Commutative Diagrams with XY-pic I. Kernel
Contents
1. lt dir end xy 3 2 Commands for arrows To simplify the syntax for directionals which has the tendency of becoming quite complicated Xy pic also provides composite directionals i e arrows To use them you have to activate the option arrow For instance to draw an arrow from a position to another we have to use something like ar de ar gt fe ar gt gt fo ar lt a ar lt gt a ar 2 gt we ar gt Zo ar 3 gt Go ar lt Pa ar a ar va ar o a ar gt P ar _ gt JO ar gt a WE a ar 3 gt a ar gt a ar 2 gt a ar 3 gt a ar gt a ar gt gt 7 Table 3 Arrows B begin xy je ar 0 0 A 10 10 B A end xy ar is the simplest notation for an arrow The complete syntax is of the form ar type see the table 3 ar is a shorthand for ar gt The table 3 contains some of the most commonly used arrows We have to mention however that Xy pic actually offers the possibility of constructing a lot more arrows by using one of the following commands 1 ar variant tail shaft head 2 ar f head The head and the tail i e the tips can be any of the combinations from the table 4 In the table they are shown as heads of arrows but they can be used equally as tails The shafts of the arrows can be any of the arguments of the dir command from the first column of the table 1 As for the variant they2. ar type _length or ar type length depending on which part we want the curving to take place Here the length can be expressed in any accepted TEX unit 11 A B begin xy ar _ 1pc 0 0 A 15 0 B end xy It is even possible to omit the length and then the program will curve the arrow by a default quantity A B begin xy ar gt 7 0 0 A 15 0 B end xy 3 4 Labelling arrows Putting a label on an arrow is not difficult There are more ways of doing it If you construct the arrow by indicating separately the body and the tips then you can add a label in the following way A B begin xy 0 0 A 20 0 B dir gt dir gt _2mm alpha end xy The core of the label is alpha If we would use only that the label would stay on the arrow The part _2mm represent a shift Don t get confused in Xy pic the underscore _ indicates that the label is on the left of the arrow when looking from tail to head while the sign indicates that the label should be on the right By default labels are on the left If in the previous example we omit the part _2mm keeping the exclamation mark we get A gt B begin xy 0 0 A 20 0 B dir gt dir gt alpha end xy 12 If we draw an arrow by using a specific command then we have two opportu nities to add labels A B b
3. access to each of the component of the arrow and we can apply the chosen variant to each By doing so we can produce for instance monsters like 10 AS B begin xy ar F2 _ gt 0 0 A 20 0 B end xy as well as perfectly normal and very useful arrows as AF 8 begin xy C gt D ar gt 0 0 A 20 0 B n ar _ gt 0 5 C 20 5 D ar _ gt 0 10 E 20 10 F end xy From the way our monster looks like we can draw one conclusion although it is technically possible to use a variant only for the shaft or for the shaft and one or both of the tips in practice this possibility is useless because the corresponding tips are not enlarged to encompass a double or a triple of the shaft Let us mention finally that it is possible to indicate only the head of the ar row and in this case the program produces an arrow without tail and with the standard shaft A B begin xy ar gt 0 0 A 20 0 B end xy See the reference manual or the user s guide for even more possibilities 3 3 Curved arrows Sometimes we have to use curved arrows for different reasons for instance to avoid a large label or for connecting two distant vertices of the diagram It is possible to do that with Xy pic in a very simple manner Beware first of all that the option curve has to be active The syntax of the command is
4. gt 18 hole v9 v8 ar gt v10 v8 ar gt v11 v9 ar gt gt v8 v6 ar gt gt v9 v7 ar gt gt v6 v2 ar gt gt v7 v3 ar gt 35 hole v4 v2 ar0 gt v5 v3 ar gt v8 v1 ar _ gt 725 hole v3 v2 ar gt gt _ gq_1 p_1 v2 v1 end xy 24 References 1 Campani C A P Tutorial de Xy pic in portuguese 2006 available from CTAN 2 Dodson C TJ Parker P E A User s Guide to Algebraic Topology Kluwer Aca demic Publishers 1997 3 Goosens M Rahtz S Mittelbach F The TEX Graphics Companion Addison Wesley 1997 4 Rose K H How to typeset pretty diagram arrows with TeX design decisions used in Xy pic in Jiri Zlatuska editor EuroTpX 92 Proceedings of the 7th Euro pean TpXConference pages 183 190 Prague Czechoslovakia September 1992 Czechoslovak TEX Users Group 5 Rose K H Xy pic User s Guide version 3 7 1999 available from CTAN 6 Rose K H Moore R Xy pic Reference Manual version 3 7 1999 available from CTAN 7 Schubert H Kategorien vol I II Springer Verlag 1970 8 Valiente Feruglio G Typesetting Commutative Diagrams TUGboat 15 4 1994 466 484 25
