Knowledge (XXG)

Canonical map

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between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention).
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For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.
69:; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. 515: 484: 382: 50: 73: 269: 161: 113: 477: 165: 123: 424: 418: 365: 307: 172: 461: 261: 81: 31: 280:
is then called the structure map (for the algebra structure). The corresponding map on the
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For the canonical map of an algebraic variety into projective space, see
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of canonical maps or canonical isomorphisms; for a typical example, see
146: 84:). In some contexts, it might be necessary to address an issue of 465: 485: 8: 492: 478: 394: 171:, then there is a canonical surjective 423:. BoD - Books on Demand. p. 274. 32:Canonical bundle § Canonical maps 205:, then there is a canonical map from 7: 446: 444: 76:is a canonical map that is also an 464:. You can help Knowledge (XXG) by 25: 295:is also called the structure map. 448: 328:, a canonical map is a function 313:, then the projection map from 417:Vialar, Thierry (2016-12-07). 404:"Grothendieck Conference Talk" 352:is an equivalence relation on 61:A closely related notion is a 1: 119:, then there is a canonical 532: 443: 260:is a homomorphism between 29: 511:Mathematical terminology 276:. The ring homomorphism 186:, that sends an element 420:Handbook of Mathematics 460:-related article is a 383:Natural transformation 141:that sends an element 18:Canonical homomorphism 321:is the structure map. 74:canonical isomorphism 268:can be viewed as an 217:that sends a vector 356:, that takes each 124:group homomorphism 67:structure morphism 516:Mathematics stubs 473: 472: 366:equivalence class 308:topological space 262:commutative rings 223:linear functional 173:ring homomorphism 16:(Redirected from 523: 494: 487: 480: 452: 445: 435: 434: 414: 408: 407: 402:Buzzard, Kevin. 399: 294: 259: 45:, also called a 21: 531: 530: 526: 525: 524: 522: 521: 520: 501: 500: 499: 498: 441: 439: 438: 431: 416: 415: 411: 401: 400: 396: 391: 379: 284: 250: 241: 232: 110:normal subgroup 101: 35: 28: 23: 22: 15: 12: 11: 5: 529: 527: 519: 518: 513: 503: 502: 497: 496: 489: 482: 474: 471: 470: 453: 437: 436: 429: 409: 393: 392: 390: 387: 386: 385: 378: 375: 374: 373: 332:mapping a set 322: 296: 247: 237: 228: 209:to the second 195: 154: 149:determined by 132:quotient group 100: 97: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 528: 517: 514: 512: 509: 508: 506: 495: 490: 488: 483: 481: 476: 475: 469: 467: 463: 459: 454: 451: 447: 442: 432: 430:9782955199008 426: 422: 421: 413: 410: 405: 398: 395: 388: 384: 381: 380: 376: 371: 367: 363: 359: 355: 351: 347: 343: 339: 335: 331: 327: 323: 320: 316: 312: 309: 305: 304:vector bundle 301: 297: 292: 288: 283: 282:prime spectra 279: 275: 271: 267: 263: 258: 254: 248: 245: 240: 236: 231: 227: 224: 220: 216: 212: 208: 204: 200: 196: 193: 190:to its coset 189: 185: 182: 181:quotient ring 178: 174: 170: 167: 163: 159: 155: 152: 148: 144: 140: 136: 133: 129: 125: 122: 118: 115: 111: 107: 103: 102: 98: 96: 93: 91: 87: 83: 79: 75: 70: 68: 64: 63:structure map 59: 56: 52: 48: 44: 43:canonical map 40: 33: 19: 466:expanding it 455: 440: 419: 412: 397: 369: 361: 357: 353: 349: 345: 341: 337: 333: 329: 318: 314: 310: 299: 290: 286: 277: 273: 265: 256: 252: 243: 238: 234: 229: 225: 218: 214: 206: 203:vector space 198: 191: 187: 183: 176: 168: 157: 150: 142: 138: 134: 127: 116: 105: 94: 85: 71: 66: 62: 60: 46: 42: 36: 458:mathematics 233:defined by 78:isomorphism 47:natural map 39:mathematics 505:Categories 389:References 211:dual space 121:surjective 82:invertible 27:Math topic 348:), where 289:) → Spec( 179:onto the 377:See also 326:topology 285:f: Spec( 242:(λ) = λ( 99:Examples 90:prestack 55:morphism 368:modulo 364:to the 344:modulo 306:over a 270:algebra 264:, then 221:to the 145:to the 130:to the 86:choices 80:(i.e., 49:, is a 427:  160:is an 456:This 302:is a 272:over 201:is a 175:from 164:of a 162:ideal 147:coset 126:from 114:group 112:of a 108:is a 462:stub 425:ISBN 166:ring 41:, a 360:in 338:X/R 324:In 317:to 298:If 251:f: 249:If 213:of 197:If 192:I+r 184:R/I 156:If 104:If 65:or 53:or 51:map 37:In 507:: 336:→ 255:→ 246:). 215:V, 139:N, 92:. 72:A 493:e 486:t 479:v 468:. 433:. 406:. 372:. 370:R 362:X 358:x 354:X 350:R 346:R 342:X 340:( 334:X 330:f 319:X 315:E 311:X 300:E 293:) 291:R 287:S 278:f 274:R 266:S 257:S 253:R 244:v 239:v 235:f 230:v 226:f 219:v 207:V 199:V 194:. 188:r 177:R 169:R 158:I 153:. 151:g 143:g 137:/ 135:G 128:G 117:G 106:N 34:. 20:)

Index

Canonical homomorphism
Canonical bundle § Canonical maps
mathematics
map
morphism
canonical isomorphism
isomorphism
invertible
prestack
normal subgroup
group
surjective
group homomorphism
quotient group
coset
ideal
ring
ring homomorphism
quotient ring
vector space
dual space
linear functional
commutative rings
algebra
prime spectra
vector bundle
topological space
topology
equivalence class
Natural transformation

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