1 | % This is pdfTeX, Version 3.14159265-2.6-1.40.18 (MiKTeX 2.9.6350) |
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2 | % LaTeX2e <2017-04-15> |
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3 | |
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4 | % Package: amsbsy 1999/11/29 v1.2d Bold Symbols |
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5 | % Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support |
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6 | % Package: amsmath 2017/09/02 v2.17a AMS math features |
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7 | % Package: amsopn 2016/03/08 v2.02 operator names |
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8 | % Package: amstext 2000/06/29 v2.01 AMS text |
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9 | % Package: amsthm 2017/10/31 v2.20.4 |
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10 | % Package: bm 2017/01/16 v1.2c Bold Symbol Support (DPC/FMi) |
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11 | % Package: cleveref 2018/03/27 v0.21.4 Intelligent cross-referencing |
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12 | % Package: enumitem 2011/09/28 v3.5.2 Customized lists |
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13 | % Package: expl3 2018-06-01 L3 programming layer (code) |
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14 | % Package: expl3 2018-06-01 L3 programming layer (loader) |
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15 | % Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) |
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16 | % Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) |
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17 | % Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX |
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18 | % Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch |
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19 | % Package: ifthen 2014/09/29 v1.1c Standard LaTeX ifthen package (DPC) |
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20 | % Package: keyval 2014/10/28 v1.15 key=value parser (DPC) |
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21 | % Package: mathtools 2018/01/08 v1.21 mathematical typesetting tools |
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22 | % Package: nameref 2016/05/21 v2.44 Cross-referencing by name of section |
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23 | % Package: pgf 2015/08/07 v3.0.1a (rcs-revision 1.15) |
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24 | % Package: scalerel 2016/12/29 v1.8 Routines for constrained scaling and stretchi |
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25 | % Package: showlabels 2015/12/08 v1.7 |
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26 | % Package: stix2 2018/04/02 v2.0.0-latex STIX Two fonts support package |
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27 | % Package: thmtools 2014/04/21 v66 |
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28 | % Package: tikz 2015/08/07 v3.0.1a (rcs-revision 1.151) |
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29 | % Package: tikz-cd 2014/10/30 v0.9e Commutative diagrams with tikzx |
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30 | |
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31 | % Package: xparse 2018-05-12 L3 Experimental document command parser |
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32 | % Package: xstring 2013/10/13 v1.7c String manipulations (C Tellechea) |
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33 | % Package: xy 2013/10/06 Xy-pic version 3.8.9 |
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34 | % Version 2 |
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35 | % 1. Define \into |
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36 | % 2. Define m-chart at a point and subchart at a point |
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37 | % 3. Change definition of M-atlas morphism |
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38 | % 3. Change definition of Ck-atlas morphism |
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39 | |
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40 | \documentclass{article} |
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41 | \usepackage{amsmath} |
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42 | %\usepackage{amssymb} |
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43 | \usepackage{amsthm} |
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44 | \usepackage{bm} |
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45 | \usepackage{enumitem} |
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46 | \usepackage{ifthen} |
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47 | %\usepackage{mathrsfs} |
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48 | \usepackage{mathtools} |
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49 | \usepackage{scalerel} |
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50 | \usepackage{stix2}[notext,not1] %reqires XeTeX or luaTeX |
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51 | \usepackage{thmtools} |
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52 | \usepackage{tikz-cd} |
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53 | \usepackage{xparse} % loads expl3 |
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54 | %See interface3.pdf |
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55 | \usepackage{xstring} |
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56 | |
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57 | \usepackage[cmtip,all,barr]{xy} |
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58 | %Remove when xybarr.tex bug fixed |
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59 | \newdir_{ (}{{ }*!/-.5em/@_{(}} |
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60 | |
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61 | %Morphisms of category |
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62 | \newcommand \Ar {\mathrm{Ar}} |
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63 | |
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64 | \DeclareMathOperator \arin {\stackrel{\Ar}{\in}} |
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65 | |
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66 | %Category of atlases from E to C |
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67 | %\DeclareMathOperator \Atl {\mathscr{A\!t\!l}} |
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68 | \newcommand \Atl [1] [] |
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69 | { \ifthenelse {\equal{#1}{}} |
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70 | {\mathscr{A\!t\!l}} |
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71 | {\mathscr{A\!t\!l}^{\mathrm{#1}}} |
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72 | } |
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73 | |
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74 | \DeclareMathOperator \Atlas {\mathrm{Atlas}} |
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75 | |
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76 | \newcommand \Bun {\mathrm{Bun}} |
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77 | |
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78 | \newcommand \BunProd {\mathrm{Bun}-\mathrm{prod}} |
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79 | |
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80 | \newcommand \C {\mathrm{C}} |
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81 | |
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82 | % Select font for standard categories |
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83 | \newcommand \Cat [1] {\mathbf{#1}} |
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84 | |
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85 | % Select font for categories and sequences of categories |
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86 | \newcommand \catname [1] {\mathscr{#1}} |
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87 | \newcommand \catseqname [1] {\bm{\mathscr{#1}}} |
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88 | % Can't use \bm without stix and XeTeX |
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89 | |
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90 | \newcommand \Ck {\mathrm{C^k}} |
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91 | |
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92 | \newcommand \Classic {\mathrm{Classic}} |
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93 | |
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94 | \DeclareMathOperator \codomain {\mathrm{codomain}} |
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95 | |
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96 | % Extended composition operators for function sequences |
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97 | \newcommand {\compose} [1] [def] |
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98 | { \ifthenelse {\equal{#1}{def}} |
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99 | {\mathbin \circ} |
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100 | {\mathbin{\overset{#1} {\circ}}} |
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101 | } |
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102 | |
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103 | % Compose head |
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104 | \newcommand {\composeh} [1] [def] |
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105 | { \ifthenelse {\equal{#1}{def}} |
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106 | {\mathbin \odot} |
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107 | {\mathbin{\overset{#1} {\odot}}} |
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108 | } |
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109 | |
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110 | % Compose tail |
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111 | \newcommand {\composet} [1] [def] |
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112 | { \ifthenelse {\equal{#1}{def}} |
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113 | {\mathbin \cdot} |
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114 | {\mathbin{\overset{#1} {\cdot}}} |
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115 | } |
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116 | |
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117 | \DeclareMathOperator \defeq {\stackrel{\mathrm{def}}{=}} |
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118 | |
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119 | \DeclareMathOperator \domain {\mathrm{domain}} |
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120 | |
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121 | \DeclareMathOperator \Domain {\seqname{domain}} |
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122 | |
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123 | \newcommand \false {\mathrm{False}} |
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124 | |
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125 | \newcommand \Fib {\mathrm{Fib}} |
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126 | |
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127 | \newcommand \full [2] [] {\underset{\mathrm {{#1}full}}{#2}} |
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128 | |
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129 | \newcommand \fullcref [1] |
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130 | { \ifthenelse {\equal{\nameref{#1}}{}} |
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131 | {\cref{#1}} |
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132 | {\cref{#1} (\nameref{#1})} |
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133 | } |
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134 | |
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135 | % Select font for functions and sequences of functions |
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136 | % \newcommand \funcname [1] {\mathit{#1}} |
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137 | % \newcommand \funcseqname [1] {\bm{#1}} |
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138 | |
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139 | \DeclareMathOperator \Functor {\mathop{\mathscr{F}}} |
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140 | |
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141 | \DeclareMathOperator \head {\mathrm{head}} |
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142 | |
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143 | \DeclareMathOperator \Hom {\mathrm{Hom}} |
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144 | |
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145 | \DeclareMathOperator \Id {\mathrm{Id}} |
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146 | |
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147 | \DeclareMathOperator \ID {\mathbf{Id}} |
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148 | |
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149 | \newcommand \into {\negthickspace\mon} |
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150 | |
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151 | %Propositional function isCk_{f,A}(x,y,...) |
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152 | \DeclareMathOperator \isCk {\mathrm{isCk}} |