5. v7 end xy Here follows another diagram from Schubert s book vol II p 39 q O 60 N ae which was produced by the following code P J ie a hj LJP AY e begin xy vertices 20 20 x Q v1 40 20 x x P_ e_j v2 60 20 x bigsqcup P_e v3 0 0 Q v4 20 0 bigoplus Q_j v5 40 0 N v6 70 0 A v7 arrows ar gt _ k_j yl eyo a arei gt Ay Ae jy yi sea ee ar gt w_j vi v6 ar gt h_j v2 v6 ar 0 gt i_ e_j v2 v3 ar gt c_j lt lt lt lt lt lt lt lt lt lt lt hole v2 v7 ar 0 gt g v3 v7 ar gt Delta v4 v5 ar gt _ 1 5pc l w v4 v6 ar 0 gt w v5 v6 ar gt _ i_D v6 v3 ar gt _ n v6 v7 end xy The next diagram figure 1 was taken from a book of algebraic topology A User s Guide to Algebraic Topology by Dodson and Parker 2 p 269 and was produced by the code newcommand sbol small bfseries begin xy hvertices 27 0 E 0 B v1 0 20 E 1 v2 70 20 F 1 v3 5 25 35 K left pi_1 F 1 right v4 95 35 K left pi_1 F 1 right v5 0 45 vdots v6 70 45 vdots v7 0 70 E7 n 1 v8 70 70 F n 1 v9 5 25 85 K left pi_ n 1 F n 1 right v10
6. 95 85 K left pi_ n 1 F n 1 right v11 20 100 E v12 5 50 100 F v13 0 115 E n v14 70 115 F In v15 5 25 130 K left pi_ n F n right v16 95 130 K left pi_ n F n right v17 0 150 vdots v18 70 150 vdots v19 arrows ar gt gt p_ n 1 v18 v14 ar 0 gt gt v19 v15 ar 7 gt v16 v14 ar 0 7 gt v17 v15 21 ar 0 _ gt v15 v14 ar 0 gt h_ n v12 v14 ar gt h gt _ n v13 v15 ar _ gt v13 v12 ar 0 gt gt p_ n 33 hole v14 v8 ar gt h_ n 1 v12 v8 ar 0 gt h_ 1 v12 v2 ar gt gt v15 v9 ar gt h _ n 1 v13 v9 ar gt _ h gt _ 1 v13 v3 ar _ gt 18 hole v9 v8 ar 0 gt v10 v8 ar gt v11 vo are gt gt v8 v6 ar gt gt v9 v7 ar gt gt v6 v2 ar gt gt v7 v3 ar 7 gt 35 hole v4 v2 ar0 gt v5 v3 ar gt v8 v1 ar _ gt 1 725 hole v3 v2 ar gt gt _ q_1 p_1 v2 v1 end xy Finally here it comes our version of the diagram from Valiente Feruglio 8 see also The Latex Graphics Companion 3 The diagram was produced by the following
7. code begin xy vertices 27 0 E 0 B v1 0 20 x E 1 v2 70 20 F 1 v3 25 35 K left pi_1 F 1 right v4 95 35 K left pi_1 F 1 right v5 0 45 vdots v6 70 45 vdots v7 0 70 E n 1 v8 70 70 F n 1 v9 25 85 K left pi_ n 1 F n 1 right v10 95 85 x K left pi_ n 1 F n 1 right v11 20 100 E v12 5 50 100 F v13 0 115 E7 In Fd v14 70 115 F7 nn F vib 5 25 130 K left pi_ n F n right v16 95 130 K left pi_ n F n right v17 0 150 vdots v18 70 150 vdots v19 arrows 22 o Moore Postnikov s tower Ell B of a fibration Figure 1 Moore Postnikov s tower of a fibration 2 23 Lin 2 Krm Rinx ok Y 7 ge m zH AG T va a AH G Gp m H ar 0 gt gt p_ n 1 v18 v14 ar gt gt v19 v15 ar gt v16 v14 ar gt v17 v15 ar _ gt v15 v14 ar gt h_ n v12 v14 ar 0 gt h _ n v13 v15 ar _ gt v13 v12 ar gt gt p_ n 33 hole v14 v8 ar gt h_ n 1 v12 v8 ar gt h_ 1 v12 v2 ar gt gt v15 v9 ar gt h _ n 1 v13 v9 ar gt _ h _ 1 v13 v3 ar _
8. either indicate how the tip should be positioned above or below either that the shaft should be doubled or tripled More specifically the four available variants are aT lt gt x lt lt r x ae a SS a j lt gt A C 3 2 C l o A N a Table 4 Arrow tips the above variant You should experiment with it The action depends on the tip It either indicates that the tip should be placed eccentrically above with the convention that this means actually on the left when we look at the arrow from tail to head either that we should use only the upper part of the tip See the examples below _ the below variant The same remarks as above are in order if we substitute above with below and left with right 2 the double variant The effect is that the shaft is doubled See examples below 3 the triple variant The effect is of course that the shaft is tripled The examples bellow show how the four variant work for the selection lt gt of the triple tail shaft head lt gt 4 B lt gt A B lt gt A 2p 2 lt gt 4B 3 lt gt A lt B These kind of variants work well with the kind of head and tail from the example above In other situation however we might want to raise or lower only the head or the tail not both Fortunately this is also possible because we can have
