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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap3_mj.html">[Previous Chapter]</a>    <a href="chap5_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p>
<p><a id="X7B2650D983390045" name="X7B2650D983390045"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X7B2650D983390045">4 <span class="Heading">Category 2-Cells</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X793836877FCD5EC6">4.1 <span class="Heading">Attributes for the Type of 2-Cells</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8061EB36809022A4">4.1-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7C16D44B8639016B">4.1-2 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X780D00C482CA707D">4.1-3 Target</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7CB51AC67E08E3A1">4.2 <span class="Heading">Adding 2-Cells to a Category</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FFCA6D82FC0ED9">4.2-1 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E3D6C538354F1D9">4.2-2 AddTwoCell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B449CCB7C8050A3">4.2-3 CreateCapCategoryTwoCellWithAttributes</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7B780EDB7E466121">4.3 <span class="Heading">Identity 2-Cell and Composition of 2-Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78ED1F327D902924">4.3-1 IdentityTwoCell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DD9EB1F80921939">4.3-2 HorizontalPreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87C942B87C6A161D">4.3-3 HorizontalPostCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D89385B7834B086">4.3-4 VerticalPreCompose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79D9FBEA851C4F00">4.3-5 VerticalPostCompose</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X84A60F017943A943">4.4 <span class="Heading">Well-Definedness for 2-Cells</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8240AF367C87663F">4.4-1 IsWellDefinedForTwoCells</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Category 2-Cells</span></h3>

<p><a id="X793836877FCD5EC6" name="X793836877FCD5EC6"></a></p>

<h4>4.1 <span class="Heading">Attributes for the Type of 2-Cells</span></h4>

<p><a id="X8061EB36809022A4" name="X8061EB36809022A4"></a></p>

<h5>4.1-1 Source</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Source</code>( <var class="Arg">c</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism</p>

<p>The argument is a <span class="SimpleMath">\(2\)</span>-cell <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span>. The output is its source <span class="SimpleMath">\(\alpha\)</span>.</p>

<p><a id="X7C16D44B8639016B" name="X7C16D44B8639016B"></a></p>

<h5>4.1-2 Range</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Range</code>( <var class="Arg">c</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism</p>

<p>The argument is a <span class="SimpleMath">\(2\)</span>-cell <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span>. The output is its range <span class="SimpleMath">\(\beta\)</span>.</p>

<p><a id="X780D00C482CA707D" name="X780D00C482CA707D"></a></p>

<h5>4.1-3 Target</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Target</code>( <var class="Arg">c</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism</p>

<p>The argument is a <span class="SimpleMath">\(2\)</span>-cell <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span>. The output is its target <span class="SimpleMath">\(\beta\)</span>.</p>

<p><a id="X7CB51AC67E08E3A1" name="X7CB51AC67E08E3A1"></a></p>

<h4>4.2 <span class="Heading">Adding 2-Cells to a Category</span></h4>

<p><a id="X82FFCA6D82FC0ED9" name="X82FFCA6D82FC0ED9"></a></p>

<h5>4.2-1 Add</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Add</code>( <var class="Arg">category</var>, <var class="Arg">twocell</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Adds <var class="Arg">twocell</var> as a <span class="SimpleMath">\(2\)</span>-cell to <var class="Arg">category</var>.</p>

<p><a id="X7E3D6C538354F1D9" name="X7E3D6C538354F1D9"></a></p>

<h5>4.2-2 AddTwoCell</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddTwoCell</code>( <var class="Arg">category</var>, <var class="Arg">twocell</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Adds <var class="Arg">twocell</var> as a <span class="SimpleMath">\(2\)</span>-cell to <var class="Arg">category</var>. If <var class="Arg">twocell</var> already lies in the filter <code class="code">IsCapCategoryTwoCell</code>, the operation <code class="func">Add</code> (<a href="chap4_mj.html#X82FFCA6D82FC0ED9"><span class="RefLink">4.2-1</span></a>) can be used instead.</p>

<p><a id="X7B449CCB7C8050A3" name="X7B449CCB7C8050A3"></a></p>

<h5>4.2-3 CreateCapCategoryTwoCellWithAttributes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CreateCapCategoryTwoCellWithAttributes</code>( <var class="Arg">category</var>, <var class="Arg">source</var>, <var class="Arg">range</var>[, <var class="Arg">attr1</var>, <var class="Arg">val1</var>, <var class="Arg">attr2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a twocell</p>

