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Quasi-isomorphism

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358: 22: 281: 145: 399: 32: 90: 276:{\displaystyle H_{n}(A_{\bullet })\to H_{n}(B_{\bullet })\ ({\text{respectively, }}H^{n}(A^{\bullet })\to H^{n}(B^{\bullet }))} 62: 309:
when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of
69: 428: 306: 76: 47: 58: 423: 418: 392: 433: 385: 310: 287: 83: 373: 43: 8: 112: 326: 314: 369: 302: 412: 136: 39: 291: 116: 339: 21: 365: 139:(respectively, cochain complexes) such that the induced morphisms 357: 305:, quasi-isomorphisms are sometimes used as the class of 148: 275: 410: 290:groups (respectively, of cohomology groups) are 393: 48:introducing citations to additional sources 400: 386: 38:Relevant discussion may be found on the 411: 352: 15: 13: 14: 445: 356: 31:relies largely or entirely on a 20: 343:Methods of Homological Algebra 270: 267: 254: 241: 238: 225: 207: 201: 188: 175: 172: 159: 1: 332: 372:. You can help Knowledge by 7: 320: 10: 450: 351: 429:Equivalence (mathematics) 345:, 2nd ed. Springer, 2000. 311:Bousfield localization 277: 278: 338:Gelfand, Sergei I., 146: 44:improve this article 424:Homological algebra 212:respectively,  113:homological algebra 59:"Quasi-isomorphism" 419:Algebraic topology 273: 381: 380: 307:weak equivalences 301:In the theory of 213: 206: 121:quasi-isomorphism 109: 108: 94: 441: 402: 395: 388: 366:topology-related 360: 353: 327:Derived category 303:model categories 282: 280: 279: 274: 266: 265: 253: 252: 237: 236: 224: 223: 214: 211: 204: 200: 199: 187: 186: 171: 170: 158: 157: 104: 101: 95: 93: 52: 24: 16: 449: 448: 444: 443: 442: 440: 439: 438: 409: 408: 407: 406: 349: 335: 323: 315:homotopy theory 261: 257: 248: 244: 232: 228: 219: 215: 210: 195: 191: 182: 178: 166: 162: 153: 149: 147: 144: 143: 137:chain complexes 105: 99: 96: 53: 51: 37: 25: 12: 11: 5: 447: 437: 436: 434:Topology stubs 431: 426: 421: 405: 404: 397: 390: 382: 379: 378: 361: 347: 346: 340:Manin, Yuri I. 334: 331: 330: 329: 322: 319: 284: 283: 272: 269: 264: 260: 256: 251: 247: 243: 240: 235: 231: 227: 222: 218: 209: 203: 198: 194: 190: 185: 181: 177: 174: 169: 165: 161: 156: 152: 127:is a morphism 115:, a branch of 107: 106: 42:. Please help 28: 26: 19: 9: 6: 4: 3: 2: 446: 435: 432: 430: 427: 425: 422: 420: 417: 416: 414: 403: 398: 396: 391: 389: 384: 383: 377: 375: 371: 368:article is a 367: 362: 359: 355: 354: 350: 344: 341: 337: 336: 328: 325: 324: 318: 316: 312: 308: 304: 299: 297: 293: 289: 262: 258: 249: 245: 233: 229: 220: 216: 196: 192: 183: 179: 167: 163: 154: 150: 142: 141: 140: 138: 134: 130: 126: 122: 118: 114: 103: 100:November 2022 92: 89: 85: 82: 78: 75: 71: 68: 64: 61: –  60: 56: 55:Find sources: 49: 45: 41: 35: 34: 33:single source 29:This article 27: 23: 18: 17: 374:expanding it 363: 348: 342: 300: 295: 292:isomorphisms 285: 132: 128: 124: 120: 110: 97: 87: 80: 73: 66: 54: 30: 117:mathematics 413:Categories 333:References 70:newspapers 263:∙ 242:→ 234:∙ 197:∙ 176:→ 168:∙ 40:talk page 321:See also 294:for all 288:homology 84:scholar 205:  86:  79:  72:  65:  57:  364:This 125:quism 91:JSTOR 77:books 370:stub 119:, a 63:news 313:in 286:of 135:of 123:or 111:In 46:by 415:: 317:. 298:. 131:→ 401:e 394:t 387:v 376:. 296:n 271:) 268:) 259:B 255:( 250:n 246:H 239:) 230:A 226:( 221:n 217:H 208:( 202:) 193:B 189:( 184:n 180:H 173:) 164:A 160:( 155:n 151:H 133:B 129:A 102:) 98:( 88:· 81:· 74:· 67:· 50:. 36:.

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"Quasi-isomorphism"
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homological algebra
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chain complexes
homology
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Bousfield localization
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Manin, Yuri I.
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