Preface |
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xiii | |
Acknowledgments |
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xxi | |
List of Notations |
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xxiii | |
Part 1. Graphs and Trees |
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Chapter 1 Directed Graphs |
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3 | (16) |
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3 | (2) |
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5 | (3) |
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8 | (2) |
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1.4 Directed (m, n)-Graphs |
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10 | (6) |
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16 | (1) |
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17 | (2) |
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Chapter 2 Extra Structures on Graphs |
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19 | (10) |
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19 | (1) |
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20 | (1) |
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21 | (3) |
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2.4 Incoming Edge Labeling |
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24 | (2) |
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2.5 Isomorphisms with Extra Structures |
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26 | (1) |
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27 | (2) |
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29 | (14) |
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30 | (1) |
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31 | (1) |
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3.3 Profile of a Rooted Tree |
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31 | (1) |
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3.4 Exceptional Edge and Corollas |
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32 | (2) |
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34 | (2) |
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36 | (4) |
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40 | (1) |
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40 | (3) |
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Chapter 4 Collapsing an Internal Edge |
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43 | (10) |
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43 | (1) |
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4.2 Defining the Quotient |
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44 | (2) |
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46 | (2) |
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48 | (1) |
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4.5 Compatibility with Extra Structures |
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48 | (3) |
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51 | (2) |
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Chapter 5 Grafting of Rooted Trees |
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53 | (22) |
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54 | (1) |
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55 | (2) |
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57 | (3) |
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60 | (2) |
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5.5 Horizontal Associativity |
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62 | (2) |
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5.6 Vertical Associativity |
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64 | (2) |
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5.7 Grafting Decomposition of Rooted Trees |
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66 | (6) |
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72 | (3) |
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Chapter 6 Grafting and Extra Structures |
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75 | (14) |
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75 | (1) |
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76 | (1) |
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77 | (2) |
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6.4 Incoming Edge Labeling |
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79 | (1) |
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6.5 Canonical Vertex Labeling |
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80 | (2) |
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6.6 Canonical Input Labeling |
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82 | (3) |
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85 | (4) |
Part 2. Category Theory |
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Chapter 7 Basic Category Theory |
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89 | (32) |
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90 | (8) |
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98 | (2) |
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7.3 Natural Transformations |
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100 | (5) |
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105 | (2) |
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107 | (2) |
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109 | (2) |
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111 | (5) |
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116 | (3) |
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119 | (2) |
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Chapter 8 Symmetric Monoidal Categories |
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121 | (20) |
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8.1 Motivation for Monoidal Categories |
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121 | (1) |
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122 | (3) |
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125 | (3) |
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128 | (2) |
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130 | (2) |
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8.6 Permuting Iterated Tensor Products |
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132 | (3) |
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8.7 Symmetric Monoidal Closed Categories |
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135 | (2) |
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8.8 Standing Categorical Assumptions |
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137 | (2) |
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139 | (1) |
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140 | (1) |
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Chapter 9 Colored Symmetric Sequences and Objects |
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141 | (20) |
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141 | (2) |
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143 | (4) |
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9.3 Colored Symmetric Sequences |
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147 | (6) |
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153 | (3) |
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156 | (2) |
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158 | (3) |
Part 3. Operads and Algebras |
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Chapter 10 Motivation for Colored Operads |
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161 | (12) |
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10.1 Categories via Linear Graphs |
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161 | (3) |
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10.2 Colored Operads via Trees |
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164 | (9) |
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Chapter 11 Colored Operads |
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173 | (30) |
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11.1 Block Permutations and Block Sums |
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173 | (2) |
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11.2 Defining Colored Operads |
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175 | (6) |
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11.3 Examples of the Axioms |
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181 | (4) |
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11.4 Initial and Terminal Colored Operads |
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185 | (5) |
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11.5 Changing the Base Category |
