar adjar Type a first upper case letter to name them and give successively the name of the upper arrow and that of the lower arrow thus Seadjar f g will produce a south east pair of adjoint arrows with upper name f and lower name g Ve
8. d admit a variable length option e The four secondary plain arrows named or unamed admit a variable length option the other secondary arrows are not provided with the vari able length option e The ternary arrows do not have the variable length option To take advantage of the variable length option it suffices to follow the name of the corresponding command with the lower case letter v and give to that command an additional last argument which will be e The new length of the arrow in the case of a primary arrow e The new horizontal extent of the arrow in the case of a secondary arrow those two lengthes must be expressed in points The new arrow will still be centered with respect to the corresponding point of the pattern For example typing Earv f 130 for the east arrow of the previous diagram will produce the desired result as in the first diagram Typing smon0Ov g 30 will pro duce a south monomorphism with right name g and length 30 points typing Wadjarv f g 100 will produce a west pair of adjoint arrows with names f g and length 100 points and typing near 82 5 will produce an unamed north east arrow with horizontal extent 82 5 points The standard length of a primary arrow is 50pt since each interval of the pattern has length 40pt the new length of a primary arrow will be 50 4 n x 40 where n is the number of intervals the arrow crosses In the same way the horizontal extent of a secondary arrow is 58pt so th
9. e nis the name of the arrow e a b are the coordinates in points of the place where you want the lower left corner of that name to be positioned 23 e x y are the coordinates of the origin of the arrow e u v is the slope of the arrow it must be one of the slopes just indicated e 1 is the horizontal extent in points of the arrow or its length in the case of a vertical arrow If you do not want to name the arrow just type following the usual Diagram Macros convention freear x y u v 1 Clearly the free slope arrow option is not a Diagram Macros feature it is just plain IATEX job 24 16 The scaling option The variable length option has allowed us to design properly the diagram of 812 But you can possibly not be satisfied with the final product because the resulting diagram is unusually big for the few arrows it does contain This is easily corrected Replacing the initial command DIAG by the corresponding variable scale command DIAGV n will scale the original pattern of points by n and adjust correspondingly the length of every arrow in the diagram For example typing DIAGV 50 A n n Earv 130 n n C nn n seepI p n n nemon0 i nn F n n B diag will produce the following result M Since text characters and various other things are unaffected by the scaling process you should avoid reducing exagerately a diagram In most cases a reduction up t
10. in Math italic and you should not type any sign to introduce a mathematical symbol in the name of an arrow 5 Emphasizing an arrow The dotted arrows are a first type of emphasized arrows their construction is rather slow It will be easier for IXTEX to print an arrow in bold face style to realize this just preceed the command for the arrow by the command B thus A B Mono f B produces A B That bolding process applies to both the in text and the diagram arrows plain arrow fMr dotted arrow fNDotar f monomorphism f Mono f f epimorphism fNEpi f L bimorphism f Bimo f te isomorphism f Iso amp pair of parallel arrows f g Biar f g a ir of adjoint 3 Adj f f pair of adjoint arrows f gNAdjart Hgl backward plain arrow f Bkar lt backward monomorphism f Bkmono f aa backward epimorphism f Bkepi f lt backward bimorphism f Bkbimo f 4 backward isomorphism f Bkiso f lt pair of parallel backward arrows f g Bkbiar f g gas backward pair of adjointarrows f g NBkadjar fMg _f triple of parallel arrows f g h Triar f g h f backward triple of parallel arrows f g h Bktriar f g h f triple of adjoint arrows f g h NTriadj ar fHgHh f backward triple of adjoint arrows f g h NBktriadj ar f g h 6 Primary diagram arrows The diagram arrows are those arrows specially designed to
11. in text arrow is one which you include in a line of text like for example when you write Let f A4 B be a monomorphism Here are the available in text arrows and the corresponding commands The length of an in text arrow is 20pt plain arrowNar gt dotted arrow dotar monomorphismWnono epimorphismNepi bimorphism bimo gt isomorphism iso gt pair of parallel arrows biar E equality eq1 pair of adjoint arrows adjar gt triple of parallel arrows triar triple of adjoint arrowsNtriadjar quadruple of parallel arrowsNquadriar quadruple of adjoint arrows quadriadjar 4 quintuple of parallel arrows quintiar quintuple of adjoint arrows quintiadjar When you insert an in text arrow the Diagram Macros will take care of the spacing at both ends of the arrow The in text arrow commands can be used both in paragraph and in Math mode but normally you should use them in Math mode to print the surrounding characters in Math italic Compare indeed Let f A epi B be an epimorphism Let fA B be an epimorphism with Let f A epi B be an epimorphism Let f A B be an epimorphism 3 Backward in text arrows It is a general rule of the Diagram Macros that to draw an arrow in some given direction it suffices to preceed the name of the arrow by a code for the direction To draw an in text arrow in the backward direction it suffices to add the prefix bk to the command for that
