GDCT User Guide - Mount Allison University
Contents
1. Displaying the Functor Comments If Settings Show Functor Comments is selected it displays the comments in the functor display If deselected it hides them Page 24 of 35 APPENDICES APPENDIX A EXAMPLE FILES Example CAT File This is an example category category sumone objects A B C D arrows f A gt C g B gt C h C gt D i A gt D j B gt D relations h f i h g j Example CGL File A commutative square category csquare objects A B C D arrows f A gt B g A gt C h B gt D k C gt D relations h f k g gml graph directed 1 node id 0 label A graphics Image Type Location center x 48 30000000000001 y 49 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 vgi labelPosition in shape Oval nn nn Page 25 of 35 node id 1 label B graphics Image Type Location center x 48 0 y 53 0 z 0 0 width 17 36 height 27 299999999999997 depth 20 0 vgj labelPosition in shape Oval e node id2 label C graphics Image Type Location ss center x 51 69999999999999 y 49 0 z 0 0 E width 18 78 height 27 299999999999997 depth 20 0 vgj labelPosition in shape Oval Z node id3 label D graphics Image Type Location center x 50 59999999999995 y 53 0 KAL an nn nn KAL KLALI Page 26 of 35 z 0 0 width 18 78 height 27 29999999999992. Check Category for Initial Object s Current Category i ANOT INITIAL cube 11 5 0 and 7 3 0 are both paths to object H Select Objects to Check B NOT INITIAL 11 5 and 7 3 are both paths to object H C NOT INITIAL 10 6 and 7 2 are both paths to object H D NOT INITIAL no path to object A a Select All Check ovject s OK WARNING This command will not operate correctly unless the category has a confluent set of relations If in doubt use Make Confluent tool first Sum Selecting Tools Sum opens a dialogue window allowing selection of a possible sum object from the current category from a drop down and specification of possibly composite first and second injections then testing the cospan for being a sum WARNING The codomain of each injection must be the possible sum object The test is only valid on a confluent category If in doubt use the Make Confluent tool NOTE This feature has not yet been implemented Terminal Object Select Tools Terminal Object This command tests objects of the current category chosen by a dialogue for being terminal For each selected object which is not terminal a reason is given WARNING This command will not operate correctly unless the category has a confluent set of relations If in doubt use Make Confluent Tool first Product Selecting Tools Product opens a dialogue window allowing selection of a possible product object from the cu
3. oa a Prd name in the Category name box E23 Graphical Database for Category Theory G type Enter File Categories Functors Tools Settings Help Type the names of objects in the Object Name box which now appears completing each with Enter Press the Objects Complete button when done Category Name TestCategory During creation of the category the objects and arrows are displayed in the Category Display Window Objects may be moved about by mouse action at any time Click on an object and drag it to a new position WARNING No two objects may have the same name If a duplicate object name is entered the program will display an error message and the new object will not be included in the category Page 6 of 35 Entering arrows is very similar to entering objects Type the names of arrows in the Arrow Name box which now appears completing each with Enter After the name of an arrow is entered the user is prompted for the domain and then the codomain of the arrow Press the Arrows Complete button when done As arrows are entered they are displayed in the Category Display window WARNINGS No two arrows may have the same name The domain and codomain of each arrow must be a declared object An arrow cannot be named 1 since this symbol reserved by the program for the identity arrow Next the user enters the equations among composites of the generating arrows The LHS of equation box appears and the user must e
4. objects 0 1 2 arrows d0 0 gt 1 d10 1 gt 2 d11 1 gt 2 s10 2 gt 1 relations d11 d0 d10 d0 s10 d11 1 s10 d10 1 gml graph Page 28 of 35 directed 1 node id 0 label 0 graphics Image Type Location center x 25 0 y 120 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 veil labelPosition in shape Oval a node id 1 label 1 graphics Image Type Location 3 center x 60 0 y 69 0 z 0 0 KLEI nn AL wie width 15 94 height 27 299999999999997 depth 20 0 vai labelPosition in shape Oval node id 2 label 2 graphics Image Type Location a center x 170 wn an Page 29 of 35 y 68 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 vgi labelPosition in shape Oval node id 4 label relations nd11 d0 d10 d0 s10 d11 1 s10 d10 1 graphics Image Type Location genter x 0 0 y 150 0 z 0 0 width 336 85999999999996 height 48 599999999999994 depth 20 0 vgi labelPosition in shape Oval edge linestyle solid label d11 d10 source 1 target 2 edge linestyle solid label d0 source 0 target 1 edge linestyle solid label s10 source 2 target 1 nn nn This is the second category Page 30 of 35 category del3 objects 0 1 2 3 arrows d0 0 gt 1 d10 1 gt 2 d11 1 gt 2