9. put these signs immediately after the signs _ or whichever is used Here are some examples B begin xy ar 0 0 A 20 0 B gt _2Qmm alpha end xy B begin xy ar7 lt alpha 0 0 A 20 0 B end xy If you put several gt or lt signs in a row the label will move away from the corresponding end of the diagram a A B begin xy ar lt lt lt lt alpha 0 0 A 20 0 B end xy A simpler way of doing that is to use instead of the characters lt and gt a single decimal number between 0 and 1 0 is equivalent to lt and 1 is equivalent to gt while 0 5 will correspond again to the middle of the construction not the middle of the arrow unless the source and the destination of the arrow have equal width A B rC begin xy ar7 4 alpha 0 0 A 20 0 B otimes_ R C end xy 14 If you want to put the label in the middle of the arrow when the two vertices are not of the same size you can do that by adding a minus sign before the label as in the following example A gt B rC begin xy ar alpha 0 0 A 20 0 B otimes_ R C end xy You can put more then one label on the same arrow why would anyone want to do this 2 B begin xy ar 0 3 alpha _ 0 7 beta 0 0 A 20 0 B end xy A You may want to use both commands for producing arrows in the same dia gr
10. For submission to The PracIFX Journal Draft of November 16 2006 Commutative Diagrams with Xy pic I Kernel Functions and Arrows Paul A Blaga Address Babe Bolyai University of Cluj Napoca Faculty of Mathematics and Computer Science 1 Kog lniceanu Street 400609 Cluj Napoca Romania Abstract This the first of two papers aiming to describe the use of the facilities of the package Xy pic for constructing commutative diagrams We tried to use in a systematic way the learning by examples approach without entering into the details of different constructions or trying to describe in an exhaustive way all the possibilities The final goal is to provide the reader with enough knowledge to be able to construct by himself complicated diagrams This first paper describes the basic possibilities of Xy pic which are provided by the kernel of the associated language but it also explores many of the opportunities provided by the extension arrow 1 Introduction Commutative diagrams lie at the very heart of mathematics and therefore it is very fortunate that both TRX and IAIFX provide ways of construction for such objects There are many packages devoted entirely or partially to such a purpose Usually they work both under plain TEX and TEX with minimal modifications Most of them are examined and compared by G Valiente Feruglio 8 in an old issue of TUGboat Unfortunately the general purpose introductions to TEX or IATEX typ
11. allowing one to put frames of several shapes around objects the xymatrix environment enclosed in the matrix extension allowing the representation of the diagrams as arrays matrices of objects providing an easier and more straightforward way of constructing diagrams without using coordinates to identify the position but also losing some of the flex ibility the 2cell extension devoted to the construction of a special class of dia grams very common in the theory of categories several other fine points of the program This article is largely based on the original documentation of the founding fa thers K Rose and R Moore the reference manual 6 and user s guide 5 but also on a recent excellent tutorial in Portuguese by Carlos Campani 1 and the chapter on Xy pic from The TeX Graphics Companion 3 2 How Xy pic works Xy pic is a package designed to be used both for Plain TEX and TEX Therefore in the tree of any TEX distribution it lies in the generic section of the texmf tree If used by Plain T X it is loaded as usual by the command input xy at the beginning of the file If it is used in a ATEX document then it is loaded by a command usepackage options xy used of course in the preamble of the document The options simply means in most cases the activation of some of the extensions If you are not sure whether an extension has to be activated or not you can play it saf