12. ires a comment For east and west multiples of adjoint arrows the direction is that of the lower arrow for north and south pairs of adjoint arrows the direction is that of the left arrow The standard length of a primary arrow is 50pt but the length of an hori zontal arrow is automatically adjusted when the adjacent vertices have too long names 7 Secondary diagram arrows The secondary diagram arrows are those which are oriented in the secondary directions of the compass card here are thus the possible directions and the corresponding abbreviations north east ne Isouth east se north west nw south west sw The variety of available arrows is reduced to the simple and double arrows plain arrow ar dotted arrow dotar monomorphism mono epimorphism epi bimorphism bimo isomorphism iso pair of parallel arrows biar equality eql and pair of adjoint arrows adjar Just as for primary arrows a secondary arrow is obtained by typing the abbreviation for its direction followed by the abbreviation for the type of arrow For example the command swbiar will produce a pair of sout west pointing arrows Here are some examples of what you can produce as secondary diagram arrows m e e nwbimo neadjar e e e 7 sedotar The case of a pair of adjoint arrows requires again a comment The direction is always that of the lower arrow The secondary diagram arrows have a standard horizontal extent equal to
13. ms will thus appear as in your input Separate two consecutive items in a same row by the next item com mand which is n The command n may not appear before the first item of a row nor after the last item of a row Separate two consecutive rows of items by the next row command which is nn The command nn may not appear before the first row of items nor after the last row of items Type the command diag to indicate the end of the diagram typing When you type a diagram using the procedure just indicated notice that between the two commands DIAG and diag e You can put as much space as you want in a line to make your input file easier to read e You should never use a sign to type a mathematical symbol except if you did include that symbol in a box With all that in mind the diagram of the previous section can be typed as follows DIAG X nn n Sedotar z n Esearfy nn n sseaR x n P n Eepi k n B nn n n Smonofh n n smon0 g nn AO n A n eepI f n C diag or equivalently as DIAG X n n n n nn n Sedotar z n Esear y n n nn n sseaR x n P n Eepi k n B nn n n Smonof h n n smon0 g nn n n A n eepI f n C diag 16 The Diagram Macros will automatically center the diagram horizontally on the page and will take care of the spacing with the preceeding and the following text When a diagram is
14. nore completely all those differences and just treat equally all those items this allows a maxi mum of flexibility and simplicity To realize this the Diagram Macros consider a formal pattern of points at a 40pt distance from each other both horizontally and vertically Then each item of the diagram is centered at one of the points of the pattern using the picture environment Let us for example visualize the pattern of points on which the previous diagram has been constructed 14 The complete pattern needs not be a square nor even a rectangle It is just compulsory to start all the lines from a same left column and leave no hole at all in any line but lines need not be of equal length To fill in the holes which could appear inside a line it suffices to attach empty items at the corresponding points of the pattern Here is for example the set of items corresponding to the previous diagram Doe emm 11 15 Typing a diagram Let us assume you have determined the nature and the position of each item of your diagram Here is what you should type in order to give that information to the Diagram Macros 1 5 Type the command DIAG to start the typing of the diagram this com mand will in particular start a center environment it can be given both in paragraph and in math mode Type between curly brackets the description of every item this description will automatically be processed in Math mode The empty ite
15. not include any sign for putting a mathematical symbol in those names Here is how those symbols will be positioned in a line of text