5. Category Graphical Settings Functor Graphical Settings Functor Animation Settings Server Settings Endomorphism Limit Displaying the VGJ Controls Displaying the Category Comments Displaying the Functor Comments EXAMPLE FILES CAT File Example CGL File Example FUN File Example FGL File Example MENU ITEM SHORTCUTS List of Shortcuts MISC INFORMATION Version Information Bug Report Additional Information about GDCT can be found at http mathcs mta ca research rosebrugh gdct USER GUIDE FOR GDCT SECTION 1 INTRODUCTION About GDCT Graphical Database for Category Theory was developed in the summer of 1999 under an NSERC summer research grant Some of the algorithms used in GDCT were originally developed in A Database of Categories a C program written by Ryan Gunther and Michael Fleming under the supervision of Dr Robert Rosebrugh and Category Theory Database Tools written by Jeremy Bradbury under the supervision of Dr R Rosebrugh This application allows for the creation editing and storage of finitely presented categories Categories can be opened and saved from local files as well as loaded from a specified server Once a category file is in memory it can be printed or tools for testing properties of objects and arrows can be applied to it The tools available in this version of GDCT are Make Confluent Equality of Composites Make Dual Initial Object Sum Terminal Object and Product This program also allows
6. Currently the below list of known problems is being dealt with and will be fixed by September 2000 It is the goal of the GDCT Development Team to have a stable version of the application available for download by that date Version 1 0 Released March 20 2000 This is the preliminary version of Graphical Database for Category Theory The release information for this version is available at http mathcs mta ca research rosebrugh gdct v 1 release htm Version 1 1 Release Date Summer 2000 This version of GDCT will contain the following fixs and enhancements The help file system has been rewrote and now includes a complete set of help files which will better enable the user to use the software Contact information in the bug report dialog has been updated e Upon closing the application a dialog window now prompts the user to see if they want to save changes in modified categories This dialog window also tells the user what modifications were made so that they can more accuratly judge if they want to save these changes e The display of categories in the Make Confluent Tool was changed so that a category isn t redrawn randomly Instead GDCT will only update the label of the node containing the list of relations e In Add Data for categories the category is no longer redrawn randomly New data is now added to the existing graphical representation as opposed to creating a new random representation In Remove Data the probl
7. Graphical Database for Category is available for download at http mathcs mta ca research rosebrugh gdct download htm There are four installation options The first two options are available only to Windows 95 98 NT 2000 users Options three and four are available to users of all platforms Option One InstallShield Setup Windows Only Download the InstallShield setup program that will automatically install GDCT for you This option is best for user who will require help with initial setup This setup includes a Java Runtime Environment for Windows 95 98 NT 2000 only The InstallShield version of the GDCT Setup is 3 42 MB in size Option Two Non InstallShield Setup Windows Only Download the non InstallShield version of the GDCT setup program For this option the user should unzip the GDCT zip file to the desire directory and then run the application from there This setup includes a Java Runtime Environment for Windows 95 98 NT 2000 only The non InstallShield version of the GDCT setup program is 2 56 MB For users who are concerned with download size this is the best version Option Three Download Compiled Java Source Files and Java Runtime Environment Under this option the complete Java source files are already compiled into class files and stored in a java archive jar file The jar file can be downloaded as a zip file as a tar gz file or in uncompressed form This download also includes all necessary help files as well as s
8. Help Open Category CGL Ctrl 0 Save Category Open Category CAT Categories Functors Tools Open Functor gt Download Category CGL Ctrl L Save Functor gt Download Category CAT Print Exit E GDCT Categories Cube cgl E GDCT Categories Del3 cql E GDCT Categories Del3 cat E GDCT Categories Del1 cgl E GDCT Categories dualdelt cgl E GDCT Categories Csquare cgl E GDCT Categories del1 b cgl E GDCT Categories del1 b cgl elations I a If there is any error in the format of the selected file an error message is displayed and the open is aborted The most recent category files can also be opened by clicking on one of their names in the File menu Downloading Categories In order to download a a text category file cat select File Open Category Download Category CAT or Categories Open Category Download Category CAT Downloading a graphical category file cgl is similar select File Open Category Download Category CGL or Categories Open Category Download Category CGL An alternative method to download graphical category files is to press CTRL D Choosing any of the above options will display a list of categories in a selection box in a new frame The categories listed are located on the server specified in the frame s title bar To select a category to download Page 8 of 35 simply double click on the category s name or click once on the category name and
9. Removing Categories Selecting Categories Remove Category opens a list of the categories currently in memory Select from the list and click OK to remove a category Page 9 of 35 Printing Categories To print the text version of the currently displayed category select File Print Print Category Text To print the graphical version select File Print Print Category Graphical To save the graphical version of the currently displayed category to a postscript file select File Print Print Category Postscript Adding Data Selecting Categories Edit Category Add Data opens a dialogue which allows addition of objects arrows and R ey SC relations to the current category To add objects type their i names in the box When finished click Objects Complete then arrows may be added type the name then the domain and codomain when finished click Arrows Complete To add relations the left and right sides must Arrows Complete be specified when finished click Relations Complete Add Status WARNING The domain and codomain of a new aii E GDCT Add Data to Category of x Add Data Arrow Name e arrow must already exist and the sides of a relation must be composable paths An arrow can be removed only if it does Atow a A gt B not appear in any current relations An g object can be removed only if it is neither the domain nor codomain of a current arrow Adding data only affects the ve
