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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346<?xml version="1.0" encoding="UTF-8"?>12<!-- This is an automatically generated file. -->3<Chapter Label="Chapter_Category_2-Cells">4<Heading>Category 2-Cells</Heading>56<P/>7<Section Label="Chapter_Category_2-Cells_Section_Attributes_for_the_Type_of_2-Cells">8<Heading>Attributes for the Type of 2-Cells</Heading>910<ManSection>11<Attr Arg="c" Name="Source" Label="for IsCapCategoryTwoCell"/>12<Returns>a morphism13</Returns>14<Description>15The argument is a <Math>2</Math>-cell <Math>c: \alpha \rightarrow \beta</Math>.16The output is its source <Math>\alpha</Math>.17</Description>18</ManSection>192021<ManSection>22<Attr Arg="c" Name="Range" Label="for IsCapCategoryTwoCell"/>23<Returns>a morphism24</Returns>25<Description>26The argument is a <Math>2</Math>-cell <Math>c: \alpha \rightarrow \beta</Math>.27The output is its range <Math>\beta</Math>.28</Description>29</ManSection>303132</Section>33343536<Section Label="Chapter_Category_2-Cells_Section_Identity_2-Cell_and_Composition_of_2-Cells">37<Heading>Identity 2-Cell and Composition of 2-Cells</Heading>3839<ManSection>40<Attr Arg="alpha" Name="IdentityTwoCell" Label="for IsCapCategoryMorphism"/>41<Returns>a <Math>2</Math>-cell42</Returns>43<Description>44The argument is a morphism <Math>\alpha</Math>.45The output is its identity <Math>2</Math>-cell <Math>\mathrm{id}_{\alpha}: \alpha \rightarrow \alpha</Math>.46</Description>47</ManSection>484950<ManSection>51<Oper Arg="C, F" Name="AddIdentityTwoCell" Label="for IsCapCategory, IsFunction"/>52<Returns>nothing53</Returns>54<Description>55The arguments are a category <Math>C</Math> and a function <Math>F</Math>.56This operations adds the given function <Math>F</Math>57to the category for the basic operation <C>IdentityTwoCell</C>.58<Math>F: \alpha \mapsto \mathrm{id}_{\alpha}</Math>.59</Description>60</ManSection>616263<ManSection>64<Oper Arg="c,d" Name="HorizontalPreCompose" Label="for IsCapCategoryTwoCell, IsCapCategoryTwoCell"/>65<Returns>a <Math>2</Math>-cell66</Returns>67<Description>68The arguments are two <Math>2</Math>-cells69<Math>c: \alpha \rightarrow \beta</Math>,70<Math>d: \gamma \rightarrow \delta</Math>71between morphisms <Math>\alpha, \beta: a \rightarrow b</Math> and <Math>\gamma, \delta: b \rightarrow c</Math>.72The output is their horizontal composition73<Math>d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)</Math>.74</Description>75</ManSection>767778<ManSection>79<Oper Arg="C, F" Name="AddHorizontalPreCompose" Label="for IsCapCategory, IsFunction"/>80<Returns>nothing81</Returns>82<Description>83The arguments are a category <Math>C</Math> and a function <Math>F</Math>.84This operations adds the given function <Math>F</Math>85to the category for the basic operation <C>HorizontalPreCompose</C>.86<Math>F: (c,d) \mapsto d \ast c</Math>.87</Description>88</ManSection>899091<ManSection>92<Oper Arg="d,c" Name="HorizontalPostCompose" Label="for IsCapCategoryTwoCell, IsCapCategoryTwoCell"/>93<Returns>a <Math>2</Math>-cell94</Returns>95<Description>96The arguments are two <Math>2</Math>-cells97<Math>d: \gamma \rightarrow \delta</Math>,98<Math>c: \alpha \rightarrow \beta</Math>99between morphisms <Math>\alpha, \beta: a \rightarrow b</Math> and <Math>\gamma, \delta: b \rightarrow c</Math>.100The output is their horizontal