Typesetting 2-cells in Diagrams

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#>Arrow<#11#>. Labels may be placed on all three components.

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

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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 ||.

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).

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

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

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

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

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

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.

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

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

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

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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].

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.

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

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

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