10. press the OK button Either of these methods will display the downloaded category If the program cannot locate the files on the server an error message will be displayed This could result from the user not being connected to the Internet or as a result of a server error E GDCT Load File from http cs mta ca res EI EG CSQUARE CGL NOTE Files can only be loaded from the server when when connected to the Internet DISCFOUR CGL DISCTWO CGL EMPTY CGL EQUALIZE CGL IDENTITY CGL To change the settings for downloading categories select Settings Set Server This displays a dialog for changing the default server and directories from which categories and functors can be loaded using the appropriate load command Saving Categories In order to save a category file cgl select File Save Category Save Category CGL or Categories Save Category Save Category CGL This opens a file selection window Saving text category files cat or simply the VGJ graph file gml is similar A shortcut to save a cgl category file is to press CTRL S To save a category to an existing file simply double click on the category file name A new file is created by typing its name in the dialogue and clicking OK E Graphical Database for Category Theory E GDCT Categories d Functors Tools Settings Help Create Category Ctrl C Edit Category Open Category Remove Category Save Category CAT Ctril S Print Category
11. 97 depth 20 0 veil labelPosition in shape Oval i node id 4 label relations nh f k g graphics Image Type Location l center x 169 0 y 4 0 z0 0 width 20 0 height 20 0 depth 20 0 vgj labelPosition center shape Oval we wie linestyle solid label k source 2 target 3 edge linestyle solid label h source 1 target 3 edge linestyle solid label g source 0 target 2 ae edge linestyle solid label f source 0 Page 27 of 35 target 1 Example FUN File This is an example of an fun file This is the first category category del2 objects 0 1 2 arrows d0 0 gt 1 d10 1 gt 2 d11 1 gt 2 s10 2 gt 1 relations d11 d0 d10 d0 s10 d11 1 s10 d10 1 This is the second category category del3 objects 0 1 2 3 arrows d0 0 gt 1 d10 1 gt 2 d11 1 gt 2 s10 2 gt 1 d20 2 gt 3 d21 2 gt 3 d22 2 gt 3 s20 3 gt 2 21 3 gt 2 relations s10 d10 1 s10 d11 1 d11 d0 d10 d0 d21 d10 d20 d10 d22 d10 d20 d11 d22 d11 d21 d11 s10 s21 s10 s20 s21 d20 d10 s10 s20 d20 1 s21 d21 1 s20 d21 1 s21 d22 1 s20 d22 d11 s10 This is the functor information functor del23 objects 0 l gt 0 1 l gt 1 2 gt 2 arrows dO gt dO d10 l gt d10 d11 gt d11 s10 l gt s10 Example FGL File This is an example of an fgl file This is the first category category del2
12. Version 1 1 GRAPHICAL DATABASE FOR CATEGORY THEORY DEVELOPED 1998 2000 MOUNT ALLISON UNIVERSITY J BRADBURY R ROSEBRUGH USER GUIDE for GRAPHICAL DATABASE for CATEGORY THEORY Written By Jeremy Bradbury Edited By Robert Rosebrugh June 20 2000 TABLE OF CONTENTS SECTION 1 gt gt gt gt SECTION 2 gt YY vY YV vY V V V V V Vv yC SECTION 3 gt gt gt SECTION 4 gt Yy v vY vY v y SECTION 5 gt Yy vY vV vV v y INTRODUCTION About GDCT Development Team Installation Directory Structure CATEGORIES Introduction Category File Types CAT File Format CGL File Format Creating Categories Opening Categories Downloading Categories Saving Categories Removing Categories Printing Categories Adding Data Removing Data Changing a Category Name GRAPHICAL DISPLAY OF CATEGORIES Introduction Visual Display Controls View GML CATEGORY TOOLS Make Confluent Equality of Composites Make Dual Category Initial Object Sum Terminal Object Product FUNCTORS Introduction Functor File Types FUN File Format FGL File Format Creating Functors Opening Functors Downloading Functors vY v v yv Saving Functors Removing Functors Printing Functors Functor Animation SECTION 6 SETTINGS gt yY vY vY vY V Vv V v y APPENDIX A gt gt gt gt APPENDIX B gt APPENDIX C gt gt Switching Between Category and Functor Displays Font Settings
13. ample category and functor files In order to interpret the compiled code in the jar file a Java Run time Environment must be downloaded JREs are used by an operating system to interpret Java class files so that they can be run on the local machine Option Four Download Java Source Files and Java Developer s Kit The complete Java source files are available for download and may be modified under the terms of the GNU General Public License Version 2 This download also includes all necessary help files as well as sample category and functor files It is available as a zip file or as a tar gz file In order to compile and interpret Java source files the Java Developers Kit must be downloaded JDKs are used by an operating system to compile Java source files and interpret Java class files so that they can be run on the local machine Page 3 of 35 Directory Structure Upon installation the Graphical Database for Category Theory unless changed by the user will be installed in a directory called GDCT Inside this directory are the subdirectories cat cgl fgl fun Help Images and J re1 1 The following files are also located inside the main directory GDCT bat This is the batch file that the user selects to interpret the java archive file jar GDCT jar This is the java archive file that is interpreted by the Java Runtime Environment This file contains all of the compiled Java source files GDCT ico This is an icon file that