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153 | |
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154 | %Propositional function isAtl(A,E,C) |
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155 | \newcommand \isAtl [1] [] |
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156 | { \ifthenelse {\equal{#1}{}} |
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157 | {\mathrm{isAtl}} |
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158 | {\mathrm{isAtl}^{\mathrm{#1}}} |
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159 | } |
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160 | |
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161 | %Propositional function isLCS(L, M, A, F, Sigma) |
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162 | \DeclareMathOperator \isLCS {\mathrm{isLCS}} |
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163 | |
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164 | \DeclareMathOperator \iso {\stackrel{\sim}{=}} |
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165 | |
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166 | % Join two tuples |
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167 | \DeclareMathOperator \join {\mathrm{join}} |
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168 | |
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169 | %Category of M-Sigma local coordinate spaces |
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170 | \newcommand \LCS {\mathrm{LCS}} |
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171 | |
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172 | \DeclareMathOperator \length {\mathrm{length}} |
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173 | |
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174 | \DeclareMathOperator \lengtho {\mathrm length0} |
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175 | |
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176 | \newcommand \M {\mathrm{M}} |
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177 | |
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178 | \newcommand \Man {\mathrm{Man}} |
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179 | |
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180 | % Adjust : spacing for f maps a to b |
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181 | \newcommand \maps {\!\colon} |
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182 | |
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183 | %Set long names upright, short names italic |
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184 | \newcommand{\mathvarname}[1]{% |
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185 | \begingroup\noexpandarg |
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186 | \StrLen{#1}[\temp]% |
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187 | \ifnum\temp>1 |
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188 | \mathrm{#1}% |
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189 | \else |
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190 | #1% |
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191 | \fi |
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192 | \endgroup |
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193 | } |
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194 | |
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195 | \newcommand \maxfull [2] [] {\underset{\mathrm {{#1}max-full}}{#2}} |
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196 | \newcommand \maximal [2] [] {\underset{\mathrm {{#1}max}}{#2}} |
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197 | |
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198 | \newcommand \minimal [2] [] {\underset{{#1}\mathrm {min}}{#2}} |
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199 | |
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200 | %Model category |
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201 | \newcommand \Mod {\mathrm{Mod}} |
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202 | |
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203 | %Morphisms between two objects |
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204 | \newcommand \Mor {\mathrm{Mor}} |
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205 | |
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206 | % Morphism of category |
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207 | \DeclareMathOperator \morphin {\stackrel{\Ar}{\in}} |
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208 | |
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209 | %Object of category |
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210 | \newcommand \Ob {\mathrm{Ob}} |
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211 | \DeclareMathOperator \objin {\stackrel{\Ob}{\in}} |
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212 | |
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213 | \newcommand \onto {\epi} |
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214 | |
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215 | % All non-null open sets |
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216 | \newcommand \op [2] [] {\underset{{#1}\mathrm {op}}{#2}} |
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217 | |
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218 | \newcommand \optriv [2] [] {\underset{{#1}\mathrm {op-triv}}{#2}} |
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219 | |
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220 | \newcommand \pagecref [1] |
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221 | { \ifthenelse {\equal{\nameref{#1}}{}} |
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222 | {\cref{#1} on \cpageref{#1}} |
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223 | {\cref{#1} (\nameref{#1}) on \cpageref{#1}\!} |
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224 | } |
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225 | |
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226 | \newcommand \Pagecref [1] |
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227 | { \ifthenelse {\equal{\nameref{#1}}{}} |
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228 | {\Cref{#1} on \cpageref{#1}} |
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229 | {\Cref{#1} (\nameref{#1}) on \cpageref{#1}} |
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230 | } |
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231 | |
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232 | \DeclareMathOperator \range {\mathrm{range}} |
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233 | |
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234 | \DeclareMathOperator \Range {\seqname{range}} |
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235 | |
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236 | \newcommand \restrictto {\!\restriction} |
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237 | |
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238 | \DeclareMathOperator \seqeq {\stackrel{()}{=}} |
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239 | |
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240 | \DeclareMathOperator \seqin {\stackrel{()}{\in}} |
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241 | |
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242 | %Font for sequence |
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243 | %\newcommand \seqname [1] {\bm{\mathit{\mathsf{#1}}}} |
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244 | |
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245 | \newcommand \Set {\mathbf{Set}} |
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246 | % |
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247 | %\newcommand \sing [2] [] {\underset{{#1}\mathrm{Sing}{#2}} |
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248 | %\newcommand \Sing [2] [] {\underset{\bm{{#1}\mathrm{Sing}}{#2}} |
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249 | |
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250 | \newcommand \singcat [2] [] {\underset{{#1}\mathscr{S\!i\!n\!g}}{#2}} |
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251 | \newcommand \Singcat [2] [] {\underset{{#1}\mathscr{S\!i\!n\!g}}{#2}} |
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252 | % Can't use \bm without stix and XeTeX |
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253 | |
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254 | \newcommand \strict [2] [] {\underset{\mathrm {{#1}strict}}{#2}} |
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255 | |
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256 | \newcommand \subcat [1] [] {\overset{\mathrm{{#1}cat}}{\subseteq}} |
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257 | \newcommand \SUBCAT [1] [] {\overset{\mathbf{{#1}cat}}{\subseteq}} |
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258 | |
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259 | \newcommand \submod [1] [] {\overset{\mathrm{{#1}mod}}{\subseteq}} |
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260 | |
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261 | \DeclareMathOperator \SUBSETEQ {\stackrel{()}{\subseteq}} |
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262 | |
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263 | \DeclareMathOperator \tail {\mathrm{tail}} |
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264 | |
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265 | \newcommand \toiso {\,\to/{>}->>/^{\iso}} |
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266 | |
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267 | \newcommand \Top {\mathrm{Top}} |
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268 | \newcommand \Topcat {\mathscr{T\!o\!p}} |
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269 | |
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270 | % Font for topology |
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271 | \newcommand \topname [1] {\mathfrak{#1}} |
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272 | |
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273 | \newcommand \Topology {\mathfrak{Top}} |
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274 | |
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275 | \newcommand \triv [2] [] {\underset{{#1}\mathrm {triv}}{#2}} |
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276 | \newcommand \Triv [2] [] {\underset{{#1}\mathbf {triv}}{#2}} |
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277 | |
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278 | \newcommand \trivcat [2] [] {\underset{{#1}\mathscr{T\!r\!i\!v}}{#2}} |
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279 | \newcommand \Trivcat [2] [] {\underset{{#1}\bm{\mathscr{T\!r\!i\!v}}}{#2}} |
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280 | % Can't use \bm without stix and XeTeX |
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281 | |
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282 | \newcommand \true {\mathrm{True}} |
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283 | % Can't use long name because \mathscr doesn't support lower case. |
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284 | \newcommand \truthcat {\mathscr{T}} |
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285 | \newcommand \truthset {\mathbb{T}} |
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286 | \newcommand \truthspace {\mathrm{Truthspace}} |
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287 | \newcommand \truthtop {\mathfrak{Truthtop}} |
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288 | |
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289 | \newcommand \unioncat [1] [] {\overset{\mathrm{{#1}cat}}{\cup}} |
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290 | \newcommand \UNIONCAT [1] [] {\overset{\mathbf{{#1}cat}}{\cup}} |
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291 | |
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292 | % Existential and universal quamtifiers |
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293 | % Set former |
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294 | % Union and intersection |
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295 | |
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296 | % -------------------------------------------------------------------- |
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297 | % | From here to closing --- belongs in package | |
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298 | % | | |
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299 | |
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300 | % Copyright 2016 Shmuel (Seymour J.) Metz |
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301 | % I grant permission to the AMS, arXiv.org, the LATEX3 project and the |
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302 | % TeX users group to incorporate these commands in any LaTeX package |
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303 | % that may be freely redistributed, provided that they attribute the |
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304 | % source. |
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305 | |
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306 | % These commands are intended to allow semantic markup for some |
