Pullbacks

xholonWorkbook.xml

<?xml version="1.0" encoding="UTF-8"?>
<!--Xholon Workbook http://www.primordion.com/Xholon/gwt/ MIT License, Copyright (C) Ken Webb, Mon Nov 27 2017 10:46:24 GMT-0500 (EST)-->
<XholonWorkbook>

<Notes><![CDATA[
Xholon
------
Title: Pullbacks
Description: 
Url: http://www.primordion.com/Xholon/gwt/
InternalName: 6ff8d2978db7a168c5b0250b220c537f
Keywords: 

My Notes
--------
November 24, 2017

Exercise 3.2.1.3.  [1]
a. Draw a set X with five elements and a set Y with three elements. Color each element of X and each element of Y red, blue, or green,2 and do so in a random-looking way. Considering your coloring of X as a function X → C, where C = {red, blue, green}, and similarly obtaining a function Y → C, draw the fiber product X ×C Y.
b. The universal property for products guarantees a function X ×C Y → X × Y, which will be an injection. This means that the drawing you made of the fiber product can be embedded into the 5 × 3 grid. Draw the grid and indicate this subset.

Solution 3.2.1.3.
a. Let X = {1, 2, 3, 4, 5} and Y = {a, b, c}.

TODO
- use XholonSpreadsheet to implement this

References
----------
(1) http://category-theory.mitpress.mit.edu/chapter003.html
    3.2.1 Pullbacks, exercise 3.2.1.3

]]></Notes> 

<_-.XholonClass>
  <PhysicalSystem/>
  
  <RandomColorer/>
  
</_-.XholonClass>

<xholonClassDetails>
  <RandomColorer implName="org.primordion.xholon.base.Behavior_gwtjs"/>
</xholonClassDetails>

<PhysicalSystem>
  
  <XholonSet roleName="X">
    <Attribute_String roleName="1"/>
    <Attribute_String roleName="2"/>
    <Attribute_String roleName="3"/>
    <Attribute_String roleName="4"/>
    <Attribute_String roleName="5"/>
  </XholonSet>
  
  <XholonSet roleName="Y">
    <Attribute_String roleName="a"/>
    <Attribute_String roleName="b"/>
    <Attribute_String roleName="c"/>
  </XholonSet>
  
  <XholonSet roleName="C">
    <Attribute_String roleName="red"/>
    <Attribute_String roleName="blue"/>
    <Attribute_String roleName="green"/>
  </XholonSet>
  
  <RandomColorer><![CDATA[
  var me;
  var beh = {
    postConfigure: function() {
      me = this.cnode;
      me.println(me);
      var colorsetNode = me.xpath("../XholonSet[@roleName='C']");
      var colorsetSize = this.size(colorsetNode);
      me.println(colorsetNode);
      // X
      var node = me.xpath("../XholonSet[@roleName='X']").first();
      while (node) {
        this.colorize(node, colorsetNode, colorsetSize);
        node = node.next();
      }
      // Y
      node = me.xpath("../XholonSet[@roleName='Y']").first();
      while (node) {
        this.colorize(node, colorsetNode, colorsetSize);
        node = node.next();
      }
      me.remove();
    },
    
    colorize: function(node, colorsetNode, colorsetSize) {
      var colorIx = Math.ceil(Math.random() * colorsetSize);
      me.println(colorIx);
      var colorNode = colorsetNode.xpath("*[" + colorIx + "]");
      me.println(colorNode);
      me.println(colorNode.role());
      node.color(colorNode.role());
      node.couleur = colorNode; // this is a port
    },
    
    // return size of a Set node
    size: function(node) {
      var count = 0;
      var cnode = node.first();
      while (cnode) {
        count++;
        cnode = cnode.next();
      }
      return count;
    }
  }
]]></RandomColorer>
  
</PhysicalSystem>

<SvgClient><Attribute_String roleName="svgUri"><![CDATA[data:image/svg+xml,
<svg width="100" height="50" xmlns="http://www.w3.org/2000/svg">
  <g>
    <title>Set X</title>
    <rect id="PhysicalSystem/XholonSet[@roleName='X']" fill="#98FB98" height="50" width="50" x="25" y="0"/>
    <g>
      <title>Set C</title>
      <rect id="PhysicalSystem/XholonSet[@roleName='C']" fill="#6AB06A" height="50" width="10" x="80" y="0"/>
    </g>
  </g>
</svg>
]]></Attribute_String><Attribute_String roleName="setup">${MODELNAME_DEFAULT},${SVGURI_DEFAULT}</Attribute_String></SvgClient>

</XholonWorkbook>