14. an decrease the Scale 2 V Show Controls scale by a factor of two increase the Seale r scale by a factor of two set the scale SeA to one or set a user defined scale Set Font The font size and face can be changed by selecting Settings Category Graphical Settings and clicking the check box in front of Set Font This will display a dialog in which the user can type in the desired face and size for the font Click Apply to save these changes or click Cancel to keep the current settings Page 13 of 35 View GML GML is the text representation of the graphical display of the current category To view the GML code select Categories Edit Category View GML Choosing this menu option will display a frame containing the GML code Each object is stored as a node The relations are stored as a single node The arrows are stored as edges between nodes Multiple arrows between nodes are stored as one edge with commas separating the names of the individual arrows E4 Graphical Database for Category Theory E GDCT Categories Cub Efe Ea Search for Apply Done File Edit graph directed 1 node id 0 label A graphics Image Type Location center v rf Exit There are several tools to help viewing of the GML Edit GoToTop Edit GoToBottom Edit GoToLine Edit Search Page 14 of 35 SECTION 4 CATEGORY TOOLS Make Confluent Select Tools Make Confluent This command
15. animate a functor Auto Animate lt lt Back Forward gt Reset The first way is to use the Back and Forward buttons below the graphical display of the functor to step though all of the mappings The animation can be restarted by pressing the Reset button The second way to animate a functor is to press the Auto Animate button which will animate all of the mappings at the animation speed indicated in Settings Functor Animation Settings y Y EI by LLP 0s OLP OZP ozs IzS TTP LZP LLP d0 gt m Jrs e o OLP m L relations relations d11 d0 d10 d0 s10 d11 1 510 d10 1 10 d10 1 10 d11 1 011 d0 d10 d0 d21 d10 d22 d11 d21 d11 10 S21 10 S20 s217d20 20 d21 1 921 d22 1 920 d22 d11 510 Page 21 of 35 SECTION 6 SETTINGS Switching Between Category and Functor Displays Selecting either of Settings Category Display or Settings Functor Display allows the user to switch between the display of categories and the display of functors fe Graphical Database for Category Theory E GDCT Categories Cube cgl Settings A B C D E F G H O A B 1 4 gt C 2 C gt D 3 B D 4 A gt E B gt F 6 C gt G 7 D H 8 E gt F 9 E gt G 10 G H 11 F H Font Settings Selecting Settings Category Settings and then checking the box in front of Set Font This displays a dialog in which the user can change the font settings within the progra
16. applies the Knuth Bendix algorithm to the currently displayed category with arrows ordered in their storage order No relations added to category The set of reductions is now confluent g i cancel Relations which are added to the current category are reported To add these relations to the current category click Ok or click Cancel to keep the category in its current from Equality of Composites To test two possibly composite arrows for equality in the current category select Tools Equality of Composites A dialogue allows entry of two paths E GDCT Equality of Composites Category Name cube Results 2 1 and 3 0 are NOT equal First Path 271 Second Path 3 0 fa Equality OK WARNING The category must be confluent for valid results Domains and codomains of paths must match Make Dual Category This tool will create the opposite category of the currently displayed category Select Tools Make Dual A dialogue window opens prompting for the name of the dual After the user provides the name the dual category is displayed WARNING The opposite category is NOT SAVED automatically when it is created use File Save Page 15 of 35 Category if you wish to save the dual Initial Object Select Tools Initial Object This command tests objects of the current category chosen by a dialogue for initiality For each selected object which is not initial a reason is given 3 GDCT
17. at Part two consists of a second category stored in the CAT file format Part three consists of the functor information In this part any line that is a comment must start with the character On the line following the word functor is the functor name The mapping of objects to objects is listed between the words objects and arrows The mapping of arrows to paths is listed after the word arrows n wow NOTE The words category functor objects arrows and gml are key words and cannot be used as the name of the categories functors between them or as the name of an object or arrow within categories An example of a typical FUN file can be found in Appendix A FGL File Format The text representation of functors is stored in the following three part format Part one consists of a category stored in the CGL file format Part two consists of a second category stored in the CGL file format Page 18 of 35 Part three consists of the functor information In this part any line that is a comment must start with the character On the line following the word functor is the functor name The mapping of objects to objects is listed between the words objects and arrows The mapping of arrows to paths is listed after the word arrows NOTE The words category functor objects arrows and gml are key words and cannot be used as the name of the categories functors between them or as the name of an objec
