Knowledge

En-ring

Source 📝

33: 714: 287: 588: 619: 495: 437: 131: 643: 324: 351: 179: 377: 62: 755: 207: 84: 774: 664: 535: 518: 136: 748: 448: 510: 45: 779: 150: 55: 49: 41: 784: 498: 384: 66: 741: 593: 469: 411: 105: 681: 646: 456: 444: 333:
subject to the requirements that the multiplication maps are compatible with composition, and that
676: 628: 502: 139: 659: 529: 302: 336: 155: 356: 725: 768: 622: 296: 440: 192: 713: 182: 98: 694: 17: 721: 396: 682:
http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf
282:{\displaystyle \mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)} 677:
http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf
26: 600: 476: 418: 112: 729: 631: 596: 583:{\displaystyle X\mapsto C_{*}(\Omega ^{n}X;\Lambda )} 538: 472: 414: 359: 339: 305: 210: 158: 108: 637: 613: 582: 489: 431: 371: 345: 318: 281: 173: 125: 54:but its sources remain unclear because it lacks 749: 8: 756: 742: 630: 605: 599: 598: 595: 562: 549: 537: 481: 475: 474: 471: 423: 417: 416: 413: 358: 338: 310: 304: 255: 227: 209: 157: 117: 111: 110: 107: 85:Learn how and when to remove this message 621:-algebra in the infinity category of 7: 710: 708: 379:. An equivalent definition is that 632: 614:{\displaystyle {\mathcal {E}}_{n}} 574: 559: 490:{\displaystyle {\mathcal {E}}_{n}} 432:{\displaystyle {\mathcal {E}}_{n}} 126:{\displaystyle {\mathcal {E}}_{n}} 25: 712: 145:consists of the following data: 31: 665:Highly structured ring spectrum 457:commutative associative algebra 577: 555: 542: 276: 270: 264: 261: 248: 233: 220: 168: 162: 1: 728:. You can help Knowledge by 326:contained in some open disk 519:symmetric monoidal category 801: 707: 449:unital associative algebra 511:braided monoidal category 638:{\displaystyle \Lambda } 40:This article includes a 69:more precise citations. 775:Higher category theory 724:-related article is a 639: 615: 584: 491: 433: 373: 347: 320: 283: 202:A multiplication map: 175: 127: 640: 616: 585: 492: 434: 374: 353:is an equivalence if 348: 321: 319:{\displaystyle U_{j}} 284: 176: 128: 629: 594: 536: 470: 412: 357: 346:{\displaystyle \mu } 337: 303: 208: 174:{\displaystyle A(U)} 156: 106: 372:{\displaystyle m=1} 635: 611: 580: 487: 455:= 1, and a unital 429: 369: 343: 316: 279: 171: 137:symmetric monoidal 123: 42:list of references 737: 736: 503:monoidal category 140:infinity category 95: 94: 87: 16:(Redirected from 792: 758: 751: 744: 716: 709: 701: 660:Categorical ring 644: 642: 641: 636: 620: 618: 617: 612: 610: 609: 604: 603: 589: 587: 586: 581: 567: 566: 554: 553: 530:commutative ring 496: 494: 493: 488: 486: 485: 480: 479: 438: 436: 435: 430: 428: 427: 422: 421: 391:over the little 378: 376: 375: 370: 352: 350: 349: 344: 325: 323: 322: 317: 315: 314: 288: 286: 285: 280: 260: 259: 232: 231: 180: 178: 177: 172: 132: 130: 129: 124: 122: 121: 116: 115: 90: 83: 79: 76: 70: 65:this article by 56:inline citations 35: 34: 27: 21: 800: 799: 795: 794: 793: 791: 790: 789: 780:Homotopy theory 765: 764: 763: 762: 705: 693: 690: 673: 656: 627: 626: 623:chain complexes 597: 592: 591: 558: 545: 534: 533: 473: 468: 467: 415: 410: 409: 405: 355: 354: 335: 334: 306: 301: 300: 251: 223: 