Knowledge (XXG)

Generalized metric space

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Bonsangue, M.M.; Van Breugel, F.; Rutten, J.J.M.M. (1998). "Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding".
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namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite.
283: 101: 217: 93:. The notion was introduced in 1973 by Lawvere who noticed that a metric space can be viewed as a particular kind of a category. 320: 222: 100:, a generalized metric space can be embedded into a much larger category in which, for instance, one can construct the 309: 349: 354: 276: 74: 269: 17: 227: 218:
https://golem.ph.utexas.edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html#more
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but without the symmetry property and some other properties. Precisely, it is a category
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https://golem.ph.utexas.edu/category/2022/01/optimal_transport_and_enriched_2.html#more
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Lawvere, F. William (1973). "Metric spaces, generalized logic, and closed categories".
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https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace
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Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Rendiconti del Seminario Matematico e Fisico di Milano
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to it so that it can be listed with similar articles.
77: 45: 85: 63: 164:Reprints in Theory and Applications of Categories 96:The categorical point of view is useful since by 277: 8: 173:Borceux, Francis; Dejean, Dominique (1986). 284: 270: 79: 78: 76: 44: 113: 175:"Cauchy completion in category theory" 7: 238: 236: 71:, the one-point compactification of 308:needs additional or more specific 256:. You can help Knowledge (XXG) by 55: 14: 297: 240: 23:In mathematics, specifically in 58: 46: 1: 204:10.1016/S0304-3975(97)00042-X 192:Theoretical Computer Science 86:{\displaystyle \mathbb {R} } 371: 235: 15: 29:generalized metric space 16:Not to be confused with 155:Lawvere, F. W. (2002). 252:-related article is a 87: 65: 88: 66: 75: 43: 148:10.1007/BF02924844 83: 61: 18:Generalised metric 350:Mathematics stubs 338: 337: 321:adding categories 265: 264: 102:Cauchy completion 362: 333: 330: 324: 301: 293: 286: 279: 272: 244: 237: 207: 186: 167: 161: 151: 121: 118: 92: 90: 89: 84: 82: 70: 68: 67: 64:{\displaystyle } 62: 370: 369: 365: 364: 363: 361: 360: 359: 355:Category theory 340: 339: 334: 328: 325: 314: 302: 291: 290: 233: 214: 212:Further reading 189: 172: 159: 154: 133: 130: 125: 124: 119: 115: 110: 73: 72: 41: 40: 25:category theory 21: 12: 11: 5: 368: 366: 358: 357: 352: 342: 341: 336: 335: 329:September 2024 305: 303: 296: 289: 288: 281: 274: 266: 263: 262: 245: 231: 230: 225: 220: 213: 210: 209: 208: 187: 170: 169: 168: 129: 126: 123: 122: 112: 111: 109: 106: 104:of the space. 98:Yoneda's lemma 81: 60: 57: 54: 51: 48: 13: 10: 9: 6: 4: 3: 2: 367: 356: 353: 351: 348: 347: 345: 332: 322: 318: 312: 311: 306:This article 304: 300: 295: 294: 287: 282: 280: 275: 273: 268: 267: 261: 259: 255: 251: 246: 243: 239: 234: 229: 226: 224: 221: 219: 216: 215: 211: 205: 201: 198:(1–2): 1–51. 197: 193: 188: 185:(2): 133–146. 184: 180: 176: 171: 165: 158: 153: 152: 149: 145: 141: 137: 132: 131: 127: 117: 114: 107: 105: 103: 99: 94: 52: 49: 38: 34: 30: 26: 19: 326: 307: 258:expanding it 247: 232: 195: 191: 182: 178: 163: 139: 135: 116: 95: 33:metric space 28: 22: 250:mathematics 142:: 135–166. 344:Categories 310:categories 166:(1): 1–37. 128:References 56:∞ 317:help out 37:enriched 315:Please 248:This 160:(PDF) 108:Notes 39:over 31:is a 254:stub 27:, a 319:by 200:doi 196:193 144:doi 346:: 194:. 183:27 181:. 177:. 162:. 140:43 138:. 331:) 327:( 313:. 285:e 278:t 271:v 260:. 206:. 202:: 150:. 146:: 80:R 59:] 53:, 50:0 47:[ 20:.

Index

Generalised metric
category theory
metric space
enriched
Yoneda's lemma
Cauchy completion
doi
10.1007/BF02924844
"Metric spaces, generalized logic and closed categories"
"Cauchy completion in category theory"
doi
10.1016/S0304-3975(97)00042-X
https://golem.ph.utexas.edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html#more
https://golem.ph.utexas.edu/category/2022/01/optimal_transport_and_enriched_2.html#more
https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace
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mathematics
stub
expanding it
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e

categories
help out
adding categories
Categories
Mathematics stubs
Category theory

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