18. cal representation The text representation is the same as the file format for CAT files Any line that is a comment must start with the character On the line following the word category is the category name Objects are listed between the words objects and arrows Arrows are listed between the words arrows and relations Relations are listed after the word relations NOTE The words category objects arrows relations and gml are key words and cannot be used as the name of the category or as the name of an object or arrow within the category CGL File Format Objects and arrows can be a string of any length and can contain any characters excluding aes eae ma aa T pns S we CAT File Format Relations are composed using the composition symbol The graphical representation uses a format known as gml which was developed at Auburn University for use with the Visualizing Graphs with Java VGJ project It is incorporated into the cgl format by placing it at the end of the file after the keyword gml For more information consult the GML File VGJ Documentation Format An example of a typical CGL file can be found in Appendix A Creating Categories To create a Category in memory the user selects Categories Create Category The names of objects and arrows can be strings of any length that is not a reserved word and can contain any characters excluding nee mt Sr me H aoe te E G 1
19. dden from view by selecting Settings Show VGJ Controls The visual controls available are Viewing Angles Viewing Offset Plane KY KZ Scale 1 Viewing Angles The viewing angle can be changed by using the control below the left side of the graphical display Click and hold the X to rotate the view to any 3D angle The current orientation of the category displayed can be determined by the XYZ axis displayed in the top left corner of the category display area Viewing Plane The plane that the category is being viewed in can be changed by pressing one of the three buttons below the Viewing Angles control window in the bottom right corner of the main screen The user can select the XY plane XZ plane or the YZ plane Viewing Offset The viewing offset is located in the bottom right hand corner of the main window It is basically a virtual screen that allows the user to click and hold on the box to move the viewable canvas area without using the scroll bars The viewing offset can be automatically centered by selecting Settings Category Graphical Settings and clicking the check box by Center Offset Scale i Sa GDCT Graphical Settings Iof x The scale that the category is being p viewed at is displayed directly below atel slewing Cleat the Viewing Offset controls The Scalef2 Center offset scale can be changed by selecting Gaee Settings Category Graphical Display Settings The user c
20. em where not all data is removed in some categories has been fixed Also in Remove Data for categories the text display is updated and the graphical display is updated as data is removed e In View GML the problem with the search feature not working right away has been corrected e The Make Confluent algorithm has been extensively error checked and all known bugs have been Page 34 of 35 fixed e Bugs in Make Dual Category have been fixed and the make display of a newly created dual category is no longer determined randomly Instead the display is based on the display of the category that was originally used to create the dual category The Equality of Composites tool has been tested and all problems have been fixed In the Initial Object and Terminal Object tools a warning is now displayed if the endomorphism limit has been reached e GDCT 1 0 was developed using Borland JBuilder 2 0 and used some borland libraries such as borland jbcl control and borland jbcl layout This caused certain compiling problems in non Borland environments To correct this situation all elements of borland libraries have been removed Known Bugs Below is a list of known bugs in the current version of the GDCT application If you have encountered additional bugs please fill out a comment form at http mathcs mta ca research rosebrugh gdct contact htm The Initial Object algorithm needs to be rechecked when it is applied to categor
21. functor using the buttons below the category display windows WARNING The newly created functor is stored in memory To save the functor to disk select File Save Functor or Functors Save Functor The functor will be saved in the format selected using a save dialog Opening Functors In order to open a text functor file fun select File Open Functor Open Functor FUN or Functors Open Functor Open Functor FUN This opens a file selection window Opening graphical functor files fgl is similar A shortcut to open a fgl functor file is to press CTRL F To open a functor simply double click on the functor s name or click once on the functor name and press the OK button If there is any error in the format of the selected file an error message is displayed and the open is aborted NOTE Opening FGL functor files is not yet implemented Downloading Functors In order to download a text functor file fun select File Open Functor Download Functor FUN or Functors Open Functor Download Functor FUN Downloading a graphical functor file fgl is similar select File Open Functor Download Functor FGL or Functors Open Functor Download Functor FGL Choosing any of the above options will display a list of functors in a selection box in a new frame The functors listed are located on the server specified in the frame s title bar To select a functor to download simply double click on the functor s name or click once on