16. o 50 gives excellent results On the other hand an enlargement can be helpful when handling vertices with very long names This variable scale option is particularly useful when designing three dimensional diagrams which require very often a rather dense pattern of refer ence points and many long arrows produced via the variable length option For example to design the following cube B B D A A h g F H M a Es f 1 CET A e E 25 it suffices to type DIAGV 50 n n B n n Earv b 130 n n D Ann n Near a n n n n Near d n nn A n n cross Earv movename c 10 0 130 Sarv movename f 0 13 130 n n C n n saRv h 130 nn nn Sarv e 130 n n F n n cross eaRv movename j 10 0 130 saRv movename g 0 13 130 n n H nn n Near i n n n n neaR 1 nn E n n eaRv k 130 n n G diag where the movename command is explained in the next section 26 17 Final adjusments The Diagram Macros are intended to provide you with a powerful tool for designing diagrams so in normal circumstances no adjusment at all should be necessary But in some unusual cases for example when you use the special features of the Diagram Macros you can want to interact for correcting some imperfections Some Macros have been designed to help you realizing this If the vertice
17. o items The length of a curved arrow will generally be a multiple of 80 points since this is the normal distance between two vertices in a primary direction example of a diagram containing curved arrows l A a B b C e Fg h k D E F It should thus be typed as follows DIAG A n Earf a n cross B Necurve 1 160 n Ear b n C nn cross Wncurve e 80 Sbiar f g n n saR h n n saR k nn D n eaR c n E n eaR d n F diagv 25 13 0 Here is an 22 15 The free slope arrows When you have to produce a given diagram of vertices and arrows you must clearly choose adequately the position of each vertex so that all the ar rows will be in one of the sixteen directions considered by the Diagram Macros Anybody who has read this user s guide will easily produce an example of a diagram where this requirement cannot be fulfilled Nevertheless such a situ ation is rather unusual since for example the most popular book on category theory Saunders MAC LANE Categories for the working mathematician Springer 1971 does not contain any such diagram Well I have been lucky Saunders did not draw the connecting morphism in the snake lemmal The slope of an arrow is a pair n m of integers when you move n points left you move m points up as usual a negative value of n or m indicates a movement in the opposite direction For example here are the slopes of the variou
18. rather big you should make it a figure or a table see INTEXbook 3 5 1 page 60 to improve the quality of the page making If you want to align horizontally various small diagrams consider these just as a unique wide diagram where some columns are completely empty 17 12 The variable length option Suppose you want to produce the following diagram A f oe B Following the instructions of the previous sections you will type DIAG A n n Ear f n n C nn n seepI p n n nemon0 i nn AO n B nn diag and you will get B which is not exactly what you wanted The Diagram Macros assume that a primary arrow connects two vertices attached at the two horizontally adjacent points of the pattern and a corre sponding assumption is made for the secondary and the ternary arrows The east arrow of the previous diagram does not satisfy that assumption it connects vertices situated at a four intervals distance The Diagram Macros provide you with a variable length option which allows you to modify the length of an arrow in order to take care of such particularities The variable length option is a special feature which means that the Diagram Macros need your help to handle the problem In fact what you have to do is to decide the new length of the arrow Not all arrows have been provided with a variable length option here are the possibilities 18 e All the primary arrows named or uname
19. rtical triar triadjar A first upper case letter produces left names and a last upper case letter produces right names give successively the names of the left central and right arrow thus Ntriar f g h draws a triple of north arrows with left names f g h 12 Horizontal triar triadjar A first upper case letter produces upper names and a last upper case letter produces lower names give successively the names of the upper central and lower arrow thus Wtriar f g h draws a triple of west arrows with left names f g h The name of an arrow will always be processed in Math mode at the textstyle size therefore the text characters will be printed in Math italic and you should not type any sign to introduce a mathematical symbol in the name of an arrow The name of an arrow will automatically be positioned in order not to bump into the corresponding arrow 13 10 Conceiving a diagram We are now arriving at the central section of this user s guide the one which describes the basic principle for realizing a diagram This will be better explained on an example Let us therefore suppose you want to construct the following diagram C This diagram contains several items five vertices A B C P X four pri mary arrows one secondary arrow and two ternary arrows one of the arrows is emphasized All those items are very different in nature size orientation position The first basic rule for conceiving a diagram is to ig