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307 | % common mathermtical constructs: |
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308 | % \equant{variables}{proposition} Existential qunatifier |
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309 | % \uquant{variables}{proposition} Universal quantifier |
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310 | % \union[indices]{set} Union |
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311 | % \intersection[indices]{set} Intersection |
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312 | % \set {elements}[propositions] Set of |
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313 | % \set{elements}[propositions]* Set of, split |
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314 | % \setupquant{optionstring} Style of \equant, \uquant |
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315 | % \setupset{optionstring} Style of \set, \union, \intersection |
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316 | % \seqname{name} Render sequence/set/tuple name |
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317 | |
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318 | % Because LaTeX has problems with parameters containg \\, the \set command |
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319 | % has code to split the line between the elements and the proposition if |
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320 | % the invocation is \set* and the environment is multline. |
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321 | |
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322 | % Surround individual indices in \intersection, individual variables in |
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323 | % quantifiers, individual propositions in \set and individual indices in |
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324 | % \union with braces, e.g., \equant {{x \in X},{y \in Y}}{P(x,y)}, |
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325 | % \set{x}[{P(x)}, {Q(x)}], \union[{i \in I},{j \in J}]{O(i.j)}. |
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326 | |
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327 | % Setup keywords for \equant, \uquant |
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328 | % subscript =none Default |
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329 | % \exists var1 ... \exists varn prop |
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330 | % stacked \exists_\substack{var1 \\ ... varn} prop |
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331 | % multiple \exists_{var1,...,varn} prop |
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332 | % parentheses=none Default |
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333 | % single (\exists var1 ... \exists varn) prop |
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334 | % multiple (\exists var1) ... (\exists varn) prop |
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335 | % separator= Default {} |
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336 | |
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337 | % Setup keywords for \intersection, \set, \union |
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338 | % subscript =none \bigcap ix1, ..., ixn set |
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339 | % stacked Default |
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340 | % \bigcap__\substack{ix1 \\ ... ixn} set |
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341 | % multiple \bigcap_{ix1,...,ixn} set |
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342 | % separator= Default {\mid} |
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343 | % {elements separator prop1 ^ ... propn} |
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344 | |
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345 | \ExplSyntaxOn |
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346 | |
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347 | \int_gzero_new:N \g_style_quant_parens_int |
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348 | \int_gzero_new:N \g_style_quant_subscr_int |
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349 | \int_gzero_new:N \g_style_set_subscr_int |
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350 | |
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351 | \NewDocumentCommand{\equant}{mm} |
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352 | { |
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353 | \quant:nnn {\exists} {#1} {#2} |
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354 | } |
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355 | |
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356 | \NewDocumentCommand{\uquant}{mm} |
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357 | { |
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358 | \quant:nnn {\forall} {#1} {#2} |
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359 | } |
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360 | |
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361 | \NewDocumentCommand \setupquant {m} |
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362 | { |
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363 | \keys_set:nn {shmuel / quant} {#1} |
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364 | } |
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365 | |
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366 | \keys_define:nn {shmuel / quant} |
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367 | { |
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368 | subscript .choices:nn = |
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369 | { |
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370 | { |
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371 | none, |
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372 | stacked, |
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373 | multiple |
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374 | } |
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375 | { |
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376 | \int_gset:Nn \g_style_quant_subscr_int {\l_keys_choice_int-1} |
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377 | } |
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378 | }, |
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379 | subscript .default:n = multiple, |
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380 | subscript .initial:n = none, |
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381 | parentheses .choices:nn = |
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382 | { |
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383 | { |
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384 | none, |
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385 | single, |
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386 | multiple |
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387 | } |
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388 | { |
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389 | \int_gset:Nn \g_style_quant_parens_int {\l_keys_choice_int - 1} |
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390 | } |
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391 | }, |
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392 | parentheses .default:n = multiple, |
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393 | parentheses .initial:n = none, |
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394 | separater .tl_set:N = \g_style_quant_sep_tl, |
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395 | separater .default:n = {.}, |
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396 | separater .initial:n = {} |
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397 | } |
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398 | |
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399 | \cs_new:Npn \quant:nnn #1 #2 #3 |
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400 | { |
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401 | %\int_show:N \g_style_quant_parens_int |
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402 | %\int_show:N \g_style_quant_subscr_int |
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403 | % g_style_quant_parens_int \ \int_use:N \g_style_quant_parens_int \ |
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404 | % g_style_quant_subscr_int \ \int_use:N \g_style_quant_subscr_int \ |
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405 | \clist_set:Nn \l_tmpa_clist {#2} |
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406 | \int_case:nn |
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407 | {\g_style_quant_subscr_int} |
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408 | { |
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409 | {0} |
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410 | { |
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411 | % No subscript |
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412 | % Set separater to ) ( quantifier or just quantifier |
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413 | \int_compare:nTF {\g_style_quant_parens_int = 2} |
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414 | {\tl_set:Nn \l_tmpa_tl {\right ) \left ( #1}} |
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415 | {\tl_set:Nn \l_tmpa_tl {#1}} |
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416 | \int_compare:nT {\g_style_quant_parens_int > 0} {\left (} |
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417 | #1 |
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418 | \clist_use:Nn \l_tmpa_clist {\l_tmpa_tl} |
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419 | \int_compare:nT {\g_style_quant_parens_int > 0} {\right )} |
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420 | \g_style_quant_sep_tl #3 |
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421 | } |
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422 | {1} |
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423 | { |
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424 | % Stacked subscript on single quantifier |
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425 | \fp_set:Nn |
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426 | \l_tmpa_fp |
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427 | {ceil{\clist_count:N{\l_tmpa_clist} - 1} * .1 + 1} |
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428 | \int_compare:nT {\g_style_quant_parens_int > 0} {\left (} |
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429 | \scaleobj{\fp_to_decimal:N \l_tmpa_fp}{#1} \sb |
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430 | { \substack { \clist_use:Nn \l_tmpa_clist { \\ } } } |
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431 | \int_compare:nT {\g_style_quant_parens_int > 0} {\right )} |
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432 | #3 |
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433 | } |
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434 | {2} |
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435 | { |
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436 | % Subscripts on separate quantifiers |
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437 | \clist_map_inline:Nn |
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438 | \l_tmpa_clist |
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439 | { |
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440 | % (quantifier \sb predicate) or quantifier \sb predicate |
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441 | { |
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442 | \int_compare:nT |
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443 | {\g_style_quant_parens_int > 0} |
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444 | {\left (} |
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445 | \scaleobj{1.2}{#1} \sb |
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446 | {##1} |
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447 | \int_compare:nT |
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448 | {\g_style_quant_parens_int > 0} |
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449 | {\right )} |
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450 | } |
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451 | } |
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452 | #3 |
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453 | } |
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454 | } |
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455 | } |
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456 | |
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457 | \NewDocumentCommand{\set}{mos} |
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458 | { |
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459 | \IfBooleanTF {#3} |
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460 | { |
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461 | % \msg_term:n {set with star} |
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462 | % \msg_term:n{{P1 #1}} |
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463 | % \msg_term:n{{P2 #2}} |
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464 | \bool_set_true:N \l_tmpa_bool |
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465 | % \bool_show:N \l_tmpa_bool |
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466 | } |