22. ies with identity arrows in relations e The Terminal Object algorithm needs to be rechecked when it is E aa ae Version 1 1 applied to categories with identity arrows in relations e The Sum algorithm needs to be fixed e Using a modified Sum algorithm the Product algorithm needs to be completely developed In Graphical Functor Settings when the Category layout display is changed the category graphical representations can no longer be scrolled GRAPHICAL DATABASE and objects and arrows can no longer be selected Animation of functor FOR CATEGORY THEORY and other settings still work Proposed Enhancements Below is a list of enhancements that will be made to the GDCT application If you have ideas for additional enhancements please fill out a comment form at http mathcs mta ca research rosebrugh gdct contact htm Open for FGL files e Download for FGL files Open Recent for local FGL FUN files Open Recent for server CAT CGL FUN FGL files e Server Settings should be saved in a file so that the user doesn t have to change them every time the program is started The current default settings should remain in a class file to be displayed in the event that the file containing the current settings is unavailable e Endomorphism Limit should be saved in a file as above for Server Settings Page 35 of 35
23. is used on any shortcuts to the GDCT application GDCT ini This initialization file contains information regarding internal settings category graphical settings functor graphical settings functor animation settings server settings as well as a list of the most recently opened files The subdirectories Cat and cgl contain sample category files The subdirectories 5 C Gdet fun and fgl contain sample functor files C cat C cal The subdirectory Help contains all of the help files while the subdirectory Images C fal contains all of the images used by the Java application C3 fun R 0 Help The J re1 1 subdirectory contains the Java Runtime Environment Version 1 1 Ga Images which is called by GDCT bat to interpret GDCT jar EC Jre1 1 WARNING The user should not modify any files in the subdirectories Help s Images i g C security Page 4 of 35 SECTION 2 CATEGORIES Introduction A finitely presented category can be considered a directed graph with relations between paths of its edges Our notation for category data is Objects A B C D Maps Arrows e A gt A f A gt B g B gt C h A OC Identity Maps Arrows 1 A gt A one for each object Composition of Maps Arrows defined by Relations e 1 g f h path of arrows File Types Categories In GDCT there are three different file types associated with categories cat These files are a text only representati
24. m E Font Category Graphical Settings 3 GDCT Graphical Settings Selecting Settings Category Graphical Settings displays a dialog for changing the graphical settings in the category display Page 22 of 35 Functor Graphical Settings Selecting Settings Functor Graphical Settings displays a dialog for changing the graphical settings in the functor display 3 GDCT Graphical Functor Settings Functor Animation Settings Selecting Settings Functor Animation Settings displays a dialog for changing the animation delay time default 2 seconds and the animation colors for the domain and codomain categories E GDCT Functor Animation Settings OF x Selecting Settings Set Server displays a dialog for changing the default server and directories from which categories and functors can be loaded using the appropriate load command Server Settings Page 23 of 35 Endomorphism Limit Selecting Settings Endomorphism Limit displays a dialog for changing the endomorphism limit B GDCT Endomorphism Order OF x Displaying the VGJ K Graphical Database for Category Theory GDCT 1 0 Controls Settings If Settings Show VGJ Controls is selected it displays the VGJ controls in the category display If deselected it hides them Displaying the Category Show Comments If Settings Show Category Comments is selected it displays the comments in the category display If deselected it hides them
25. nter a composable path of arrows separated by the composition symbol In the RHS of equation box another composable path is entered After the equations have been entered press the Equations complete button E Graphical Database for Category Theory GDCT 1 0 TestCategory De a Category Name TestCategory Object A Object B Object C Object D Arrow a A gt B Arrow b B C Arrow c A C Arrow d D gt D WARNING Only valid equations are accepted An equation is invalid if it contains arrows that are not in the category or an undefined composition or if the domain and codomain of the left side do not equal the domain and codomain of the right side This completes the creation of the new category Page 7 of 35 WARNING The newly created category is stored in memory To save the category to disk select File Save Category and select graphical cgl or text cat format A save dialogue window opens Opening Categories In order to open a text category file CAT select File Open Category Open Category CAT or Categories Open Category Open Category CAT This opens a file selection window Opening graphical category files CGL is similar An alternative method to open a cgl category file is to press CTRL O To open a category simply double click on the category s name or click once on the category name and press the OK button Graphical Database for Category Theory GDCT 1 0 Settings