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190 | (3) |
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193 | (4) |
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11.7 Colored Non-Symmetric Operads |
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197 | (1) |
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198 | (2) |
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200 | (3) |
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Chapter 12 Operads in Arity 1 |
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203 | (12) |
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12.1 Motivation for Monoids |
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203 | (2) |
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205 | (3) |
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12.3 Small Enriched Categories |
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208 | (4) |
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212 | (3) |
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Chapter 13 Algebras over Colored Operads |
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215 | (22) |
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215 | (2) |
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217 | (2) |
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13.3 Examples of the Axioms |
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219 | (2) |
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13.4 Visualizing Algebras |
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221 | (1) |
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13.5 Algebras over a 1-Colored Operad |
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222 | (2) |
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13.6 Algebras over a Colored Non-Symmetric Operad |
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224 | (1) |
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13.7 Motivation for Endomorphism Operads |
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225 | (1) |
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13.8 Colored Endomorphism Operads |
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226 | (5) |
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231 | (1) |
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232 | (3) |
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235 | (2) |
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Chapter 14 Examples of Algebras |
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237 | (28) |
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14.1 Initial and Terminal Algebras |
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237 | (2) |
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239 | (7) |
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14.3 Colored Operad for Monoid Maps |
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246 | (7) |
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14.4 Colored Operad for Colored Monoids |
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253 | (5) |
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258 | (7) |
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Chapter 15 Motivation for Partial Compositions |
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265 | (10) |
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15.1 Simplifying the Operadic Composition |
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265 | (2) |
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15.2 Operadic Composition to Partial Composition |
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267 | (3) |
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15.3 Partial Composition to Operadic Composition |
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270 | (1) |
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271 | (1) |
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272 | (1) |
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273 | (2) |
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Chapter 16 Colored Pseudo-Operads |
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275 | (46) |
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16.1 Partial Compositions of Profiles |
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275 | (3) |
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16.2 Defining Colored Pseudo-Operads |
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278 | (7) |
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16.3 Examples of the Axioms |
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285 | (4) |
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16.4 Colored Pseudo-Operads Are Colored Operads |
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289 | (4) |
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16.5 1-Colored Pseudo-Operads |
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293 | (3) |
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16.6 Colored Non-Symmetric Pseudo-Operads |
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296 | (2) |
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16.7 Algebras via Partial Compositions |
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298 | (3) |
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301 | (4) |
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16.9 Little Square Operad |
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305 | (4) |
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309 | (8) |
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317 | (4) |
Part 4. Free Colored Operads |
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Chapter 17 Motivation for Free Colored Operads |
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321 | (12) |
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321 | (6) |
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17.2 Heuristic Free Colored Non-Symmetric Operads |
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327 | (3) |
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17.3 Heuristic Free Colored Operads |
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330 | (2) |
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332 | (1) |
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Chapter 18 General Operadic Composition |
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333 | (18) |
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18.1 Decoration by a Colored Object |
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333 | (7) |
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18.2 Defining General Operadic Composition |
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340 | (7) |
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347 | (3) |
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350 | (1) |
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Chapter 19 Free Colored Non-Symmetric Operads |
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351 | (30) |
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19.1 Entries and Structure Maps |
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352 | (5) |
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19.2 Colored Non-Symmetric Operad Structure |
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357 | (4) |
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361 | (9) |
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19.4 Motivation for Tree Operads |
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370 | (2) |
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19.5 Colored Non-Symmetric Operad Generated by a Tree |
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372 | (7) |
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379 | (2) |
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Chapter 20 Free Colored Operads |
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381 | (28) |
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20.1 Colored Non-Symmetric Operads to Colored Operads |
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382 | (7) |
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389 | (5) |
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20.3 Free Colored Operad of a Colored Object |
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394 | (10) |
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20.4 Colored Operad Generated by a Tree |
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404 | (4) |
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408 | (1) |
Further Reading |
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409 | (4) |
Bibliography |
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413 | (6) |
List of Main Facts |
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419 | (4) |
Index |
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423 | |