206: 205: 154: 153: 109: 104: 103: 91: 80: 74: 71: 60: 46:related reading 36: 32: 23: 22: 15: 12: 11: 5: 798: 796: 788: 787: 782: 777: 767: 766: 761: 760: 753: 746: 738: 735: 734: 717: 703: 702: 689: 688:External links 686: 685: 684: 679: 672: 669: 668: 667: 662: 655: 652: 651: 650: 634: 608: 602: 579: 576: 573: 570: 565: 561: 557: 552: 548: 544: 541: 526: 484: 478: 464: 426: 420: 404: 401: 368: 365: 362: 342: 331: 330: 313: 309: 292: 291: 290: 289: 278: 275: 272: 269: 266: 263: 258: 254: 250: 247: 244: 241: 238: 235: 230: 226: 222: 219: 216: 213: 200: 170: 167: 164: 161: 120: 114: 93: 92: 50:external links 39: 37: 30: 24: 14: 13: 10: 9: 6: 4: 3: 2: 797: 786: 785:Algebra stubs 783: 781: 778: 776: 773: 772: 770: 759: 754: 752: 747: 745: 740: 739: 733: 731: 727: 723: 718: 715: 711: 706: 700: 696: 692: 691: 687: 683: 680: 678: 675: 674: 670: 666: 663: 661: 658: 657: 653: 648: 624: 606: 571: 568: 563: 550: 546: 539: 531: 527: 524: 520: 516: 512: 508: 504: 500: 482: 465: 462: 458: 454: 450: 446: 442: 441:vector spaces 424: 407: 406: 402: 400: 398: 394: 390: 386: 382: 366: 363: 360: 340: 329: 311: 307: 298: 294: 293: 273: 267: 256: 252: 245: 242: 239: 236: 228: 224: 217: 214: 211: 204: 203: 201: 198: 194: 191: 187: 184: 165: 159: 152: 148: 147: 146: 144: 141: 138: 134: 118: 100: 89: 86: 78: 68: 64: 58: 57: 51: 47: 43: 38: 29: 28: 19: 730:expanding it 719: 704: 698: 695:"En-algebra" 522: 514: 506: 497:-algebra in 460: 452: 439:-algebra in 392: 388: 380: 332: 327: 196: 193:homeomorphic 189: 185: 142: 102: 96: 81: 72: 61:Please help 53: 699:ncatlab.org 590:defines an 517:= 2, and a 299:open disks 183:open subset 99:mathematics 67:introducing 769:Categories 671:References 528:If Λ is a 499:categories 633:Λ 575:Λ 560:Ω 551:∗ 543:↦ 341:μ 265:→ 243:⊗ 240:⋯ 237:⊗ 212:μ 654:See also 403:Examples 297:disjoint 295:for any 181:for any 133:-algebra 75:May 2024 18:E n-ring 722:algebra 647:modules 532:, then 509:= 1, a 443:over a 395:-disks 385:algebra 63:improve 397:operad 383:is an 199:-disk. 195:to an 151:object 720:This 501:is a 447:is a 445:field 135:in a 101:, an 48:, or 726:stub 525:≥ 3. 463:≥ 2. 625:of 521:if 513:if 505:if 466:An 459:if 451:if 408:An 387:in 188:of 149:An 97:In 771:: 697:, 399:. 52:, 44:, 757:e 750:t 743:v 732:. 649:. 645:- 607:n 601:E 578:) 572:; 569:X 564:n 556:( 547:C 540:X 523:n 515:n 507:n 483:n 477:E 461:n 453:n 425:n 419:E 393:n 389:C 381:A 367:1 364:= 361:m 328:V 312:j 308:U 277:) 274:V 271:( 268:A 262:) 257:m 253:U 249:( 246:A 234:) 229:1 225:U 221:( 218:A 215:: 197:n 190:R 186:U 169:) 166:U 163:( 160:A 143:C 119:n 113:E 88:) 82:( 77:) 73:( 59:. 20:)

Index

E n-ring
list of references
related reading
external links
inline citations
improve
introducing
Learn how and when to remove this message
mathematics
symmetric monoidal
infinity category
object
open subset
homeomorphic
disjoint
algebra
operad
vector spaces
field
unital associative algebra
commutative associative algebra
categories
monoidal category
braided monoidal category
symmetric monoidal category
commutative ring
chain complexes
modules
Categorical ring
Highly structured ring spectrum

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.