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467 | { |
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468 | \bool_set_false:N \l_tmpa_bool |
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469 | } |
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470 | \set_of:nnn {#1} {#2} {\l_tmpa_bool} |
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471 | } |
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472 | |
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473 | %\tl_new:N \g_style_set_sep_tl |
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474 | %\tl_gset:Nn \g_style_set_sep_tl {\mid} |
---|
475 | |
---|
476 | \NewDocumentCommand \setupset {m} |
---|
477 | { |
---|
478 | \keys_set:nn {shmuel / set} {#1} |
---|
479 | } |
---|
480 | |
---|
481 | \keys_define:nn {shmuel / set} |
---|
482 | { |
---|
483 | separater .tl_set:N = \g_style_set_sep_tl, |
---|
484 | subscript .choices:nn = |
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485 | { |
---|
486 | { |
---|
487 | stacked, |
---|
488 | multiple |
---|
489 | } |
---|
490 | { |
---|
491 | \int_gset:Nn \g_style_set_subscr_int {\l_keys_choice_int-1} |
---|
492 | } |
---|
493 | }, |
---|
494 | separater .initial:n = {\mid}, |
---|
495 | subscript .initial:n = stacked |
---|
496 | } |
---|
497 | |
---|
498 | \cs_new:Npn \set_of:nnn #1 #2 #3 |
---|
499 | { |
---|
500 | \IfValueTF {#2} |
---|
501 | { |
---|
502 | % \msg_term:n {set_of:nn \ has \ predicates \ #2} |
---|
503 | } |
---|
504 | { |
---|
505 | % \msg_term:n {set_of:nn \ has \ no \ predicates} |
---|
506 | } |
---|
507 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
508 | % \msg_term:n {l_tmpa_clist \ set} |
---|
509 | \tl_gset:Nn \g_tmpa_tl {\clist_use:Nn \l_tmpa_clist {\land}} |
---|
510 | % \msg_term:n {g_tmpa_tl \ set \ to \ \g_tmpa_tl} |
---|
511 | \IfValueTF {#2} |
---|
512 | { |
---|
513 | \bool_if:nTF {#3} |
---|
514 | { |
---|
515 | \scalerel*{\{}{#1\g_tmpa_tl} |
---|
516 | #1 |
---|
517 | % \msg_term:n {scalerel returns \scalerel{\g_style_set_sep_tl}{\g_tmpa_tl}} |
---|
518 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
519 | \scalerel*{\mathbin{\g_style_set_sep_tl}}{#1\g_tmpa_tl} |
---|
520 | \\ |
---|
521 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
522 | \g_tmpa_tl |
---|
523 | % \tl_show:N \g_tmpa_tl |
---|
524 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
525 | \scalerel*{\}}{#1\g_tmpa_tl} |
---|
526 | } |
---|
527 | { |
---|
528 | \left \{ |
---|
529 | #1 |
---|
530 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
531 | \scalerel*{\mathbin{\g_style_set_sep_tl}}{#1\g_tmpa_tl} |
---|
532 | \g_tmpa_tl |
---|
533 | \right \} |
---|
534 | } |
---|
535 | } |
---|
536 | { |
---|
537 | \left \{ #1 \right \} |
---|
538 | } |
---|
539 | } |
---|
540 | |
---|
541 | \NewDocumentCommand{\funcname}{m} |
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542 | { |
---|
543 | \funcname:n {#1} |
---|
544 | } |
---|
545 | |
---|
546 | \cs_new:Npn \funcname:n #1 |
---|
547 | { |
---|
548 | %code here \tl_count:n |
---|
549 | \int_compare:nTF {\tl_count:n{#1} > 1} |
---|
550 | { |
---|
551 | {\mathrm{#1}} |
---|
552 | } |
---|
553 | { |
---|
554 | {\mathit{#1}} |
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555 | } |
---|
556 | } |
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557 | |
---|
558 | \NewDocumentCommand{\funcseqname}{m} |
---|
559 | { |
---|
560 | \funcseqname:n {#1} |
---|
561 | } |
---|
562 | |
---|
563 | \cs_new:Npn \funcseqname:n #1 |
---|
564 | { |
---|
565 | %code here \tl_count:n |
---|
566 | \int_compare:nTF {\tl_count:n{#1} > 1} |
---|
567 | { |
---|
568 | {\bm{\mathrm{#1}}} |
---|
569 | } |
---|
570 | { |
---|
571 | {\bm {\mathit{#1}}} |
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572 | } |
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573 | } |
---|
574 | |
---|
575 | \NewDocumentCommand{\seqname}{m} |
---|
576 | { |
---|
577 | \seqname:n {#1} |
---|
578 | } |
---|
579 | |
---|
580 | \cs_new:Npn \seqname:n #1 |
---|
581 | { |
---|
582 | %code here \tl_count:n |
---|
583 | \int_compare:nTF {\tl_count:n{#1} > 1} |
---|
584 | { |
---|
585 | {\mathbf{#1}} |
---|
586 | } |
---|
587 | { |
---|
588 | {\mathbfit{#1}} |
---|
589 | } |
---|
590 | } |
---|
591 | |
---|
592 | \NewDocumentCommand{\intersection}{om} |
---|
593 | { |
---|
594 | \unint_of:nnn \bigcap {#1} {#2} |
---|
595 | } |
---|
596 | |
---|
597 | \NewDocumentCommand{\union}{om} |
---|
598 | { |
---|
599 | \unint_of:nnn \bigcup {#1} {#2} |
---|
600 | } |
---|
601 | |
---|
602 | \cs_new:Npn \unint_of:nnn #1 #2 #3 |
---|
603 | { |
---|
604 | \IfValueTF {#2} |
---|
605 | { |
---|
606 | % \int_show:N \g_style_set_subscr_int |
---|
607 | \clist_set:Nn \l_tmpa_clist {#2} |
---|
608 | \int_case:nn |
---|
609 | {\g_style_set_subscr_int} |
---|
610 | { |
---|
611 | {0} |
---|
612 | { |
---|
613 | % Stacked subscript |
---|
614 | % \msg_term:n {\clist_count:N{\l_tmpa_clist} \ tokens \ stacked} |
---|
615 | % \clist_show:N \l_tmpa_clist |
---|
616 | \fp_set:Nn |
---|
617 | \l_tmpa_fp |
---|
618 | {ceil{\clist_count:N{\l_tmpa_clist} - 1} * .1 + 1} |
---|
619 | \scaleobj{\fp_to_decimal:N \l_tmpa_fp}{#1} \sb |
---|
620 | {\substack { \clist_use:Nn \l_tmpa_clist { \\ } }} |
---|
621 | #3 |
---|
622 | } |
---|
623 | {1} |
---|
624 | { |
---|
625 | % Subscripts comma separated |
---|
626 | % \msg_term:n {\clist_count:N{\l_tmpa_clist} \ tokens \ comma \ separated} |
---|
627 | % \clist_show:N \l_tmpa_clist |
---|
628 | #1 \sb |
---|
629 | {\clist_use:Nn \l_tmpa_clist {,}} |
---|
630 | #3 |
---|
631 | } |
---|
632 | } |
---|
633 | } |
---|
634 | { |
---|
635 | % No subscript |
---|
636 | % \msg_term:n {No~subscript} |
---|
637 | % \clist_show:N \l_tmpa_clist |
---|
638 | #1 #3 |
---|
639 | } |
---|
640 | } |
---|
641 | |
---|
642 | \ExplSyntaxOff |
---|
643 | |
---|
644 | % | | |
---|
645 | % | From opening to here --- belongs in package | |
---|
646 | % -------------------------------------------------------------------- |
---|
647 | |
---|
648 | |
---|
649 | \newtheorem{theorem}{Theorem}[section] |
---|
650 | \newtheorem{lemma}[theorem]{Lemma} |
---|
651 | \def\lemmaautorefname{Lemma} % Needed for \autoref |
---|
652 | \newtheorem{corollary}[theorem]{Corollary} |
---|
653 | \def\corollaryautorefname{Corollary} % Needed for \autoref |
---|
654 | |
---|
655 | \theoremstyle{definition} |
---|
656 | \newtheorem{definition}[theorem]{Definition} |
---|
657 | \def\definitionutorefname{Definition} % Needed for \autoref |
---|
658 | \newtheorem{example}[theorem]{Example} |
---|
659 | \newtheorem{xca}[theorem]{Exercise} |
---|
660 | |
---|
661 | \theoremstyle{remark} |
---|
662 | \newtheorem{remark}[theorem]{Remark} |
---|
663 | |
---|
664 | \numberwithin{equation}{section} |
---|
665 | |
---|
666 | \newcommand \Alpha A |
---|
667 | \newcommand \Beta B |
---|
668 | \newcommand \Epsilon E |
---|
669 | \newcommand \Zeta Z |
---|
670 | \newcommand \Eta H |
---|
671 | \newcommand \Iota I |
---|
672 | \newcommand \Kappa K |
---|
673 | \newcommand \Mu M |
---|
674 | \newcommand \Nu N |
---|
675 | \newcommand \Omicron O |
---|
676 | \newcommand \Rho P |
---|
677 | \newcommand \Tau T |
---|
678 | \newcommand \Chi S |
---|
679 | |
---|
680 | \usepackage[colorlinks,hidelinks,draft=false]{hyperref} |
---|
681 | |
---|
682 | \usepackage{cleveref} |
---|
683 | \def\corollaryautorefname{Corollary} % Needed for \autoref |
---|
684 | \def\definitionutorefname{Definition} % Needed for \autoref |
---|
685 | \def\lemmaautorefname{Lemma} % Needed for \autoref |
---|
686 | \hypersetup { |
---|
687 | bookmarksnumbered=true, |
---|
688 | colorlinks, |
---|
689 | pdfinfo={ |
---|
690 | Author={Shmuel (Seymour J.) Metz}, |
---|
691 | Keywords={fiber bundles,manifolds}, |
---|
692 | Subject={Topology}, |
---|
693 | Title={Local Coordinate Spaces: a proposed unification of manifolds with fiber bundles, and associated machinery} |
---|
694 | } |
---|
695 | } |
---|
696 | \def\corollaryautorefname{Corollary} % Needed for \autoref |
---|
697 | \def\definitionutorefname{Definition} % Needed for \autoref |
---|
698 | \def\lemmaautorefname{Lemma} % Needed for \autoref |
---|
699 | % \usepackage[final]{showlabels} |
---|
700 | \usepackage[draft]{showlabels} |
---|
701 | \showlabels{cite} |
---|
702 | \showlabels{cref} |
---|
703 | \showlabels{crefrange} |
---|
704 | |
---|
705 | %\DeclareMathAlphabet{\mathbdit}{T1}{\itdefault}{\mddefault}{\sldefault} |
---|
706 | %\SetMathAlphabet{\mathbdit}{bold}{T1}{\itdefault}{\bfdefault}{\sldefault} |
---|
707 | % |
---|
708 | %\DeclareMathAlphabet{\mathsfbd}{T1}{\sfdefault}{\mddefault}{\sldefault} |
---|
709 | %\SetMathAlphabet{\mathsfbd}{bold}{T1}{\sfdefault}{\bfdefault}{\sldefault} |
---|
710 | % |
---|
711 | %\DeclareMathAlphabet{\mathsfit}{T1}{\sfdefault}{}{\sldefault} |
---|
712 | %\SetMathAlphabet{\mathsfit}{normal}{T1}{\sfdefault}{}{\sldefault} |
---|
713 | % |
---|
714 | %\DeclareMathAlphabet{\mathsfbdit}{T1}{\sfdefault}{\mddefault}{\sldefault} |
---|
715 | %\SetMathAlphabet{\mathsfbdit}{bold}{T1}{\sfdefault}{\bfdefault}{\sldefault} |
---|
716 | |
---|
717 | \begin{document} |
---|
718 | \hyphenation{near pres-en-ta-tion} |
---|
719 | |
---|
720 | \title% |
---|
721 | {% |
---|
722 | Local Coordinate Spaces: |
---|
723 | a proposed unification of manifolds and fiber bundles, |
---|
724 | and associated machinery\thanks% |
---|
725 | { |
---|
726 | I wish to gratefully thank Walter Hoffman (z"l), |
---|
727 | Milton Parnes, Dr. Stanley H. Levy, the Mathematics department of |
---|
728 | Wayne State University, the Mathematics department of the State |
---|
729 | University of New York at Buffalo and others who guided my education. |
---|
730 | } |
---|
731 | } |
---|
732 | \author{Shmuel (Seymour J.) Metz |
---|
733 | } |
---|
734 | \maketitle |
---|
735 | |
---|
736 | % \address{4963 Oriskany Drive\\Annandale, VA 22003-5141} |
---|
737 | % \email{smetz3@gmu.edu} |
---|
738 | % \urladdr{http://mason.gmu.edu/~smetz3} |
---|
739 | |
---|
740 | % \subjclass[2010]{Primary 18F15; Secondary 55R65,57N99,58A05} |
---|
741 | % 18F15 Abstract manifolds and fiber bundles |
---|
742 | % 32Qxx Complex manifolds |
---|
743 | % 53- Differential geometry |
---|
744 | % 53Axx Classical differential geometry |
---|
745 | % 53A99 None of the above, but in this section |
---|
746 | % 55Rxx Fiber spaces and bundles |
---|
747 | % 55R10 Fiber bundles |
---|
748 | % 55R65 Generalizations of fiber spaces and bundles |
---|
749 | % 57Nxx Topological manifolds |
---|
750 | % 57N99 None of the above, but in this section |
---|
751 | % 57Rxx Differential topology |
---|
752 | % 58Axx General theory of differentiable manifolds |
---|
753 | % 58A05 Differentiable manifolds, foundations |
---|
754 | % 58Bxx Infinite-dimensional manifolds |
---|
755 | |
---|
756 | % Primary 18F15 Abstract manifolds and fiber bundles |
---|
757 | % Secondary 55R65,57N99,58A05 |
---|
758 | % \keywords{fiber bundles,manifolds} |
---|
759 | |
---|
760 | \begin{abstract} |
---|
761 | This paper presents a unified view of manifolds and fiber bundles, |
---|
762 | which, while superficially different, have strong parallels. It |
---|
763 | introduces the notions of an m-atlas and of a local coordinate space, |
---|
764 | and shows that special cases are equivalent to fiber bundles and |
---|
765 | manifolds. Along the way it defines some convenient notation, defines |
---|
766 | categories of atlases, and constructs potentially useful functors. |
---|
767 | \end{abstract} |
---|
768 | |
---|
769 | \setupquant {parentheses=multiple,subscript=stacked} |
---|
770 | \setupset {} |
---|
771 | |
---|
772 | \tikzset{ |
---|
773 | rot90/.style={anchor=south, rotate=90, inner sep=.2mm} |
---|
774 | } |
---|
775 | |
---|
776 | \part {Introduction} |
---|
777 | \label{part;intro} |
---|
778 | |
---|
779 | Historically, the concept of pseudo-groups allowed unifying manifolds |
---|
780 | and manifolds with boundary. The definitions of fiber bundles and |
---|
781 | manifolds have strong parallels, and can be unified in a similar fashion; |
---|
782 | there are several ways to do so. The central part of this paper, |
---|
783 | \pagecref{part;lcs}\negmedspace, defines an approach using categories and |
---|
784 | commutative diagrams that is designed for easy exposition at the possible |
---|
785 | expense of abstractness and generality. In particular, I have chosen to |
---|
786 | assume the Axiom of Choice (AOC). |
---|
787 | |
---|
788 | This paper treats atlases as objects of interest in their own right, |
---|
789 | although it does not give them primacy. It introduces notions that |
---|
790 | are convenient for use here and others that, while not used here, may be |
---|
791 | useful for future work. It defines the new notions of |
---|
792 | \hyperref[def:model]{model space}\negmedspace |
---|
793 | \footnote{The phrase has been used before, but with a different |
---|
794 | meaning.}, |
---|
795 | \hyperref[def:m-atlas]{m-atlas} |
---|
796 | and of \hyperref[def:M-ATLmorph]{m-atlas morphism}. |
---|
797 | Informally, a model space is a topological space with a category |
---|
798 | specifying a family of open sets and functions satisfying specified |
---|
799 | conditions. |
---|
800 | |
---|
801 | Although this paper incidentally defines partial equivalents to |
---|
802 | manifolds and fiber bundles using model spaces and model atlases, it |
---|
803 | proposes the more general |
---|
804 | \hyperref[def:LCS]{Local Coordinate Space (LCS)} in order to explicitly |
---|
805 | reflect the role of the group in fiber bundles. |
---|
806 | |
---|