12. am A B begin xy ay ar 75pc 0 0 A 20 0 B _2mm alpha ar _ 75pc alpha 0 0 A 20 0 B end xy There is though a small problem the fonts used for the two arrows have different sizes It is not difficult to deal with this problem You have to choose first which is the size you prefer If you prefer the larger size just substitute in the inner label alpha by displaystyle alpha 15 EEA begin xy i ee ar0 7 75pc 0 0 A 20 0 B _2mm alpha ar _ 75pc displaystyle alpha 0 0 A 20 0 B end xy otherwise substitute in the outer label alpha by scriptstyle alpha A A B begin xy a ar 75pc 0 0 A 20 0 B _2mm scriptstyle alpha ar _ 75pc alpha 0 0 A 20 0 B end xy 3 5 Drawing multiple arrows To draw several arrows between the same position all we have to do is to shift them a little bit This is done with the command lt dimension gt If the arrows have opposite directions then the shifting can be made with the same positive quantity A B begin xy 0 20 A a 20 20 B b 5 ar lt 1 ex gt a b 3 ar lt 1 ex gt b a end xy otherwise one of them should be shifted with a negative quantity 16 A B begin xy 0 20 A a 20 20 B b 3 ar lt 1 ex gt a s b 3 ar lt 1 ex gt a b end
13. e and activate all the extensions by the option all We mention however that there are options which are not enclosed in this all and which have to be prescribed independently See the reference manual for details In Plain TEX you can activate an option by the command xyoptionf option With a single notable extension which will be described bellow the txt construction all Xy pic constructions should be enclosed in an environment call xy For this environment two syntaxes are provided a Plain T X version xy endxy a IATRX version begin xy end xy In a IATEX document you can use whichever syntax you like in Plain TEX however only the first one is accepted We shall use throughout this document exclusively the ATEX syntax so if you want to use any of the examples in a TEX document all you have to do is to switch to the other syntax of the xy environment Working with Xy pic is very much like programming with an object oriented programming language In fact the authors of the Xy pic system devised a quite complicated language the previously mentioned kernel of the system with which one can make very complex drawings It is by no means the aim of this paper to describe the details of this language For this the reader may consult the reference manual The philosophy of the system however is quite easy to explain For our needs at least the following elements of the kernel are crucial positio
14. egin xy ar 0 0 A 20 0 B _2mm alpha end xy which means that the label command has exactly the same syntax and it is placed immediately after the arrow was produced A second possibility is a A B begin xy ar7 alpha 0 0 A 20 0 B end xy Notice two things this time the label is on the left although we used the exponentiation sign the label is smaller The reason is that in this case the label is thought of as an exponent or an index if we use the underscore Also in this case we have an extra possibility allowing us to put the label on the arrow which is interrupted to accommodate the label A a gt B begin xy ar alpha 0 0 A 20 0 B end xy Implicitly the label is placed on the middle Be careful though is not neces sary the middle of the arrow is the middle of the entire construction objects arrow a A amp C B begin xy ar alpha 0 0 A otimes C 20 0 B end xy 13 Therefore in many situations you will have to position the label explicitly by yourself This is easy to do no matter which of the two syntaxes you use If you want to put the label at one of the two ends of the arrow just add a lt for the tail or a gt for the head at the right position meaning if you use the command starting with a immediately after the question mark If you use the other command then
15. ically ig nore commutative diagrams with the notable exception of the amslatex package amscd which is very easy to use but unfortunately allows only vertical and hor izontal arrows For the rest the user is either left at the mercy of the original documentation accompanying the package which is not always very complete let alone pedagogical or is required to consult more specialized books such as The E TeX Graphics Companion 3 which are not always available This is the first of two articles aiming to describe the commutative diagrams capabilities of one of the most complex packages Xy pic This package written both for plain T X and IAIFX is quite ambitious and offers possibilities of drawing a lot more things than just commutative diagrams We only intend however to describe the main tools for constructing such diagrams either by using the primitive elements of the graphics language of Xy pic what we call the kernel of Xy pic or by using more advanced constructions and environments We ought to mention that in fact the package was created originally by Kristoffer Rose exactly for typesetting pretty diagram arrows with TEX see 4 In this article we will discuss the basic possibilities of Xy pic namely essen tially constructing diagrams by assigning coordinates to different objects and connecting these objects with different kinds of arrows In the second article we shall describe the frame extension