20. s The first letter of the command indicates where with respect to the diagram the arrow has to stand this determines the concavity of the ar row the second letter indicates the overall direction of the arrow then follows the abbreviation curve for curved arrow As usual the two first indications are given using the compass card terminology n s e w Typing the first letter as an upper case letter will produce a named arrow the name being given as a first argument of the command It is compulsory to give the length of the arrow as a last argument By length of the arrow we mean clearly its horizontal or vertical extent according to the case expressed in points For example Necurve f 160 will produce a curved arrow with name f and length 160 points to be positioned at the north side of a diagram and with overall east direction For more clarity here are the eight unamed curved arrows of length 80 points and the corresponding commands The bullets indicate the position of the formal centers of these arrows each bullet corresponds thus to the point which will coincide with the corresponding point of the pattern hecurve nwcurve e e e e securve swcurve wscurve e wncurve Vescurve e encurve Most often you will have to include a curved arrow at a point of the pattern which is already occupied by another item a vertex or an arrow just use the 21 cross command of the previous section to superpose the tw
21. s diagram arrows e The primary arrows have slope 1 0 horizontal arrows or 0 1 ver tical arrows e The secondary arrows have slope 1 1 e The ternary arrows have slope 1 2 or 2 1 according to the case In fact IATEXallows you to draw arrows in thirty two other directions using the picture environment and the commands put and Nvector the corresponding available slopes are 1 3 3 1 1 4 4 1 2 3 3 2 43 4 4 3 The length of an arrow is always specified by giving the horizon tal extent of the arrow except in the case of vertical arrows where the actual length is given So you can just design any arrow or set of arrows using the picture environment and choose the corresponding picture as an item of your diagram the center of your picture will be positioned automatically at the cor responding point of the pattern This is exactly what the Diagram Macros are doing any time you ask them to draw an arrow You can avoid some typing but just that by using the free slope arrow Macro to produce directly a diagram arrow like for primary secondary and ternary arrows It has the following form where as a matter of convention the origin of the coordinates is the point of the pattern at which you want to include the free slope arrow Freear n a b x y u v 1 where
22. s of a same row have names of different heights those names will no longer stand at the same altitude since they are centered at the corresponding points of the pattern There is a normalizing command NN which you can apply to the high vertices in order to correct this defect instead of typing vertex type N vertex where vertex stands here for the name of the vertex You can want to move the name of an arrow especially when you have used the Ncross command to superpose two named arrows If f is the name of the arrow instead of f you just type movename f n m where n and m are integers positive or negative the name of the arrow will be moved n pt right and m pt up In the same way the commands movearrow Ear f n m and moveverterx A n m will move respectively the east arrow f and the vertex A n pt right and m pt up Because the curved arrows are very excentric with respect to their formal center they will cause problems of spacing when drawn along an outer edge of a diagram Drawing a curved arrow along the upper or the lower edge of a dia gram will generally cause the diagram to overlap the surrounding text drawing a curved arrow along just one vertical side of a diagram will cause bad horizon tal centering Those defects can be corrected by using the variable spacing option of the diagram commands Just replace the final command diag by the corresponding variable spacing command diagv t 1
23. ties pairs triples quadruples and quintuples of parallel or adjoint arrows Most arrows can be drawned in the sixteen basic directions of the compass card and this inspires the terminology Some curved arrows are also available All these arrows can be given a name on either side of the arrow For emphasizing arrows can be printed in bold face type Except for some special features the Macros will take care of choosing the correct length of each arrow as well as positioning correctly the various elements which appear in the diagram The final result will be a diagram with rows and columns at a distance of 80 points 1 1in 28mm from each other this standard distance can be changed for every individual diagram just by giving a scaling factor Diagram includes also a list of in text symbols containing in particular the same variety of arrows in both the forward and the backward directions The Diagram Macros are IXTEX Macros which make an intensive use of the IXTEX picture environment and in particular of the vector command To use them for producing a document it suffices to have in your TEX file a copy of the file diagram and to call it with the command Ninput diagram in the preamble of your new document The Diagram macros will give good results with 10pt 1lpt and 12pt styles The following sections should provide you with full information on how to use the Diagram macros for producing your documents 2 Inserting in text arrows An
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