807 | A local coordinate space (LCS) is a space (total space) with some |
---|
808 | additional structure, including a coordinate model space and an atlas |
---|
809 | whose transition functions are restricted to morphisms of the coordinate |
---|
810 | model space; one can impose, e.g., differentiability restrictions, by |
---|
811 | appropriate choice of the coordinate category. There is an equivalent |
---|
812 | paradigm that avoids explicit mention of the total space by imposing |
---|
813 | compatibility conditions on the transition functions, but that approach |
---|
814 | is beyond the scope of this paper. |
---|
815 | |
---|
816 | This paper defines functors among categories of atlases, categories of |
---|
817 | model spaces, categories of local coordinate spaces, categories of |
---|
818 | manifolds and categories of fiber bundles; it constructs more machinery |
---|
819 | than is customary in order to facilitate the presentation of those |
---|
820 | categories and functors. |
---|
821 | |
---|
822 | \Crefrange{part;conv}{part;pre} present nomenclature and give basic |
---|
823 | results. |
---|
824 | \Cref{part;m-charts} defines m-atlases, m-atlas morphisms and categories |
---|
825 | of them; \cref{lem:ATLiscat} proves that the defined categories are |
---|
826 | indeed categories. |
---|
827 | \Cref{part;lcs} defines local coordinate spaces and |
---|
828 | categories of them; \pagecref{the:LCSiscat} proves that the defined |
---|
829 | categories are indeed categories. |
---|
830 | |
---|
831 | \Pagecref{Examples} gives some examples of structures that can be |
---|
832 | represented as local coordinate spaces; \pagecref{part;man} and |
---|
833 | \pagecref{part;bun} present two of the examples in detail, showing the |
---|
834 | equivalence of manifolds and fiber bundles with special cases of local |
---|
835 | coordinate spaces by explicitly exhibiting functors to and from local |
---|
836 | coordinate spaces. |
---|
837 | \begin{remark} |
---|
838 | The unconventional definitions of manifold and fiber bundle are intended |
---|
839 | to make their relationship to local coordinate spaces more natural. |
---|
840 | \end{remark} |
---|
841 | |
---|
842 | Most of the lemmata, theorems and corollaries in this paper should be |
---|
843 | substantially identical to results that are familiar to the reader. What |
---|
844 | is novel is the perspective and the material directly related to local |
---|
845 | coordinate spaces. The presentation assumes only a basic knowledge of |
---|
846 | Category Theory, such as may be found in the first chapter of |
---|
847 | \cite{CftWM} or |
---|
848 | { |
---|
849 | \showlabelsinline |
---|
850 | \cite{JoyCat}. |
---|
851 | } |
---|
852 | |
---|
853 | \section{New concepts and notation} |
---|
854 | \label{sec:new} |
---|
855 | This paper introduces a significant number of new concepts and some |
---|
856 | modifications of the definitions for some conventional concepts. It also |
---|
857 | introduces some notation of lesser importance. The following are the |
---|
858 | most important. |
---|
859 | |
---|
860 | \begin{enumerate} |
---|
861 | \item \hyperref[def:NCD]{Nearly commutative diagram (NCD)}, NCD at a point, |
---|
862 | locally NCD and special cases with related nomenclature |
---|
863 | \item \hyperref[def:model]{Model space} and related concepts |
---|
864 | \item \hyperref[def:ModTop]{Model topology} and |
---|
865 | \hyperref[def:M-para]{M-paracompactness} |
---|
866 | \item \hyperref[sec:sig]{Signature}, |
---|
867 | \hyperref[def:Sigmacomm]{$\Sigma$-commutation} and related concepts |
---|
868 | \item |
---|
869 | \hyperref[def:LCS]{Local Coordinate Space (LCS)} and related concepts |
---|
870 | \item |
---|
871 | \hyperref[def:lin]{Linear space and related concepts} |
---|
872 | \item |
---|
873 | \hyperref[def:trivck]{Trivial $\Ck$ linear model space} and related |
---|
874 | concepts |
---|
875 | \item \hyperref[def:BunAtl]{$G$-$\rho$ bundle atlas} |
---|
876 | \footnote{Similar to coordinate bundles} |
---|
877 | and related concepts |
---|
878 | \end{enumerate} |
---|
879 | |
---|
880 | \part {Conventions} |
---|
881 | \label{part;conv} |
---|
882 | An arrow with an Equal-Tilde ($A \toiso_\phi B$) represents an |
---|
883 | isomorphism. One with a hook ($A \underset{i}{\hookrightarrow} B$) |
---|
884 | represents an inclusion map. One with a tail ($A \into_i B$) represents |
---|
885 | a monomorphism. One with a double head ($A \onto_\pi B$) represents a |
---|
886 | surjection. |
---|
887 | |
---|
888 | When a superscript ends in $-1$, e.g., $\funcname{f}^{i-1}$, it is to be |
---|
889 | taken as function inverse rather than subtraction of 1. |
---|
890 | |
---|
891 | All diagrams shown are commutative; none are exact. |
---|
892 | |
---|
893 | A definition of a base term several more restrictive terms may |
---|
894 | specify the modifiers in parenthese in the base definition and then |
---|
895 | give the restrictions for each modifier, e.g., if |
---|
896 | "$\funcseqname{f}$ is a (semi-strict, strict) prestructure morphism |
---|
897 | of $\seqname{P}^1$ to $\seqname{P}^2$ iff" is followed by the |
---|
898 | definition of prestructure morphism, then the restrictions for |
---|
899 | strict and semi-strict prestructure morphisms. |
---|
900 | |
---|
901 | When a definition defines a base propositional function and variant |
---|
902 | propositional functions with a qualifier given as a superscript, |
---|
903 | then the form $\mathrm{base}^{(qualifiers)}$ will refer to |
---|
904 | either $\mathrm{base}^{\mathrm{qualifer}}$ |
---|
905 | or $\mathrm{base}$, e.g., |
---|
906 | $\strict[semi-]{\isAtl[(classic,near)]_\Ar}$ refers to either |
---|
907 | $\strict[semi-]{\isAtl[classic]_\Ar}$, |
---|
908 | $\strict[semi-]{\isAtl[near]_\Ar}$ |
---|
909 | or $\strict[semi-]{\isAtl_\Ar}$. |
---|
910 | |
---|
911 | Alternatively, a definition may specify a numbered list of alternatives, |
---|
912 | and subsequently specify additional numbered lists with items |
---|
913 | corresponding to those in the first list, e.g., |
---|
914 | $\funcseqname{f}$ is also a |
---|
915 | \begin{enumerate} |
---|
916 | \item full |
---|
917 | \item semi-maximal |
---|
918 | \item maximal |
---|
919 | \item full semi-maximal |
---|
920 | \item full maximal |
---|
921 | \end{enumerate} |
---|
922 | $E^1$-$E^2$ $\Ck$ near morphism of |
---|
923 | $\seqname{A}^1$ to $\seqname{A}^2$ in the coordinate spaces $C^1$, |
---|
924 | $C^2$, abbreviated as |
---|
925 | \begin{enumerate} |
---|
926 | \item abbreviation for full |
---|
927 | \item abbreviation for semi-maximal |
---|
928 | \item abbreviation for maximal |
---|
929 | \item abbreviation for full semi-maximal |
---|
930 | \item abbreviation for full maximal |
---|
931 | \end{enumerate} |
---|
932 | iff |
---|
933 | \begin{enumerate} |
---|
934 | \item definition for full |
---|
935 | \item definition for semi-maximal |
---|
936 | \item definition for maximal |
---|
937 | \item definition for full semi-maximal |
---|
938 | \item definition for full maximal |
---|
939 | \end{enumerate} |
---|
940 | |
---|
941 | A Corollary, Lemma or Theorem that applies to related terms defined |
---|
942 | with the above convention will specify the modifiers in parentheses |
---|
943 | to indicate that it applies to the base term and to the more |
---|
944 | restrictive terms, e.g., "a (semi-strict, strict) |
---|
945 | $\seqname{E}^i$-$\seqname{E}^{i+1}$ m-atlas morphism" If it applies |
---|
946 | only to more restrictive terms them it will specify the first |
---|
947 | relevant modifier followed by the remaining relevant modifiers in |
---|
948 | parentheses, e.g., "If $\catname{S}^i \SUBCAT[full-] |
---|
949 | \catname{S}'^i$ and $\funcseqname{f}^i$ is a semi-strict (strict) |
---|
950 | prestructure morphism". |
---|
951 | |
---|
952 | Blackboard bold upper case will denote specific sets, e.g., the |
---|
953 | Naturals. |
---|
954 | |
---|
955 | Bold lower case italic letters will refer to sets, sequences and tuples |
---|
956 | of functions, e.g., |
---|
957 | $\funcseqname{f} \defeq (\funcname{f}_1, \funcname{f}_2)$. |
---|
958 | |
---|
959 | Bold lower case Latin letters will refer to sequence valued functions of |
---|
960 | sequences and tuple valued functions of tuples, e.g., $\Range$ yields |
---|
961 | the sequence of ranges of a sequence of functions. |
---|
962 | |
---|
963 | Bold upper case italic letters will refer to sequences or tuples, e.g., |
---|
964 | $\seqname{A}=(x,y,z)$, to sets of them, to sets of topological spaces or |
---|
965 | to sets of open sets. |
---|
966 | |
---|
967 | Bold upper case script letters will refer to |
---|
968 | sequences of categories, e.g., |
---|
969 | $\catseqname{A} \defeq (\catname{A}_\alpha, \alpha \in \Alpha)$. |
---|
970 | |
---|
971 | Fraktur will refer to topologies and to topology-valued functions, e.g., |
---|
972 | $\Topology$. |
---|
973 | |
---|
974 | Functions have a range, domain and relation, not just a relation. Unless |
---|
975 | otherwise stated, they are assumed to be continuous. |
---|
976 | |
---|
977 | Groups are assumed to be topological groups. The ambiguous notation |
---|
978 | $x^{-1}$ will be used when it is obvious from context what the group |
---|
979 | operation $\star$ and the group identity $\mathbf{1}_G$ are. |
---|
980 | |
---|
981 | Lower case Greek letters other than $\pi$, $\rho$, $\sigma$, $\phi$ and |
---|
982 | $\psi$ will refer to ordinals, possibly transfinite, and to formal |
---|
983 | labels. A letter with a Greek superscript and a letter with a Latin or |
---|
984 | numeric superscript always refer to distinct variables. |
---|
985 | |
---|
986 | Lower case $\pi$ will refer to a projection operator |
---|
987 | |
---|
988 | Lower case $\rho$ will refer to a continuous effective group action, |
---|
989 | i.e., a continuous representation of a group in a homeomorphism group. |
---|
990 | |
---|
991 | Lower case $\sigma$ will refer to a sequence of ordinals, referred to as |
---|
992 | a signature. |
---|
993 | |
---|
994 | Lower case $\phi$ will refer to a coordinate function. |
---|
995 | |
---|
996 | Lower case italic and Latin letters will refer to |
---|
997 | \begin{enumerate} |
---|
998 | \item elements of a set or sequence |
---|
999 | \item functions |
---|
1000 | \item morphisms of a category |
---|
1001 | \item natural numbers |
---|
1002 | \item objects of a category |
---|
1003 | \item ordinal numbers |
---|
1004 | \end{enumerate} |
---|
1005 | |
---|
1006 | Upper case Greek letters other than $\Sigma$ may refer to |
---|
1007 | \begin{enumerate} |
---|
1008 | \item ordinal used as the limit of a sequence of consecutive ordinals, |
---|
1009 | e.g., $x_\alpha, \alpha \preceq \Alpha$ |
---|
1010 | \item ordinal used as the order type of a sequence of consecutive |
---|
1011 | ordinals, e.g., $x_\alpha, \alpha \prec \Alpha$ |
---|
1012 | \end{enumerate} |
---|
1013 | |
---|
1014 | Upper case $\Sigma$ will refer to a sequence of signatures |
---|
1015 | |
---|
1016 | Upper case Latin letters will refer to |
---|
1017 | \begin{enumerate} |
---|
1018 | \item Natural numbers |
---|
1019 | \item Topological spaces |
---|
1020 | \item Open sets |
---|
1021 | \item Elements of a sequence or tuple of functions, e.g., |
---|
1022 | $\funcname{f}_E$ might be $\funcname{f}_0 \maps E_1 \to E_2$. |
---|
1023 | \end{enumerate} |
---|
1024 | |
---|
1025 | Upper case Script Latin letters will refer to categories and functors. |
---|
1026 | |
---|
1027 | Upright Latin letters will be used for long names. |
---|
1028 | |
---|
1029 | The term $\Ck$ includes $\mathrm{C}^\infty$ (smooth) and |
---|
1030 | $\mathrm{C}^\omega$ (analytic). |
---|
1031 | |
---|
1032 | This paper uses the term morphism in preference to arrow, but uses |
---|
1033 | the conventional $\Ar$. |
---|
1034 | |
---|
1035 | The term sequence without an explicit reference to $\mathbb{N}$ |
---|
1036 | will refer to a general ordinal sequence, possibly transfinite. |
---|
1037 | |
---|
1038 | Sequence numbering, unlike tuple numbering, starts at 0 and the |
---|
1039 | exposition assumes a von Neumann definition of ordinals, so that |
---|
1040 | $\alpha \in \beta \equiv \alpha \prec \beta$. |
---|
1041 | |
---|
1042 | Except where explicitly stated otherwise, all categories mentioned are |
---|
1043 | small categories with underlying sets, but the morphisms will often not |
---|
1044 | be set functions between the objects and there will not always be a |
---|
1045 | forgetful function to $\Set$ or $\Cat{Top}$. By abuse of language no |
---|
1046 | distinction will be made between a category $\catname{A}$ of topological |
---|
1047 | spaces and the concrete category $(\catname{A},\catname{U})$ over |
---|
1048 | $\Cat{Top}$. Similarly, no distinction will be made among the object $U |
---|
1049 | \in \Ob(\catname{A})$, the topological space $\catname{U}(U)$ and the |
---|
1050 | underlying set. |
---|
1051 | |
---|
1052 | The notation $G^V$ will refer only to the set of continuous functions |
---|
1053 | from $V$ to $G$, never to the set of all functions from $V$ to $G$. |
---|
1054 | |
---|
1055 | When defining a category, the Ordered pair $(\seqname{O}, \seqname{M})$ |
---|
1056 | refers to the smallest concrete category over $\Set$ or $\Cat{Top}$ |
---|
1057 | whose objects are the elements in $\seqname{O}$, whose morphisms include |
---|
1058 | all of the elements of $\seqname{M}$ and whose morphisms from |
---|
1059 | $o^1 \in \seqname{O}$ to $o^2 \in \seqname{O}$ are functions |
---|
1060 | $\funcname{f} \maps o^1 \to o^2$ and whose composition is function |