16. ns they are nothing but the pair of coordinates of a point of the figure The coordinate system is local it is related to each figure not to the page objects they are thought of as boxes that unlike ordinary TEX boxes have edges They can be circular elliptic or rectangular and there is an entire set of operations that can applied to them We shall discuss more about objects in the sequel of this paper For now an object will be for us just an ordinary box containing one or more TEX symbols directionals they provide ways of connecting different positions and can be of two kinds 1 connectors they are used to construct the shaft the body of the arrow 2 tips they are used to construct the tail and the head of the arrow decorations other things that are added to the diagram e g labels Thus to construct a diagram with Xy pic you have to use the following steps 1 choose the positions the coordinates of the vertices of the diagram 2 drop some objects at these positions these objects will be the origins and targets of the arrows 3 connect the positions with arrows 4 add whatever extra information you want e g labels or text Apart from the basic elements of the Xy pic which was described roughly above the system also includes some extra goodies which are called options They are somehow arbitrary divided into two classes 1 extensions they add some more objects and meth
17. ods Typical for this kind of enhancement of the kernel are the extension frame which will be de scribed in the second article and which offer the possibility of framing the objects and the extension curve which we will use in this paper to construct curved arrows 2 features they provide notations for particular applications areas In this class fall the options arrow matrix polygons lattices knots 2cell We mention in the end of this short description that everything that happens in side an xy environment is in mathematical mode If you want to drop somewhere ordinary text there is provided a special command for that Nevertheless if you want your diagrams to be numbered you can include them into an equation environment 3 Basic constructions 3 1 Constructing arrows from pieces The simplest operation we can make with Xy pic is probably dropping an object on a position It is made with the operator A begin xy 0 0 A end xy We defined a position of coordinates 0 0 and we attached to this position the label A To construct diagrams we should also be able to draw line connecting the vertices of the diagram This is done in Xy pic with the so called directionals employed by the operator The directionals are of two species 1 connectors they construct the connecting line itself the shaft of the arrow see table 1 for a list of most frequent connectors 2 tips they constr
18. sitions and then address them by their names like in the following example A us B begin xy YN 0 20 A a 20 20 B b 5 we 0 0 C c 20 0 D d 5 a ar a b _2mm alpha C D ar a c ar b d ar c d ar gt a d _2mm gamma end xy As one can see very easily we assigned to each position a name don t forget the quotation marks and then we used the names to define arrows from a position to another To draw spatial diagrams we need an opportunity to suggest the spatial character by interrupting some of the arrows Xy pic provides such an opportu nity as well A B begin xy ye 0 20 A a 20 20 B b 5 0 0 C c 20 0 D qd ye ar gt a d C D ar gt gt gt gt gt gt gt gt gt gt gt gt hole b c end xy What we are doing is to insert a hole in one of the arrows at the right place With the command gt gt put as many gt sign as it takes you just fix this position This is done by trial and error Again it is possible also to use several lt signs to indicate the distance from the other end or even simpler substitute these signs 18 with a fractional decimal number between parentheses as we explained when we talked about labels Here follows a first example of more complicated diagram adapted from the user s manual in which are applied several of