26. of 35 Image Type Location center x 0 0 y 150 0 z0 0 width 673 4 height 91 19999999999999 depth 20 0 vgj labelPosition in shape Oval edge linestyle solid label d22 d20 d21 source 2 target 3 edge linestyle solid label d11 d10 source 1 target 2 edge linestyle solid label s21 s20 source 3 target 2 edge linestyle solid label d0 source 0 target 1 edge linestyle solid label s10 source 2 target 1 nn nn This is the functor information functor del23 objects 0 l gt 0 1 l gt 1 2 l gt 2 Page 33 of 35 arrows d0 l gt dO d10 I gt d10 d11 gt d11 s10 I gt s10 APPENDIX B MENU ITEM SHORTCUTS The following short cuts to menu items are supported in Graphical Database for Category Theory CTRL C Create a Category CTRL H Display Help File for GDCT CTRL F Open a FGL Functor File CTRL G Save a FGL Functor File CTRL D Download a CGL Category File CTRL O Open a CGL Category File CTRL S Save a CGL Category File APPENDIX C MISC INFORMATION Version Information The Graphical Database for Category Theory is a research project at Mount Allison University that now includes over 80 Java source files and over 30000 lines of code Version 1 0 of the GDCT was released as a preliminary version which provides the basic functionality and demonstrates the abilities of the final version of the application
27. on of a category These files can be both opened and saved Since a graphical representation of the category is not saved one is generated randomly upon opening the cat file cgl These files contain both a text and graphical representation of a category and can be both opened and saved They are a concatenation of a cat file and a gml file gml These files contain only a graphical representation of a category These files can only be saved and may be used by the VGJ applet they can not be opened CAT File Format The text representation of categories is stored in the following format Any line that is a comment must start with the character On the line following the word category is the category name Objects are listed between the words objects and arrows Arrows are listed between the words arrows and relations Relations are listed after the word relations NOTE The words category objects arrows relations and gml are key words and cannot be used as the name of the category or as the name of an object or arrow within the category Objects and arrows can be a string of any length and can contain any characters excluding gt yen Wom non onn nn e a E Page 5 of 35 Relations are composed using the composition symbol An example of a typical CAT file can be found in Appendix A CGL File Format A CGL file contains the information for both a text representation and a graphi
28. rrent category from a drop down and specification of possibly composite first and second projections then testing the span for being a product Page 16 of 35 WARNING The domain of each projection must be the possible product object The test is only valid on a confluent category If in doubt use the Make Confluent tool NOTE This feature has not yet been implemented Page 17 of 35 SECTION 5 FUNCTORS Introduction A functor is a structure preserving interpretation of one finitely presented category into another A functor preserves the structure of a category because it sends Objects to Objects Arrows to Arrows and it preserves the composition that is provided by relations Functor File Types In Category Theory Database Tools there are two different file types that are associated with functors fun These files are a text only representation of a functor These files can be both opened and saved Since a graphical representation of the two categories involved is not saved the graphical representations are generated randomly upon opening the FUN file fgl These files contain both a text and graphical representation of the categories involved in a given functor as well as the functor information itself This type of file can be both opened and saved FUN File Format The text representation of functors is stored in the following three part format Part one consists of a category stored in the CAT file form
29. rsion of a category in memory To save any changes to disk use the Save Category dialogue Removing Data Selecting Categories Edit E3 GDCT Remove Data from Category Iof x Category Remove data opens Remove Lists a dialogue with lists of the Objects to Remove Arrow s to Remove Relation s to Remove objects arrows and relations of 0 d0 0 4 510410 1 F the current category Select 1 d10 1 gt 2 10 d11 1 items from the lists and click 2 dit d0 di 0 d0 3 d21 d10 d20 d10 Remove to delete the data from the category d22 d11 d21 d11 10 521 10 s20 21 d20 d10 s10 20 d20 1 21 d21 1 x Remove Status Current Category Relation d22 d10 d20 d11 removed Jets Arrow d10 NOT removed Arrow has relations dependent on it Remove g Close Page 10 of 35 WARNING An arrow can be removed only if it does not appear in any current relations An object can be removed only if it is neither the domain nor codomain of a current arrow Removing data only affects the version of a category in memory To save any changes to disk use the Save Category dialogue Changing a Category Name To change the name of the currently displayed category the user selects Categories Edit Category Change Name f Graphical Database for Category Theory E GDCT Categories Del3_cgl Le Cie es Change Name AMM 34 AANA AWD Added 2D Selecting this option will display a dialog in which
30. s10 2 gt 1 d20 2 gt 3 d21 2 gt 3 d22 2 gt 3 s20 3 gt 2 21 3 gt 2 relations s10 d10 1 s10 d11 1 d11 d0 d10 d0 d21 d10 d20 d10 d22 d10 d20 d11 d22 d11 d21 d11 s10 s21 s10 s20 s21 d20 d10 s10 s20 d20 1 s21 d21 1 s20 d21 1 s21 d22 1 s20 d22 d11 s10 gml graph directed 1 node id O label 0 graphics Image Type Location center x 37 0 y 131 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 vgj labelPosition in shape Oval node id 1 label 1 graphics Image Type Location center x 0 0 y 73 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 nn nn nn we Page 31 of 35 labelPosition in shape Oval i node id 2 label 2 graphics Image Type Location center x 121 0 y 139 0 z 0 0 we wit vidth 15 94 height 27 299999999999997 depth 20 0 veil labelPosition in shape Oval 1 node id 3 label 3 graphics Image Type Location Da center x 52 0 y 21 0 z 0 0 width 15 94 height 27 299999999999997 depth 20 0 vejl labelPosition in shape Oval i Fd node id 5 label relations ns10 d10 1 s10 d11 1 d11 d0 d10 d0 d21 d10 d20 d10 d22 d10 d20 d11 nd22 d11 d21 d11 s10 s21 s10 s20 s21 d20 d10 s10 s20 d20 1 s21 d21 1 ns20 d21 1 s21 d22 1 s20 d22 d11 s10 graphics wer woe Page 32