---|
1061 | composition. |
---|
1062 | |
---|
1063 | When defining a category, the Ordered triple |
---|
1064 | $(\seqname{O}, \seqname{M}, C)$ refers to the small category whose |
---|
1065 | objects are in $\seqname{O}$, whose morphisms are in $\seqname{M}$, |
---|
1066 | whose $\Hom$ is |
---|
1067 | |
---|
1068 | \begin{equation} |
---|
1069 | \Hom_{(\seqname{O}, \seqname{M}, C)} |
---|
1070 | (o_1 \in \seqname{O}, o_2 \in \seqname{O}) |
---|
1071 | \defeq |
---|
1072 | \set |
---|
1073 | { |
---|
1074 | ( |
---|
1075 | \funcseqname{f}, |
---|
1076 | o_1, |
---|
1077 | o_2 |
---|
1078 | ) |
---|
1079 | \in \seqname{M} |
---|
1080 | } |
---|
1081 | \end{equation} |
---|
1082 | and whose composition is C. |
---|
1083 | |
---|
1084 | By abuse of language I may write |
---|
1085 | ``$\catname{S}$'' for $\Ob(\catname{S})$, |
---|
1086 | ``$A \in \catname{A}$'' for $A \in\ \Ob(\catname{A})$, |
---|
1087 | ``$A \subset \catname{A}$'' for $A \subset \Ob(\catname{A})$, |
---|
1088 | ``$A \in \catname{A} \subset B \in \catname{B}$'' for |
---|
1089 | ``the underlying set of $A$ is contained in the underlying set of $B$ |
---|
1090 | and the inclusion $\funcname{i} \maps x \in A \hookrightarrow x \in B$ |
---|
1091 | is a morphism'' and |
---|
1092 | ``$\funcname{f} \maps A \to B$'' for |
---|
1093 | $\funcname{f} \in \Hom_{\catname{C}}(A,B)$, where $\catname{C}$ is |
---|
1094 | understood by context. |
---|
1095 | |
---|
1096 | By abuse of language I shall use the same nomenclature for sequences and |
---|
1097 | tuples, and will use parentheses around a single expression both for |
---|
1098 | grouping and for a tuple with a single element; the intent should be |
---|
1099 | clear from context. |
---|
1100 | |
---|
1101 | By abuse of language I will omit parenthese around the operands of |
---|
1102 | Functors when they can be assumed by context. |
---|
1103 | |
---|
1104 | By abuse of language I shall use the $\times$ and $\bigtimes$ symbols |
---|
1105 | for both Cartesian products of sets and Cartesian products of |
---|
1106 | functions on those sets. |
---|
1107 | |
---|
1108 | By abuse of language, and assuming AOC, I shall refer to some sets as |
---|
1109 | ordinal sequences, e.g., ``$(C_\alpha, \alpha \in \Alpha)$'' for |
---|
1110 | ``$\{C_\alpha \mid \alpha \in \Alpha\}$'', in cases where the order is |
---|
1111 | irrelevant. |
---|
1112 | |
---|
1113 | By abuse of language, I may omit universal quantifiers in cases where |
---|
1114 | the intent is clear. |
---|
1115 | |
---|
1116 | In some cases I define a notion similar to a conventional notion and |
---|
1117 | also need to refer to the conventional notion. In those cases I prefix |
---|
1118 | a letter or phrase to the term, e.g., m-paracompact versus paracompact. |
---|
1119 | |
---|
1120 | \part {General notions} |
---|
1121 | \label{part;notions} |
---|
1122 | This section describes nomenclature used throughout the paper. In |
---|
1123 | some cases this reflects new nomenclature or notions, in others it |
---|
1124 | simply makes a choice from among the various conventions in the |
---|
1125 | literature. |
---|
1126 | |
---|
1127 | \begin{definition}[Operations on categories] |
---|
1128 | \label{def:catprop} |
---|
1129 | If $\catname{C}$ is a category then $x \objin \catname{C}$ iff $x$ is an |
---|
1130 | object of $\catname{C}$ and $y \arin \catname{C}$ iff $y$ is a morphism |
---|
1131 | of $\catname{C}$. |
---|
1132 | |
---|
1133 | If $\catname{S}$ and $\catname{T}$ are categories then |
---|
1134 | $\catname{S} \subcat \catname{T}$ iff $S$ is a subcategory of |
---|
1135 | $\catname{T}$ and $\catname{S} \subcat[full-] \catname{T}$ iff $S$ is a |
---|
1136 | full subcategory of $\catname{T}$. |
---|
1137 | |
---|
1138 | If $\catname{S}$ and $\catname{T}$ are categories then the category |
---|
1139 | union of $\catname{S}$ and $\catname{T}$, abbreviated |
---|
1140 | $\catname{S} \unioncat \catname{T}$, is the category whose objects are |
---|
1141 | in $\catname{S}$ or in $\catname{T}$ and whose morphisms are in |
---|
1142 | $\catname{S}$ or in $\catname{T}$. |
---|
1143 | \end{definition} |
---|
1144 | |
---|
1145 | \begin{definition}[Identity] |
---|
1146 | $\Id_S$ is the identity function on the space $S$, |
---|
1147 | |
---|
1148 | $\Id_o$ is the |
---|
1149 | identity morphism for the object $o$\footnote{ |
---|
1150 | The object is often expressed as a tuple, e.g., |
---|
1151 | $\Id_{(\seqname{A}, \seqname{B})}$ is the identity morphism for the |
---|
1152 | object $(\seqname{A}, \seqname{B})$ |
---|
1153 | }, |
---|
1154 | |
---|
1155 | $\Id_{U,V}$, for $U \subseteq V$, is the inclusion map\footnote{ |
---|
1156 | $U$ and $V$ need not have the same topology. |
---|
1157 | }. |
---|
1158 | |
---|
1159 | $\Id_\catname{C}$ is the identity functor on the category $\catname{C}$. |
---|
1160 | |
---|
1161 | $\ID_{\seqname{S}^i}$, $i=1,2$, is the sequence of identity functions |
---|
1162 | for the elements of the sequence |
---|
1163 | $\seqname{S}^i \defeq (\seqname{S}^1_\alpha,\ \alpha \prec \Alpha)$. Let |
---|
1164 | $\seqname{S}^1 \SUBSETEQ \seqname{S}^2$. Then |
---|
1165 | $\ID_{\seqname{S}^1,\seqname{S}^2}$ is the sequence of inclusion maps |
---|
1166 | $(\Id_{\seqname{S}^1_\alpha,\seqname{S}^2_\alpha}),\ \alpha \prec \Alpha$ |
---|
1167 | for the elements of the sequences $\seqname{S}^i$. |
---|
1168 | |
---|
1169 | The subscript may be omitted |
---|
1170 | when it is clear from context. |
---|
1171 | \end{definition} |
---|
1172 | |
---|
1173 | \begin{definition}[Images] |
---|
1174 | $\funcname{f} [U] \defeq \set { {\funcname{f}(x)} }[x \in U]$ is the |
---|
1175 | image of $U$ under $\funcname{f}$ and |
---|
1176 | $\funcname{f}^{-1} [V] \defeq \set {x}[\funcname{f}(x) \in V]$ is the |
---|
1177 | inverse image of $V$ under $\funcname{f}$. |
---|
1178 | |
---|
1179 | \begin{remark} |
---|
1180 | This notation, adopted from \cite{GenTop}, avoids the ambiguity in |
---|
1181 | the traditional $\funcname{f}(U)$ and $\funcname{f}^{-1}(V)$. |
---|
1182 | \end{remark} |
---|
1183 | \end{definition} |
---|
1184 | |
---|
1185 | \begin{definition}[Projections] |
---|
1186 | \label{projections} |
---|
1187 | $\pi_\alpha$ is the projection function that maps a sequence into |
---|
1188 | element $\alpha$ of the sequence. $\pi_i$ is also the projection |
---|
1189 | function that maps a tuple into element $i$ of the tuple. |
---|
1190 | \end{definition} |
---|
1191 | |
---|
1192 | \begin{definition}[Topological category] |
---|
1193 | \label{def:topcat} |
---|
1194 | A topological category is a small subcategory of $\Cat{Top}$ or its |
---|
1195 | concrete category over $\Set$. |
---|
1196 | |
---|
1197 | $\catname{T}$ is a full topological category iff it is a topological |
---|
1198 | category and whenever $U^i,V^i \objin \catname{T}$, $i=1,2$, |
---|
1199 | $V^i \subseteq U^i$, |
---|
1200 | $\funcname{f} \maps U^1 \to U^2 \arin \catname{T}$ and |
---|
1201 | $\funcname{f}[V^1] \subseteq V^2$ then \\ |
---|
1202 | $\funcname{f} \maps V^1 \to V^2 \arin \catname{T}$. |
---|
1203 | \end{definition} |
---|
1204 | |
---|
1205 | \begin{lemma}[Inclusions in topological categories are morphisms] |
---|
1206 | \label{lem:topInc} |
---|
1207 | Let $\catname{T}$ be a full topological category, |
---|
1208 | $S^i \objin \catname{T}$, $i=1,2$, and $S^1 \subseteq S^2$. Then |
---|
1209 | $\Id_{S^1,S^2}$ is a morphism of $\catname{T}$ |
---|
1210 | |
---|
1211 | \begin{proof} |
---|
1212 | $\Id_{S^2} \arin \catname{T}$, $S^1 \subseteq S^2$ by hypothesis and |
---|
1213 | $S^1 \subseteq S^2$, so $\Id_{S^1,S^2} \arin \catname{T}$ by |
---|
1214 | \cref{def:topcat}. |
---|
1215 | \end{proof} |
---|
1216 | \end{lemma} |
---|
1217 | |
---|
1218 | \begin{definition}[Local morphisms] |
---|
1219 | \label{def:topLocal} |
---|
1220 | Let $\catname{S}^i$, $i=1,2$, be a full topological category and |
---|
1221 | $S^i \objin \catname{S}^i$, A continuous function |
---|
1222 | $\funcname{f} \maps S^1 \to S^2$ is locally a |
---|
1223 | $\catname{S}^1$-$\catname{S}^2$ morphism of $S^1$ to $S^2$ iff |
---|
1224 | $\catname{S}^1 \subcat[full-] \catname{S}^2$ and for every |
---|
1225 | $u \in S^1$ there is an open neighborhood $U_u$ for $u$ and an open |
---|
1226 | neighborhood $V_u$ for $v \defeq \funcname{f}(u)$ such that |
---|
1227 | $\funcname{f}[U_u] \subseteq V_u$ and $\funcname{f} \maps U_u \to V_u$ |
---|
1228 | is a morphism of $\catname{S}^2$. |
---|
1229 | |
---|
1230 | \begin{remark} |
---|
1231 | The condition $\funcname{f} \maps U_u \to V_u \arin \catname{S}^2$ |
---|
1232 | ensures that $U_u \objin \catname{S}^1$ and $V_u \objin \catname{S}^2$ |
---|
1233 | \end{remark} |
---|
1234 | |
---|
1235 | Let $\catname{T}$ be a full topological category and |
---|
1236 | $S^i \objin \catname{T}$, $i=1,2$. A continuous function |
---|
1237 | $\funcname{f} \maps S^1 \to S^2$ is locally a $\catname{T}$ morphism of |
---|
1238 | $S^1$ to $S^2$ iff it is locally a $\catname{T}$-$\catname{T}$ morphism |
---|
1239 | of $S^1$ to $S^2$. |
---|
1240 | \end{definition} |
---|
1241 | |
---|
1242 | \begin{lemma}[Local morphisms] |
---|
1243 | \label{lem:topLocal} |
---|
1244 | Let $\catname{T}^i$, $i \in [1,3]$, be a full topological category, |
---|
1245 | $S^i \objin \catname{T}^i$ and |
---|
1246 | $\catname{T}^i \subcat[full-] \catname{T}^{i+1}$. |
---|
1247 | |
---|
1248 | If $\funcname{f}^i \maps S^i \to S^{i+1} \arin \catname{T}^{i+1}$ then |
---|
1249 | $\funcname{f}^i$ is locally a $\catname{T}^i$-$\catname{T}^{i+1}$ |
---|
1250 | morphism of $S^i$ to $S^{i+1}$. |
---|
1251 | |
---|
1252 | \begin{proof} |
---|
1253 | Let $u \in S^i$ and $v \defeq \funcname{f}^i(u) \in S^{i+1}$. $S^i$ is |
---|
1254 | an open for $u$, $S^{i+1}$ is an open neighborhood for $v$ and |
---|
1255 | $\funcname{f}^i \maps S^i \to S^{i+1} \arin \catname{T}^{i+1}$ by |
---|
1256 | hypothesis. |
---|
1257 | \end{proof} |
---|
1258 | |
---|
1259 | If each $\funcname{f}^i \maps S^i \to S^{i+1}$, is locally a |
---|
1260 | $\catname{T}^i$-$\catname{T}^{i+1}$ morphism of $S^i$ to |
---|
1261 | $S^{i+1}$ then |
---|
1262 | $\funcname{f}^2 \compose \funcname{f}^1 \maps S^1 \to S^3$ is locally a |
---|
1263 | $\catname{T}^1$-$\catname{T}^3$ morphism of $S^1$ to $S^3$. |
---|
1264 | |
---|
1265 | \begin{proof} |
---|
1266 | Since $\catname{T}^1 \subcat[full-] \catname{T}^2$ and |
---|
1267 | $\catname{T}^2 \subcat[full-] \catname{T}^3$, |
---|
1268 | $\catname{T}^1 \subcat[full-] \catname{T}^3$. |
---|
1269 | Let $u \in S^1$, $v \defeq \funcname{f}^1(u)$ and |
---|
1270 | $w \defeq \funcname{f}^2(v)$. There exist an open neighborhood $U_u$ |
---|
1271 | for $u$, open neighborhoods $V_u$, $V'_v$ for $v$ and an open |
---|
1272 | neighborhood $W_v$ of $w$ such that |
---|
1273 | $\funcname{f}^1[U_u] \subseteq V_u$, |
---|
1274 | $\funcname{f}^1 \maps U_u \to V_u$ is a morphism of $\catname{T}^2$, |
---|
1275 | $\funcname{f}^2[V'_v] \subseteq W_v$ and |
---|
1276 | $\funcname{f}^2 \maps V'_v \to W_v$ is a morphism of $\catname{T}^3$. |
---|
1277 | Then $\hat{V_u} \defeq V_u \cap V'_v \neq \emptyset$, $\hat{V_u}$ is an |
---|
1278 | open neighborhood of $v$ and |
---|
1279 | $\hat{U_u} \defeq \funcname{f}^{i-1}_1[\hat{V_u}]$ is an open |
---|
1280 | neighborhood for $u$. $\funcname{f}^1 \maps \hat{U_u} \to \hat{V_u}$ |
---|
1281 | and $\funcname{f}^2 \maps \hat{V_u} \to W_v$ are morphisms of |
---|
1282 | $\catname{T}^3$ by \pagecref{def:topcat} and thus |
---|
1283 | $\funcname{f}^2 \compose \funcname{f}^1 \maps \hat{U_u} \to W_v$ is a |
---|
1284 | morphism of $\catname{T}^3$. |
---|
1285 | \end{proof} |
---|
1286 | \end{lemma} |
---|
1287 | |
---|
1288 | \begin{corollary}[Local morphisms] |
---|
1289 | \label{cor:topLocal} |
---|
1290 | Let $\catname{T}^i$, $i=1,2$, be a full topological category, |
---|
1291 | $\catname{T}^i \subcat[full-] \catname{T}^{i+1}$, |
---|
1292 | $S^i \objin \catname{T}^i$ and $S^1 \subseteq S^2$. Then $\Id_{S^1,S^2}$ |
---|
1293 | is locally a $\catname{T}^1$-$\catname{T}^2$ morphism of $S^1$ to $S^2$ |
---|
1294 | and $\Id_{S^i}$ is locally a $\catname{T}$ morphism |
---|
1295 | of $S^i$ to $S^i$. |
---|
1296 | |
---|
1297 | \begin{proof} |
---|
1298 | $S^1 \objin \catname{T}^2$ because $S^1 \objin \catname{T}^1$ and |
---|
1299 | $\catname{T}^1 \subcat \catname{T}^2$, |
---|
1300 | $S^2 \objin \catname{T}^2$ by hypothesis and |
---|
1301 | $S^1 \subseteq S^2$ by hypothesis, so |
---|
1302 | $\Id_{S^1,S^2} \arin \catname{T}^2$ by \cref{lem:topInc}. |
---|
1303 | |
---|
1304 | $\Id_{S^i} \defeq \Id_{S^i,S^i}$. |
---|
1305 | \end{proof} |
---|
1306 | \end{corollary} |
---|
1307 | |
---|
1308 | \begin{definition}[Sequence functions] |
---|
1309 | Let $\seqname{S} \defeq (s_\alpha, \alpha \prec \Alpha)$ be a sequence |
---|
1310 | of functions. Then |
---|
1311 | \begin{equation} |
---|
1312 | \Domain(\seqname{S}) \defeq \bigl ( \domain(s_\alpha), \alpha \prec \Alpha \bigr ) |
---|
1313 | \end{equation} |
---|
1314 | \begin{equation} |
---|
1315 | \Range(\seqname{S}) \defeq \bigl ( \range(s_\alpha), \alpha \prec \Alpha \bigr ) |
---|
1316 | \end{equation} |
---|
1317 | |
---|
1318 | Let $\seqname{T} \defeq (t_\alpha, \alpha \prec \Alpha)$ be a sequence |
---|
1319 | of functions with $\Range(\seqname{S}) = \Domain(\seqname{T})$. Then |
---|
1320 | their composition is the sequence |
---|
1321 | $ |
---|
1322 | \seqname{T} \compose[()] \seqname{S} \defeq |
---|
1323 | (t_\alpha \compose s_\alpha, \alpha \prec \Alpha) |
---|
1324 | $, |
---|
1325 | |
---|
1326 | Let $\seqname{S} \defeq (s_\gamma, \gamma \preceq \Gamma)$, then these |