19. the constructions described so far These kind of diagrams are common in the theory of categories especially when one wishes to describe universality properties begin xy 0 40 U u 30 20 X times_ZY xzy 3 55 20 X x 5 30 5 Y y 55 5 Z 2 3 ar gt _p xzy x 3 ar gt q xzy y 3 ar gt g y z ar gt f x z ar 0 gt l x y u xzy ar gt 0 0 5pc x u x ar gt 0 _ _ y u y end xy 5 Complex diagrams from real life In this final section of the article we shall construct several complex diagrams taken from books of the theory of categories or algebraic topology to illustrate the power of Xy pic The following diagram taken from the book of categories theory by Horst 19 Schubert vol I p 135 L M LN ey p ae hy A B was produced by the following code begin xy vertices 15 30 bigsqcup M_e v1 45 30 bigsqcup N_e v2 0 15 bullet v3 30 15 x bullet v4 60 15 bullet v5 15 0 A v6 45 0 B v7 arrows ar gt gt v1 v3 ar gt _ m v1 v6 ar gt gt p vi v2 ar 0 gt gt v1 v4 ar gt _ n v2 v7 ar gt gt v2 v5 ar gt gt m v3 v6 ar gt gt fm gt va v7 ar gt n gt v5 v7 ar gt _ f v6
20. uct the head or the tail of the arrow see table 2 Of course the tips may be absent The simplest example is the following A B begin xy 0 0 A 10 0 B dir end xy In this situation we used a single connector dir connects the two posi tions with a single solid line Notice the two signs after the s They introduce 5 dir dir dir dir dir dir gt dir lt dir dir dir dir gt gt dir lt lt dir dir dir gt dir lt lt dir dir dir fo dir2 4 dir3 Wibtp 2 Aa y dir2 7 A dir3 Pa dir2 ye dir3 s2 2 Anaea A DS Table 1 Directionals connectors a dir7 gt P dir_ gt dir lt dir_ lt dir dir_ 1 dir dir_ dir dir_ dir gt gt a dir_ gt gt Pa Pa dir_ lt lt AY AS dir7 dir_ I dir dir_ P dir gt gt we dir lt a dir ye dir o dir x cal dir_ dir7 ra Ndir dir_ we Table 2 Directionals tips Ve es some extra space around the labels otherwise the connection line would be too close to them Notice also that we have to add a semicolon before any new po sition is defined The connector provides a link between the last two positions we defined Adding a head or a tail or both to a line to produce an arrow is slightly more complicated The synta
21. x for the command producing a tip is identical to the one of the connector only the argument is of a different kind e g dir gt We might be tempted to think that it is enough to add a command of this kind to the above example The result looks like in the figure bellow and it indicates that the system interprets this command as a connector also fe begin xy 0 0 A 10 0 B dir dir gt end xy What we have to do is to use another operator to indicate first where the tip should be added The full syntax is gt if we want to add the tip at the end i e near last defined position or lt if we want to add the tip at the origin We have thus A gt B begin xy 0 0 A 10 0 B dir gt dir gt end xy or A lt B begin xy 0 0 A 10 0 B dir lt dir lt end xy or if we want to add tips at both sides A gt B begin xy 0 0 A 10 0 B dir gt dir gt lt dir lt end xy If we use several lt or gt signs between and then we can put the tip also at an intermediate position 4 begin xy 0 0 A 10 0 B dir lt lt lt dir lt end xy We can do this also 4 begin xy 0 0 A 10 0 B dir 7 5 dir lt end xy Of course we can add different kind of tips At gt B begin xy 0 0 A 10 0 B dir gt dir gt
22. xy We can even produce curved arrows pee begin xy 0 20 A a 20 20 B b ar lt 1 ex gt 5pc a b ar lt 1 ex gt _ 5pc g b a end xy 3 6 Adding text to the diagrams Ordinary text is added in a diagram through the command txt lt width gt style text string The text is typeset in centered paragraph mode i e the line breaks can be con trolled through the command Here width represents the the width of the paragraph box while style represents the style in which the text should be type set as the following example suggests A A newcommand smit small itshape begin xy This is an arrow ar 0 0 A 25 0 A 12 5 10 txt lt 4cm gt smit This is an arrow end xy As you can see from the example the text can be dropped as any other object One or both of the options for txt can be absent As we stated earlier txt works outside the xy environment but it is not really useful because in fact you want to put the text in a certain position which is only possible if you use the command inside the environment 17 4 Drawing more complicated diagrams The sequential way of constructing a diagram is not very efficient It would be nice to have the possibility to draw arrows from a position to any of the already defined positions not only to the next one or to the previous one Fortunately this is possible The trick is to give names to different po
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