31. t or arrow within categories An example of a typical FGL file can be found in Appendix A Creating Functors To create a Functor in memory the user selects Functors Create Functor FGL File Format CGL File Format 1 CGL File Format 2 FUN File Format First the domain and codomain categories must be selected from those in memory using the drop down category lists The categories will be displayed in two windows Enter a descriptive name in the Functor Name box and type Enter EE Graphical Database for Category Theory E GDCT Categories Del3 cgl File Categories Functors Tools Settings Help Functor Name TestFunctor Y a Y Domain Category aen z l x Codomain Category uer2 Functor Display ap relations Tip Ls Pp d10 d0 d11 d0 s10 d11 1 s10 d10 1 op AN fe acl me 1 afl For each object type the name of its image under the functor in the box which now appears and type Enter For each arrow type the name of its image under the functor which may be a composite arrow in the box and type Enter During creation of a functor its action is displayed in the Functor display box WARNING The image of an arrow must have domain and codomain which are the images of the domain and codomain of the arrow Page 19 of 35 This completes the creation of the new functor Following creation it is possible to animate the display of the
32. the display of categories through the use of graph classes that were developed at Auburn University for Visualizing Graphs with Java VGJ a tool for graph drawing and graph layout The original program was heavily modified for displaying categories It can be found at http www eng auburn edu department cse research graphdrawing Functors between finitely presented categories can also be created and stored Functors in memory display their domain and codomain categories and their action is shown by an animated display Some of the algorithms used in GDCT were originally developed in A Database of Categories a C program written by Ryan Gunther and Michael Fleming with supervision by Robert Rosebrugh This is available at http mathcs mta ca research rosebrugh dbc The GDCT application is also base on Category Theory Database Tools a Java applet written by Jeremy Bradbury with supervision by Robert Rosebrugh This program is available at http mathcs mta ca research rosebrugh ctdt Page 1 of 35 Development Team Jeremy Bradbury B Sc Honours in Computer Science amp Mathematics Mount Allison University e mail address jsbrdbr mta ca webpage address http cs mta ca jsbrdbr Dr Robert Rosebrugh Professor Dept Mathematics and Computer Science Mount Allison University e mail address rrosebrugh mta ca webpage address http www mta ca rrosebrugh Page 2 of 35 Installation The most current version of
33. the functor name and press the OK button Either of these methods will display the downloaded functor If the program cannot locate the files on the server an error message will be displayed This could result from the user not being connected to the Internet or as a result of a server error To change the settings for downloading functors select Settings Set Server This displays a dialog for changing the default server and directories from which categories and functors can be loaded using the appropriate load command NOTE Downloading FGL functor files is not yet implemented Saving Functors In order to save a functor in memory as a graphical file fgl select File Save Functor Save Functor FGL or Functors Save Functor Save Functor FGL This opens a file selection window To save a functor to an existing file simply double click on the functor file name A new file is created by typing its name in the dialogue and clicking OK Saving functor files as text only fun is similar Removing Functors Selecting Functors Remove Functor opens a list of the functors currently in memory Select from the list and click OK to remove a functor Page 20 of 35 Printing Functors To print the text version of the currently displayed functor select File Print Print Functor Text Functor Animation The mappings on objects and arrows defined by a functor can be highlighted through the use of animation There are two ways to
34. the user can enter the new name ey CTDT Change Category Name hs aa Page 11 of 35 SECTION 3 GRAPHICAL DISPLAY OF CATEGORIES Introduction The controls for the graphical display of categories are explained in the help files under the title Visual Display Controls This help file deals with manipulating the graphical display of a category There are two ways to select a object or arrow The first is to click directly on the object or arrow so that it become highlighted red Node 0 x 100 12571397633897 y 98 99812775197015 z 0 9372819894918791 w 15 94 h 27 29999999999999 Lx oo A j mS cts The second way is to click and hold the left mouse button down before dragging a square around the desired af object s or arrow s x 66 50000000000006 y 69 5 z 0 0 y El relations 10 d10 1 510 d11 1 d10 d0 d11 d0 d20 d10 d21 d10 d20 d11 d22 d10 d21 d11 d22 d11 510 s20 10 s21 d10 s10 21 d20 s20 d20 1 521 d21 1 20 d21 1 821 d22 1 d11 s10 s20 d22 a i i J af Once an object or arrow has been selected it can be moved by holding down on it and pulling it to another position on the graphical display canvas Page 12 of 35 Visual Display Controls The controls for the graphical display of categories were developed at Auburn University as part of the a 3D package that is used by GDCT in the display of categories The visual controls can be hi
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