---|
1327 | functions extract information about the sequence: |
---|
1328 | \begin{equation} |
---|
1329 | \head(\seqname{S},\Omega) \defeq (s_\gamma, \gamma \prec \Omega) |
---|
1330 | \end{equation} |
---|
1331 | \begin{equation} |
---|
1332 | \head(\seqname{S}) \defeq \head(\seqname{S},\Gamma) |
---|
1333 | \end{equation} |
---|
1334 | \begin{equation} |
---|
1335 | \lengtho(\seqname{S}) \defeq \Gamma |
---|
1336 | \end{equation} |
---|
1337 | \begin{equation} |
---|
1338 | \tail(\seqname{S}) \defeq S_\Gamma |
---|
1339 | \end{equation} |
---|
1340 | |
---|
1341 | Let $\seqname{S} \defeq (s_\gamma, \gamma \prec \Gamma)$, then |
---|
1342 | \begin{equation} |
---|
1343 | \length(\seqname{S}) \defeq \Gamma |
---|
1344 | \end{equation} |
---|
1345 | |
---|
1346 | \begin{remark} |
---|
1347 | If $\lengtho(\seqname{S})$ is defined then |
---|
1348 | $\length(\seqname{S}) = \lengtho(\seqname{S}) + 1$. |
---|
1349 | $\lengtho(\seqname{S})$ is the ordinal type of |
---|
1350 | $\head(\seqname{S})$, not the ordinal type of $\seqname{S}$. |
---|
1351 | \end{remark} |
---|
1352 | |
---|
1353 | Let $\catseqname{S} \defeq (\catname{S}_\alpha, \alpha \prec \Alpha)$ |
---|
1354 | and $\catseqname{T} \defeq (\catname{T}_\alpha, \alpha \prec \Alpha)$ be |
---|
1355 | sequences of categories. Then $\catseqname{S}$ is a subcategory |
---|
1356 | sequence of $\catseqname{T}$, abbreviated |
---|
1357 | $\catseqname{S} \SUBCAT \catseqname{T}$, iff every category in |
---|
1358 | $\catseqname{S}$ is a subcategory of the corresponding category in |
---|
1359 | $\catseqname{T}$, i.e., |
---|
1360 | $ |
---|
1361 | \uquant% |
---|
1362 | {\alpha \prec \Alpha} |
---|
1363 | {\catname{S}_\alpha \subcat \catname{T}_\alpha} |
---|
1364 | $, |
---|
1365 | and $\catseqname{S}$ is a full subcategory sequence of $\catseqname{T}$, |
---|
1366 | abbreviated $\catseqname{S} \SUBCAT[full-] \catseqname{T}$, iff every |
---|
1367 | category in $\catseqname{S}$ is a full subcategory of the corresponding |
---|
1368 | category in $\catseqname{T}$, i.e., |
---|
1369 | $ |
---|
1370 | \uquant% |
---|
1371 | {\alpha \prec \Alpha} |
---|
1372 | {\catname{S}_\alpha \subcat[full-] \catname{T}_\alpha} |
---|
1373 | $. |
---|
1374 | |
---|
1375 | The category sequence union of $\catseqname{S}$ and $\catseqname{T}$, |
---|
1376 | abbreviated $\catseqname{S} \UNIONCAT \catseqname{T}$, is the sequence |
---|
1377 | of category unions of corresponding categories in $\catseqname{S}$ and |
---|
1378 | $\catseqname{T}$, i.e., |
---|
1379 | $(\catseqname{S}_\alpha \unioncat \catseqname{T}_\alpha)$. |
---|
1380 | |
---|
1381 | \end{definition} |
---|
1382 | |
---|
1383 | \begin{lemma}[Sequence functions] |
---|
1384 | \label{lem:seqfunc} |
---|
1385 | Let $\funcseqname{f}^i \defeq (\funcname{f}^i_\alpha, \alpha \prec \Alpha)$, |
---|
1386 | $i \in [1,3]$, be sequences of functions with |
---|
1387 | $\Domain(\funcseqname{f}^2) = \Range(\funcseqname{f}^1)$ and |
---|
1388 | $\Domain(\funcseqname{f}^3) = \Range(\funcseqname{f}^2)$. Then |
---|
1389 | $ |
---|
1390 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 = |
---|
1391 | \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1) |
---|
1392 | $. |
---|
1393 | \begin{proof} |
---|
1394 | \begin{equation*} |
---|
1395 | \begin{split} |
---|
1396 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 |
---|
1397 | & = |
---|
1398 | \bigl ( (\funcname{f}^3_\alpha \compose \funcname{f}^2_\alpha) \compose \funcname{f}^1_\alpha, \alpha \prec A \bigr ) |
---|
1399 | \\* |
---|
1400 | & = |
---|
1401 | \bigl ( \funcname{f}^3_\alpha \compose (\funcname{f}^2_\alpha \compose \funcname{f}^1_\alpha), \alpha \prec A \bigr ) |
---|
1402 | \\* |
---|
1403 | & = |
---|
1404 | \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1) |
---|
1405 | \end{split} |
---|
1406 | \end{equation*} |
---|
1407 | \end{proof} |
---|
1408 | |
---|
1409 | Let $\funcseqname{f} \defeq (\funcname{f}_\alpha, \alpha \prec \Alpha)$ |
---|
1410 | be a sequence of functions, $\seqname{D} = \Domain(\funcseqname{f})$ and |
---|
1411 | $\seqname{R} = \Range(\funcseqname{f})$. Then $ID_\seqname{R}$ is a left |
---|
1412 | $\compose[()]$ identity for $\funcseqname{f}$ and $ID_\seqname{D}$ is a |
---|
1413 | right $\compose[()]$ identity for $\funcseqname{f}$. |
---|
1414 | |
---|
1415 | \begin{proof} |
---|
1416 | \begin{equation*} |
---|
1417 | \begin{split} |
---|
1418 | \ID_\seqname{R} \compose[()] \funcseqname{f} |
---|
1419 | & = |
---|
1420 | (\Id_{\range(\funcname{f}_\alpha)} \compose \funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1421 | \\* |
---|
1422 | & = |
---|
1423 | (\funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1424 | \\* |
---|
1425 | & = |
---|
1426 | \funcseqname{f} |
---|
1427 | \end{split} |
---|
1428 | \end{equation*} |
---|
1429 | \begin{equation*} |
---|
1430 | \begin{split} |
---|
1431 | \funcseqname{f} \compose[()] \ID_\seqname{D} |
---|
1432 | & = |
---|
1433 | (\funcname{f}_\alpha \compose \Id_{\domain(\funcname{f}_\alpha)}, \alpha \prec \Alpha) |
---|
1434 | \\* |
---|
1435 | & = |
---|
1436 | (\funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1437 | \\* |
---|
1438 | & = |
---|
1439 | \funcseqname{f} |
---|
1440 | \end{split} |
---|
1441 | \end{equation*} |
---|
1442 | \end{proof} |
---|
1443 | \end{lemma} |
---|
1444 | |
---|
1445 | \begin{definition}[Tuple functions] |
---|
1446 | Let $\seqname{S} \defeq (s_n, n \in [1,N])$ be a tuple of functions. Then |
---|
1447 | \begin{equation} |
---|
1448 | \Domain(\seqname{S}) \defeq \bigl ( \domain(s_n), n \in [1,N] \bigr ) |
---|
1449 | \end{equation} |
---|
1450 | \begin{equation} |
---|
1451 | \Range(\seqname{S}) \defeq \bigl ( \range(s_n), n \in [1,N] \bigr ) |
---|
1452 | \end{equation} |
---|
1453 | |
---|
1454 | Let $\seqname{T} \defeq (t_n, n \in [1,N])$ be a tuple of functions with |
---|
1455 | $\Range(\seqname{S})=\Domain(\seqname{T})$, |
---|
1456 | Then their composition is the tuple |
---|
1457 | $ |
---|
1458 | \seqname{T} \compose[()] \seqname{S} \defeq |
---|
1459 | (t_n \compose s_n, n \in [1,N]) |
---|
1460 | $ |
---|
1461 | |
---|
1462 | Let $\seqname{S} \defeq (s_m, m \in [1,M])$ and |
---|
1463 | $\seqname{T} \defeq (t_n, n \in [1,N])$ be tuples. |
---|
1464 | Then the following are tuple functioms |
---|
1465 | \begin{equation} |
---|
1466 | \head(S,I) \defeq (s_m, m \in [1,I]) |
---|
1467 | \end{equation} |
---|
1468 | \begin{equation} |
---|
1469 | \head(S) \defeq \head(S,M-1) |
---|
1470 | \end{equation} |
---|
1471 | \begin{equation} |
---|
1472 | \tail(T,I) \defeq (t_n, n \in [I,N]) |
---|
1473 | \end{equation} |
---|
1474 | \begin{equation} |
---|
1475 | \tail(T) \defeq t_N |
---|
1476 | \end{equation} |
---|
1477 | \begin{equation} |
---|
1478 | \join(S,T) \defeq (s_1, \dots, s_M, t_1, \dots, t_N) |
---|
1479 | \end{equation} |
---|
1480 | \end{definition} |
---|
1481 | |
---|
1482 | \begin{lemma}[Tuple functions] |
---|
1483 | \label{lem:tupfunc} |
---|
1484 | Let $\funcseqname{f}^i \defeq (\funcname{f}^i_n, n \in [1,N])$, |
---|
1485 | $i \in [1,3]$, be tuples of functions with |
---|
1486 | $\Domain(\funcseqname{f}^2) = \Range(\funcseqname{f}^1)$ and |
---|
1487 | $\Domain(\funcseqname{f}^3) = \Range(\funcseqname{f}^2)$. Then |
---|
1488 | $ |
---|
1489 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 = |
---|
1490 | \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1) |
---|
1491 | $. |
---|
1492 | |
---|
1493 | \begin{proof} |
---|
1494 | \begin{equation*} |
---|
1495 | \begin{split} |
---|
1496 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 |
---|
1497 | & = |
---|
1498 | \bigl ( (\funcname{f}^3_n \compose \funcname{f}^2_n) \compose \funcname{f}^1_n, n \in [1,N] \bigr ) |
---|
1499 | \\* |
---|
1500 | & = |
---|
1501 | \bigl ( \funcname{f}^3_n \compose (\funcname{f}^2_n \compose \funcname{f}^1_n), n \in [1,N] \bigr ) |
---|
1502 | \\* |
---|
1503 | & = |
---|
1504 | \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1) |
---|
1505 | \end{split} |
---|
1506 | \end{equation*} |
---|
1507 | \end{proof} |
---|
1508 | |
---|
1509 | Let $\funcseqname{f} \defeq (\funcname{f}_\alpha, \alpha \prec \Alpha)$ |
---|
1510 | be a sequence of functions, |
---|
1511 | $\seqname{D} \defeq \Domain(\funcseqname{f})$ and \\* |
---|
1512 | $\seqname{R} \defeq \Range(\funcseqname{f})$. Then $\ID_\seqname{R}$ is a |
---|
1513 | left $\compose[()]$ identity for $\funcseqname{f}$ and $\ID_\seqname{D}$ |
---|
1514 | is a right $\compose[()]$ identity for $\funcseqname{f}$. |
---|
1515 | |
---|
1516 | \begin{proof} |
---|
1517 | \begin{equation*} |
---|
1518 | \begin{split} |
---|
1519 | \ID_\seqname{R} \compose[()] \funcseqname{f} |
---|
1520 | & = |
---|
1521 | (\Id_{\range(\funcname{f}_\alpha)} \compose \funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1522 | \\* |
---|
1523 | & = |
---|
1524 | (\funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1525 | \\* |
---|
1526 | & = |
---|
1527 | \funcseqname{f} |
---|
1528 | \end{split} |
---|
1529 | \end{equation*} |
---|
1530 | \begin{equation*} |
---|
1531 | \begin{split} |
---|
1532 | \funcseqname{f} \compose[()] \ID_\seqname{D} |
---|
1533 | & = |
---|
1534 | (\funcname{f}_\alpha \compose \Id_{\domain(\funcname{f}_\alpha)}, \alpha \prec \Alpha) |
---|
1535 | \\* |
---|
1536 | & = |
---|
1537 | (\funcname{f}_\alpha, \alpha \prec \Alpha) |
---|
1538 | \\* |
---|
1539 | & = |
---|
1540 | \funcseqname{f} |
---|
1541 | \end{split} |
---|
1542 | \end{equation*} |
---|
1543 | \end{proof} |
---|
1544 | \end{lemma} |
---|
1545 | |
---|
1546 | \begin{definition}[Tuple composition for labeled morphisms] |
---|
1547 | \label{def:labcomp} |
---|
1548 | Let $\seqname{M}^i \defeq (\funcseqname{f}^i, o^i_1, o^i_2)$, $i=1,2$, |
---|
1549 | be tuples such that each $\funcseqname{f}^i$ is a sequence of functions |
---|
1550 | or each $\funcseqname{f}^i$ is a tuple of functions, |
---|
1551 | $\Range(\funcseqname{f}^1) = \Domain(\funcseqname{f}^2)$ and |
---|
1552 | $0^1_2=o^2_1$. Then |
---|
1553 | |
---|
1554 | \begin{equation} |
---|
1555 | \seqname{M}^2 \compose[A] \seqname{M}^1 \defeq |
---|
1556 | \bigl ( |
---|
1557 | \funcseqname{f}^2 \compose[()] \funcseqname{f}^1, |
---|
1558 | o^1_1, |
---|
1559 | o^2_2 |
---|
1560 | \bigr ) |
---|
1561 | \end{equation} |
---|
1562 | \end{definition} |
---|
1563 | |
---|
1564 | \begin{lemma}[Tuple composition for labeled morphisms] |
---|
1565 | \label{lem:labcomp} |
---|
1566 | Let $\seqname{M}^i \defeq (\funcseqname{f}^i, o^i_1, o^i_2)$, |
---|
1567 | $i \in [1,3]$, |
---|
1568 | be a tuple such that $\funcseqname{f}^i$ is a sequence or tuple of |
---|
1569 | functions, |
---|
1570 | $\Range(\funcseqname{f}^i) = \Domain(\funcseqname{f}^{i+1})$ |
---|
1571 | and $o^i_2 = o^{i+1}_1$, $i=1,2$. |
---|
1572 | Then |
---|
1573 | \begin{equation} |
---|
1574 | \seqname{M}^3 \compose[A] \bigl ( \seqname{M}^2 \compose[A] \seqname{M}^1 \bigr ) |
---|
1575 | = |
---|
1576 | \bigl ( \seqname{M}^3 \compose[A] \seqname{M}^2 \bigr ) \compose[A] \seqname{M}^1 |
---|
1577 | \end{equation} |
---|
1578 | |
---|
1579 | \begin{proof} |
---|
1580 | From \fullcref{def:labcomp}\negmedspace, \\ |
---|
1581 | { |
---|
1582 | \showlabelsinline |
---|
1583 | \pagecref{lem:seqfunc} |
---|
1584 | } |
---|
1585 | and \fullcref{lem:tupfunc}\negmedspace, we have |
---|
1586 | \begin{equation*} |
---|
1587 | \begin{split} |
---|
1588 | \seqname{M}^3 \compose[A] (\seqname{M}^2 \compose[A] \seqname{M}^1) |
---|
1589 | & = |
---|
1590 | \seqname{M}^3 \compose[A] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1, o^1_1, o^2_2) |
---|
1591 | \\* |
---|
1592 | & = |
---|
1593 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2 \compose[()] \funcseqname{f}^1, o^1_1, o^3_2) |
---|
1594 | \\* |
---|
1595 | & = |
---|
1596 | (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2, o^2_1, o^3_2) \compose[A] \seqname{M}^1 |
---|
1597 | \\* |
---|
1598 | & = |
---|
1599 | (\seqname{M}^3 \compose[A] \seqname{M}^2) \compose[A] \seqname{M}^1 |
---|
1600 | \end{split} |
---|
1601 | \end{equation*} |
---|
1602 | \end{proof} |
---|
1603 | |
---|
1604 | Let $\seqname{D}^i \defeq \Domain(\funcseqname{f}^i)$ and |
---|
1605 | $\seqname{R}^i \defeq \Range(\funcseqname{f}^i)$. Then |
---|
1606 | \begin{equation} |
---|
1607 | (\ID_{\seqname{R}^i}, o^i_2, o^i_2) \compose[A] \seqname{M}^i = |
---|
1608 | \seqname{M}^i |
---|
1609 | \end{equation} |
---|
1610 | \begin{equation} |
---|
1611 | \seqname{M}^i \compose[A] (\ID_{\seqname{D}^i}, o^i_1, o^i_1) = |
---|
1612 | \seqname{M}^i |
---|
1613 | \end{equation} |
---|
1614 | |
---|
1615 | \begin{proof} |
---|
1616 | \begin{equation*} |
---|
1617 | \begin{split} |
---|
1618 | (\ID_{\seqname{R}^i}, o^i_2, o^i_2) \compose[A] \seqname{M}^i |
---|
1619 | & = |
---|
1620 | (\ID_{\seqname{R}^i} \compose[()] \funcseqname{f}^i, o^i_1, o^i_2) |
---|
1621 | \\* |
---|
1622 | & = |
---|
1623 | (\funcseqname{f}^i, o^i_1, o^i_2) |
---|
1624 | \\* |
---|
1625 | & = |
---|
1626 | \seqname{M}^i |
---|
1627 | \end{split} |
---|
1628 | \end{equation*} |
---|
1629 | \begin{equation*} |
---|
1630 | \begin{split} |
---|
1631 | \seqname{M}^i \compose[A] (\ID_{\seqname{D}^i}, o^i_1, o^i_1) |
---|
1632 | & = |
---|
1633 | (\funcseqname{f}^i \compose[()] \ID_{\seqname{D}^i}, o^i_1, o^i_2) |
---|
1634 | \\* |
---|
1635 | & = |
---|
1636 | (\funcseqname{f}^i, o^i_1, o^i_2) |
---|
1637 | \\* |
---|
1638 | & = |
---|
1639 | \seqname{M}^i |
---|
1640 | \end{split} |
---|
1641 | \end{equation*} |
---|
1642 | \end{proof} |
---|
1643 | \end{lemma} |
---|
1644 | |
---|
1645 | \begin{definition}[Cartesian product of sequence] |
---|
1646 | \label{def:Cart} |