composition101<Math>d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta)</Math>.102</Description>103</ManSection>104105106<ManSection>107<Oper Arg="C, F" Name="AddHorizontalPostCompose" Label="for IsCapCategory, IsFunction"/>108<Returns>nothing109</Returns>110<Description>111The arguments are a category <Math>C</Math> and a function <Math>F</Math>.112This operations adds the given function <Math>F</Math>113to the category for the basic operation <C>HorizontalPostCompose</C>.114<Math>F: (d,c) \mapsto d \ast c</Math>.115</Description>116</ManSection>117118119<ManSection>120<Oper Arg="c,d" Name="VerticalPreCompose" Label="for IsCapCategoryTwoCell, IsCapCategoryTwoCell"/>121<Returns>a <Math>2</Math>-cell122</Returns>123<Description>124The arguments are two <Math>2</Math>-cells125<Math>c: \alpha \rightarrow \beta</Math>,126<Math>d: \beta \rightarrow \gamma</Math>127between morphisms <Math>\alpha, \beta, \gamma: a \rightarrow b</Math>.128The output is their vertical composition129<Math>d \circ c: \alpha \rightarrow \gamma</Math>.130</Description>131</ManSection>132133134<ManSection>135<Oper Arg="C, F" Name="AddVerticalPreCompose" Label="for IsCapCategory, IsFunction"/>136<Returns>nothing137</Returns>138<Description>139The arguments are a category <Math>C</Math> and a function <Math>F</Math>.140This operations adds the given function <Math>F</Math>141to the category for the basic operation <C>VerticalPreCompose</C>.142<Math>F: (c,d) \mapsto d \circ c</Math>.143</Description>144</ManSection>145146147<ManSection>148<Oper Arg="d,c" Name="VerticalPostCompose" Label="for IsCapCategoryTwoCell, IsCapCategoryTwoCell"/>149<Returns>a <Math>2</Math>-cell150</Returns>151<Description>152The arguments are two <Math>2</Math>-cells153<Math>d: \beta \rightarrow \gamma</Math>,154<Math>c: \alpha \rightarrow \beta</Math>155between morphisms <Math>\alpha, \beta, \gamma: a \rightarrow b</Math>.156The output is their vertical composition157<Math>d \circ c: \alpha \rightarrow \gamma</Math>.158</Description>159</ManSection>160161162<ManSection>163<Oper Arg="C, F" Name="AddVerticalPostCompose" Label="for IsCapCategory, IsFunction"/>164<Returns>nothing165</Returns>166<Description>167The arguments are a category <Math>C</Math> and a function <Math>F</Math>.168This operations adds the given function <Math>F</Math>169to the category for the basic operation <C>VerticalPostCompose</C>.170<Math>F: (d,c) \mapsto d \circ c</Math>.171</Description>172</ManSection>173174175</Section>176177178<Section Label="Chapter_Category_2-Cells_Section_Well-Definedness_for_2-Cells">179<Heading>Well-Definedness for 2-Cells</Heading>180181<ManSection>182<Oper Arg="c" Name="IsWellDefinedForTwoCells" Label="for IsCapCategoryTwoCell"/>183<Returns>a boolean184</Returns>185<Description>186The argument is a <Math>2</Math>-cell <Math>c</Math>.187The output is <C>true</C> if <Math>c</Math> is well-defined,188otherwise the output is <C>false</C>.189</Description>190</ManSection>191192193<ManSection>194<Oper Arg="C, F" Name="AddIsWellDefinedForTwoCells" Label="for IsCapCategory, IsFunction"/>195<Returns>nothing196</Returns>197<Description>198The arguments are a category <Math>C</Math> and a function <Math>F</Math>.199This operations adds the given function <Math>F</Math>200to the category for the basic operation <C>IsWellDefinedForTwoCells</C>.201<Math>F: c \mapsto \mathtt{IsWellDefinedForMorphisms}( c )</Math>.202</Description>203</ManSection>204205206</Section>207208209<P/>210</Chapter>211212213214