<p>Creates a <span class="SimpleMath">\(2\)</span>-cell in <var class="Arg">category</var> with the given attributes.</p>

<p><a id="X7B780EDB7E466121" name="X7B780EDB7E466121"></a></p>

<h4>4.3 <span class="Heading">Identity 2-Cell and Composition of 2-Cells</span></h4>

<p><a id="X78ED1F327D902924" name="X78ED1F327D902924"></a></p>

<h5>4.3-1 IdentityTwoCell</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityTwoCell</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(2\)</span>-cell</p>

<p>The argument is a morphism <span class="SimpleMath">\(\alpha\)</span>. The output is its identity <span class="SimpleMath">\(2\)</span>-cell <span class="SimpleMath">\(\mathrm{id}_{\alpha}: \alpha \rightarrow \alpha\)</span>.</p>

<p><a id="X7DD9EB1F80921939" name="X7DD9EB1F80921939"></a></p>

<h5>4.3-2 HorizontalPreCompose</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HorizontalPreCompose</code>( <var class="Arg">c</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(2\)</span>-cell</p>

<p>The arguments are two <span class="SimpleMath">\(2\)</span>-cells <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span>, <span class="SimpleMath">\(d: \gamma \rightarrow \delta\)</span> between morphisms <span class="SimpleMath">\(\alpha, \beta: a \rightarrow b\)</span> and <span class="SimpleMath">\(\gamma, \delta: b \rightarrow c\)</span>. The output is their horizontal composition <span class="SimpleMath">\(d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)\)</span>.</p>

<p><a id="X87C942B87C6A161D" name="X87C942B87C6A161D"></a></p>

<h5>4.3-3 HorizontalPostCompose</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HorizontalPostCompose</code>( <var class="Arg">d</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(2\)</span>-cell</p>

<p>The arguments are two <span class="SimpleMath">\(2\)</span>-cells <span class="SimpleMath">\(d: \gamma \rightarrow \delta\)</span>, <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span> between morphisms <span class="SimpleMath">\(\alpha, \beta: a \rightarrow b\)</span> and <span class="SimpleMath">\(\gamma, \delta: b \rightarrow c\)</span>. The output is their horizontal composition <span class="SimpleMath">\(d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)\)</span>.</p>

<p><a id="X7D89385B7834B086" name="X7D89385B7834B086"></a></p>

<h5>4.3-4 VerticalPreCompose</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VerticalPreCompose</code>( <var class="Arg">c</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(2\)</span>-cell</p>

<p>The arguments are two <span class="SimpleMath">\(2\)</span>-cells <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span>, <span class="SimpleMath">\(d: \beta \rightarrow \gamma\)</span> between morphisms <span class="SimpleMath">\(\alpha, \beta, \gamma: a \rightarrow b\)</span>. The output is their vertical composition <span class="SimpleMath">\(d \circ c: \alpha \rightarrow \gamma\)</span>.</p>

<p><a id="X79D9FBEA851C4F00" name="X79D9FBEA851C4F00"></a></p>

<h5>4.3-5 VerticalPostCompose</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VerticalPostCompose</code>( <var class="Arg">d</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(2\)</span>-cell</p>

<p>The arguments are two <span class="SimpleMath">\(2\)</span>-cells <span class="SimpleMath">\(d: \beta \rightarrow \gamma\)</span>, <span class="SimpleMath">\(c: \alpha \rightarrow \beta\)</span> between morphisms <span class="SimpleMath">\(\alpha, \beta, \gamma: a \rightarrow b\)</span>. The output is their vertical composition <span class="SimpleMath">\(d \circ c: \alpha \rightarrow \gamma\)</span>.</p>

<p><a id="X84A60F017943A943" name="X84A60F017943A943"></a></p>

<h4>4.4 <span class="Heading">Well-Definedness for 2-Cells</span></h4>

<p><a id="X8240AF367C87663F" name="X8240AF367C87663F"></a></p>

<h5>4.4-1 IsWellDefinedForTwoCells</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWellDefinedForTwoCells</code>( <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a <span class="SimpleMath">\(2\)</span>-cell <span class="SimpleMath">\(c\)</span>. The output is <code class="code">true</code> if <span class="SimpleMath">\(c\)</span> is well-defined, otherwise the output is <code class="code">false</code>.</p>


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