---|
1647 | Let $\seqname{S}^i \defeq (S^i_\alpha, \alpha \prec \Alpha)$, $i=1,2$, |
---|
1648 | be a sequence and |
---|
1649 | $\funcseqname{f} \defeq (\funcname{f}_\alpha \maps S^1_\alpha \to |
---|
1650 | S^2_\alpha, \alpha \prec \Alpha)$ |
---|
1651 | be a sequence of functions, then |
---|
1652 | $ |
---|
1653 | \bigtimes \seqname{S}^i \defeq |
---|
1654 | \bigtimes_{\alpha \prec \Alpha} S^i_\alpha |
---|
1655 | $ |
---|
1656 | is the generalized Cartesian product of the sequence $\seqname{S}^i$ and |
---|
1657 | $ |
---|
1658 | \bigtimes \funcseqname{f} \maps \seqname{S}^1 \to \seqname{S}^2 |
---|
1659 | \defeq |
---|
1660 | \bigtimes_{\alpha \prec \Alpha}\funcname{f}_\alpha |
---|
1661 | $ |
---|
1662 | is the generalized Cartesian product of the function sequence |
---|
1663 | $\funcseqname{f}$. |
---|
1664 | \end{definition} |
---|
1665 | |
---|
1666 | \begin{definition}[underline] |
---|
1667 | Let $\seqname{S}^1 \defeq (S^1_\alpha, \alpha \preceq \Alpha)$, |
---|
1668 | $\seqname{S}^2 \defeq (S^2_\alpha, \alpha \preceq \Alpha)$ be |
---|
1669 | sequences and |
---|
1670 | $ |
---|
1671 | \funcseqname{f} \defeq |
---|
1672 | ( |
---|
1673 | \funcname{f}_\alpha \maps S^1_\alpha \to S^2_\alpha, |
---|
1674 | \alpha \preceq \Alpha |
---|
1675 | ) |
---|
1676 | $ |
---|
1677 | be a sequence of functions, then |
---|
1678 | \begin{equation} |
---|
1679 | \underline{\funcname{f}} \maps head(S_1) \to head(S_2) \defeq |
---|
1680 | \bigtimes \head(\funcseqname{f}) = |
---|
1681 | \bigtimes_{\alpha \prec \Alpha} \funcname{f}_\alpha |
---|
1682 | \end{equation} |
---|
1683 | is the function mapping |
---|
1684 | $(s_\alpha \in S^1_\alpha, \alpha \prec \Alpha)$ |
---|
1685 | into $(\funcname{f}_\alpha(s_\alpha), \alpha \prec \Alpha)$. |
---|
1686 | \end{definition} |
---|
1687 | |
---|
1688 | \begin{definition}[Head and tail compositions] |
---|
1689 | Let \\ |
---|
1690 | $\funcname{f}^1 \maps (S^1_\alpha, \alpha \prec \Alpha) \to S^1_\Alpha$, |
---|
1691 | $\funcname{f}^2 \maps (S^2_\alpha, \alpha \prec \Alpha) \to S^2_\Alpha$ |
---|
1692 | and |
---|
1693 | $ |
---|
1694 | \funcseqname{g} \defeq |
---|
1695 | ( |
---|
1696 | \funcname{g}_\alpha \maps S^1_\alpha \to S^2_\alpha, |
---|
1697 | \alpha \preceq \Alpha |
---|
1698 | ) |
---|
1699 | $. |
---|
1700 | Then (see \cref{fig:fg,fig:gf}) |
---|
1701 | |
---|
1702 | \begin{subequations} |
---|
1703 | \begin{equation} |
---|
1704 | \funcname{f}^2 \composeh \funcseqname{g} \defeq |
---|
1705 | \funcname{f}^2 \compose \underline{\funcseqname{g}} |
---|
1706 | \end{equation} |
---|
1707 | \begin{equation} |
---|
1708 | \funcseqname{g} \composet \funcname{f}^1 \defeq |
---|
1709 | \tail(\funcseqname{g}) \compose \funcname{f}^1 |
---|
1710 | \end{equation} |
---|
1711 | \end{subequations} |
---|
1712 | \end{definition} |
---|
1713 | |
---|
1714 | \begin{figure} |
---|
1715 | \[ \bfig |
---|
1716 | \node s1(0,0)[{\head(S^1)}] |
---|
1717 | \node s2(0,-1000)[{\head(S^2)}] |
---|
1718 | \node s2a(1000,-1000)[{S^2_\Alpha}] |
---|
1719 | \arrow |l|[s1`s2;{\underline{\funcseqname{g}}}] |
---|
1720 | \arrow |r|[s1`s2a;{\funcseqname{f}^2 \composeh \funcseqname{g}}] |
---|
1721 | \arrow |b|[s2`s2a;{\funcseqname{f^2}}] |
---|
1722 | \efig \] |
---|
1723 | \caption{$\funcname{f}^2 \composeh \funcseqname{g}$} |
---|
1724 | \label{fig:fg} |
---|
1725 | \end{figure} |
---|
1726 | |
---|
1727 | \begin{figure} |
---|
1728 | \[ \bfig |
---|
1729 | \node s1(0,0)[{\head(S^1)}] |
---|
1730 | \node s1a(0,-1000)[{S^1_\Alpha}] |
---|
1731 | \node s2a(1000,-1000)[{S^2_\Alpha}] |
---|
1732 | \arrow |l|[s1`s1a;{\funcname{f}^1}] |
---|
1733 | \arrow |r|[s1`s2a;{\funcseqname{g} \composet \funcseqname{f}^1}] |
---|
1734 | \arrow |b|[s1a`s2a;{\tail(\funcseqname{g}) = \funcseqname{g}_\Alpha}] |
---|
1735 | \efig \] |
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1736 | \caption{$\funcseqname{g} \composet \funcname{f}^1$} |
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1737 | \label{fig:gf} |
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1738 | \end{figure} |
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1739 | |
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1740 | \begin{definition}[Topology functions] |
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1741 | Let $S$ be a topological space and $Y$ a subset. Then \\ |
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1742 | \begin{enumerate} |
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1743 | \item $\Topology(S)$ is the topology of $S$. |
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1744 | \item |
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1745 | $\Topology(Y,S) \defeq \set{U \cap Y}[{U \in \Topology(S)}]$ |
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1746 | is the relative topology of $Y$. |
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1747 | \item |
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1748 | $\Top(Y,S) \defeq \bigl ( Y, \Topology(Y,S) \bigr )$ is $Y$ with the |
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1749 | relative topology. |
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1750 | \item |
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1751 | $ |
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1752 | \op{S} \defeq |
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1753 | \set |
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1754 | {(U, \Topology(U,S))}% |
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1755 | [{{ U \in \Topology(S) \setminus \{ \emptyset \} }}] |
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1756 | $ |
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1757 | is the set of all non-null open subspaces of $S$. |
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1758 | \end{enumerate} |
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1759 | |
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1760 | Let $\seqname{S}$ be a set of topological spaces. Then |
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1761 | $\op{\seqname{S}} \defeq \union [{S \in \seqname{S}}] { {\op{S}}}$ is the set |
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1762 | of open subspaces in $\seqname{S}$. |
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1763 | |
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1764 | Let $S$ and $T'$ be spaces, $T \subseteq T'$ be a subspace and |
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1765 | $\funcname{f} \maps S \to T$ a function. Then |
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1766 | $\funcname{f} \maps S \to T' \defeq \Id_{T,T'} \compose \funcname{f}$ |
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1767 | is $\funcname{f}$ considered as a function from $S$ to $T'$. |
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1768 | |
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1769 | Let $S'$ and $T'$ be spaces, $S \subseteq S'$, $T \subseteq T'$ be |
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1770 | subspaces and $\funcname{f}' \maps S' \to T'$ a function such that |
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1771 | $\funcname{f}'[S] \subseteq T$. Then $\funcname{f}' \maps S \to T$, also |
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1772 | written $\funcname{f}' \restrictto_{S,T}$, is |
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1773 | $\funcname{f}' \restrictto_S$ considered as a function from $S$ to |
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1774 | $T$. |
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1775 | |
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1776 | Let $\seqname{S}^i \defeq (S^i_\alpha, \alpha \preceq \Alpha)$, $i-1,2$, |
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1777 | be a sequence of spaces, $\seqname{S}^1 \SUBSETEQ \seqname{S}^2$ and \linebreak |
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1778 | $\funcname{f}^2 \maps \head(\seqname{S}^2) \to \tail(\seqname{S}^2)$ a |
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1779 | function. |
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1780 | $ |
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1781 | \funcname{f}^2 \restrictto_{\head(\seqname{S}^1)} \defeq |
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1782 | \funcname{f}^2 \restrictto_{\bigtimes \head(\seqname{S}^1)} |
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1783 | $. |
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1784 | If \linebreak |
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1785 | $ |
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1786 | \funcname{f}^2 \restrictto_{\head(\seqname{S}^1)} |
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1787 | [\bigtimes \head(\seqname{S}^1)] |
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1788 | \subseteq \tail(\seqname{S}^1) |
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1789 | $ |
---|
1790 | then |
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1791 | $ \funcname{f}^2 \maps \head(\seqname{S}^1) \to \tail(\seqname{S}^1)$, |
---|
1792 | also written |
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1793 | $ |
---|
1794 | \funcname{f}^2 \restrictto_% |
---|
1795 | {\head(\seqname{S}^1),\tail(\seqname{S}^1)} |
---|
1796 | $, |
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1797 | is $\funcname{f}^2 \restrictto_{\head(\seqname{S}^1)}$ considered as a |
---|
1798 | function from $\bigtimes \head(\seqname{S}^1)$ to $\tail(\seqname{S}^1)$. |
---|
1799 | \end{definition} |
---|
1800 | |
---|
1801 | \begin{definition}[Truth space] |
---|
1802 | $\false \defeq \emptyset$, $\true \defeq \set{\emptyset}$, |
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1803 | the truth set is \linebreak |
---|
1804 | $\truthset \defeq \set{\false, \true}$, |
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1805 | $\truthtop \defeq \set {\emptyset,\truthset}$ and the truth space |
---|
1806 | $\truthspace \defeq (\truthset, \truthtop)$ is $\truthset$ |
---|
1807 | with the indiscrete topology. |
---|
1808 | \end{definition} |
---|
1809 | |
---|
1810 | \begin{definition}[Truth category] |
---|
1811 | The truth category is |
---|
1812 | $ |
---|
1813 | \truthcat \defeq \\ |
---|
1814 | \bigl ( |
---|
1815 | \truthspace, |
---|
1816 | \set{\truthspace \to \truthspace} |
---|
1817 | \bigr ) |
---|
1818 | $. |
---|
1819 | The truth model space is \\* |
---|
1820 | $\seqname{Truthspace} \defeq (\truthspace, \truthcat)$. |
---|
1821 | \end{definition} |
---|
1822 | |
---|
1823 | \begin{definition}[Constraint functions] |
---|
1824 | A constraint function is a continuous function with range |
---|
1825 | $\truthspace$ or a model |
---|
1826 | function with range $\seqname{Truthspace}$. |
---|
1827 | \end{definition} |
---|
1828 | |
---|
1829 | \begin{definition}[Sequence inclusion] |
---|
1830 | \label{def:seqin} |
---|
1831 | Let $\seqname{S} \defeq (S_\alpha, \alpha \prec \Alpha)$ and |
---|
1832 | $\seqname{T} \defeq (T_\alpha, \alpha \prec \Alpha)$ be sequences. |
---|
1833 | $\seqname{S} \seqin \seqname{T}$ iff |
---|
1834 | $\uquant{\alpha \prec \Alpha} {S_\alpha \in T_\alpha}$ or |
---|
1835 | $\uquant{\alpha \prec \Alpha} {S_\alpha \objin T_\alpha}$. |
---|
1836 | \end{definition} |
---|
1837 | |
---|
1838 | \begin{lemma}[Sequence inclusion] |
---|
1839 | \label{lem:seqin} |
---|
1840 | Let $\seqname{S} \defeq (S_\alpha, \alpha \prec \Alpha)$ |
---|
1841 | be a sequence, |
---|
1842 | $ |
---|
1843 | \catseqname{T}^i \defeq \\ |
---|
1844 | (\catname{T}^i_\alpha, \alpha \prec \Alpha) |
---|
1845 | $, |
---|
1846 | $i=1,2$, a sequence of |
---|
1847 | categories, $\catseqname{T}^1 \SUBCAT \catseqname{T}^2$ and |
---|
1848 | $\seqname{S} \seqin \catseqname{T}^1$ Then |
---|
1849 | $\seqname{S} \seqin \catseqname{T}^2$. |
---|
1850 | |
---|
1851 | \begin{proof} |
---|
1852 | If |
---|
1853 | $\uquant{\alpha \prec \Alpha} {S_\alpha \objin \catname{T}^1_\alpha}$ |
---|
1854 | then |
---|
1855 | $\uquant{\alpha \prec \Alpha} {S_\alpha \objin \catname{T}^2_\alpha}$. |
---|
1856 | \end{proof} |
---|
1857 | \end{lemma} |
---|
1858 | |
---|
1859 | \begin{thebibliography}{9} |
---|
1860 | |
---|
1861 | \bibitem[Ad\'amek, Herrlich, Strecker, 1990] {JoyCat} Ji\v{r}\'i Ad\'amek, |
---|
1862 | Horst Herrlich, |
---|
1863 | George E. Strecker. |
---|
1864 | \textit{Abstract and Concrete Categories The Joy of Cats}, |
---|
1865 | John Wiley and Sons, Inc., 1990. |
---|
1866 | |
---|
1867 | \bibitem[Kelley, 1955] {GenTop} John L. Kelley, |
---|
1868 | \textit{General Topology}, |
---|
1869 | D. Van Nostrand Company (first edition), 1955. |
---|
1870 | |
---|
1871 | \bibitem[Kobayashi, 1996] {FoundDiffGeo1} Shoshichi Kobayashi, |
---|
1872 | Katsumi Nomizu, |
---|
1873 | \textit{Foundations of Differential Geometry, Volume I}, |
---|
1874 | ISBN 0-471-15733-3, John Wiley and Sons, Inc., 1996. |
---|
1875 | |
---|
1876 | %\bibitem[Lee, 2000] {DiffPhys} Jeffrey Marc lee, |
---|
1877 | %\textit{Differential Geometry}, |
---|
1878 | %Analysis and Physics, 2000. |
---|
1879 | |
---|
1880 | \bibitem[Mac Lane, 1998] {CftWM} Saunders Mac Lane, |
---|
1881 | \textit{Categories for the Working Mathemation, 2nd edition}, |
---|
1882 | ISBN 0-387-98403-8, Springer-Verlag, 1998. |
---|
1883 | |
---|
1884 | \bibitem[Steenrod, 1999] {TopFib} Norman Steenrod, |
---|
1885 | \textit{The Topology Of Fibre Bundles}, ISBN 0-691-00548-6, |
---|
1886 | Prineton University Press (seventh printing), 1999. |
---|
1887 | |
---|
1888 | \end{thebibliography} |
---|
1889 | |
---|
1890 | \end{document} |
---|