Amiga Tools 2/Amiga Tools 2.iso/tex/macros/source/contrib/supported/xypic/src/xy2cell.doc/node2_mn.html

Text File | 1995-03-15 | 4KB | 163 lines

<H2><A ID="SECTION00001000000000000000">
Typesetting 2-cells in Diagrams</A>
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Categorical
``2-cell'' morphisms are used in the study of tensor categories and elsewhere.
The morphisms are displayed as a pair of curved arrows, symmetrically
placed, together with an orientation indicated by a short broad arrow,
or <#11#><I>Arrow</I><#11#>. Labels may be placed on all three components.

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<#12#>5pc<#12#><tex2html_comment_mark>31
<#13#>2pc<#13#><tex2html_comment_mark>32

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<BR>
<tex2html_image_mark>#code14#
<BR>
<P><tex2html_verbatim_mark>#math449#</P><DIV ALIGN="CENTER">
<tex2html_image_mark>#tex2html_wrap_indisplay1723#

</DIV><P></P>


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<BR>
<tex2html_image_mark>#code16#
<BR>
<P><tex2html_verbatim_mark>#math450#</P><DIV ALIGN="CENTER">
<tex2html_image_mark>#tex2html_wrap_indisplay1725#

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These categorical diagrams frequently
having a matrix-like layout, as with commutative diagrams.
To facilitate this there are control sequences of the form:
|| , || , || , ...
analogous to the names defined in |xyv2| for use
in diagrams produced using |xymatrix|.
As this involves the definition of 21 new control sequences, many of which
may never be used, these are not defined immediately upon loading |xy2cell|.
Instead the user must first specify ||.

<P>
As in the second example above, just the upper or lower curved
arrow may be set using control sequences of the form
|.̇uppertwocell| and |.̇lowertwocell|.
These together with the |.̇compositemap| family, in which
two abutting arrows are set with an empty object at the join,
allow for the construction of complicated ``pasting diagrams''
(see figure ??[f.pasting] for an example).

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The following initialise the families of control sequences
for use in matrix diagrams.

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<BR>
<tex2html_image_mark>#defs20#
<BR>

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2
(
@<#1091#><#22#>twocell<#22#><#869#><tex2html_comment_mark>34
@<#23#>start of 2-cell<#23#><#24#><#24#><tex2html_comment_mark>35
@<#869#><#1091#>

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@<#1092#><tex2html_comment_mark>36
<#25#>uppertwocell<#25#><#870#><tex2html_comment_mark>37
@<#26#>start of 2-cell<#26#><#27#><#27#><tex2html_comment_mark>38
@<#870#><tex2html_comment_mark>39
<#28#>lowertwocell<#28#><#871#><tex2html_comment_mark>40
@<#29#>start of 2-cell<#29#><#30#><#30#><tex2html_comment_mark>41
@<#871#><#1092#>

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@<#1093#><#31#>compositemap<#31#><#872#><tex2html_comment_mark>42
@<#32#>start composite-map<#32#><#33#><#33#><tex2html_comment_mark>43
@<#872#><#1093#>

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@<#34#><tex2html_comment_mark>44
<#34#>
)

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2
 The families of connections are defined using the utility macro...
||
which defines control sequences for morphisms between
 neighbouring cells, and next-to-neighbouring cells.

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More distant cells use  ||, ||, etc.
 with ;SPMlt;hop;SPMgt;s e.g.
      |[1,3]|  and  |[llddd]|.

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(
@#1[#2]#3<#873#>#1<#873#>
@[#2]#3<#36#><#36#>

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@#1#2<#1094#><tex2html_comment_mark>48
<#1094#>
)

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3

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Alternatively 2-cells can be set directly in -pictures without using the
matrix feature. In this case the above commands are not needed.  This is
described in~§??[standard].

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Furthermore a new directional |<#86#>=;SPMgt;<#86#>| can be used to
place an ``Arrow'' anywhere in a picture,
after the direction has been established appropriately.
It is used with all of the 2-cell types.

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(
<#87#>=;SPMgt;<#87#><#899#>!/5pt/<#88#>=<#88#>!/2.5pt/<#89#>=<#89#>*!/-5pt/2<#90#>;SPMgt;<#90#><#899#>

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@#1<#900#><#900#>
@@ <#901#><#92#>=;SPMgt;<#92#><#901#>
)

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3
<BR>
<tex2html_image_mark>#code93#
<BR>

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<DIV class="CENTER"><A ID="125"><tex2html_anchor_mark></A>
<TABLE>
<CAPTION class="BOTTOM"><STRONG><#1728#>Figure<#1728#>:</STRONG>
<#1729#>Pasting diagram.<#1729#></CAPTION>
<TR><TD><P><tex2html_verbatim_mark>#math451#</P><DIV ALIGN="CENTER">
<tex2html_image_mark>#tex2html_wrap_indisplay1727#

</DIV><P></P>
??=[f.pasting]</TD></TR>
</TABLE>
</DIV>


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2