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vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness. Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete
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Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich, and George E. Strecker, theorem 12.7, page 213
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The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
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is complete if and only if it is cocomplete. A small complete category is necessarily thin.
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A group, considered as a category with a single object, is complete if and only if it is
519: 432: 226: 140: 70: 507:, considered as a small category, is complete (and cocomplete) if and only if it is a 112:: for any two objects there can be at most one morphism from one object to the other. 649: 526: 109: 423: 105: 119:
if all finite limits exists (i.e. limits of diagrams indexed by a finite category
555: 253: 167: 31: 595: 489:, is finitely complete but has neither binary coproducts nor infinite products. 443: 577: 171: 115:
A weaker form of completeness is that of finite completeness. A category is
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Finite completeness can be characterized in several ways. For a category
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is cocomplete but not complete (since it has no terminal object).
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Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).
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Dually, a category is cocomplete if and only if it has
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is a category which is both complete and cocomplete.
562:. New York: Cambridge University Press. p. 32. 155:and binary products (consider the pullback of ( 471:is finitely complete and finitely cocomplete. 8: 147:(of all pairs of morphisms) and all (small) 27:Category with all limits of (small) diagrams 282:. Unsourced material may be challenged and 151:. Since equalizers may be constructed from 315:The following categories are bicomplete: 302:Learn how and when to remove this message 222:The dual statements are also equivalent. 624:Categories for the Working Mathematician 538: 204:has equalizers, binary products, and a 198:has equalizers and all finite products, 7: 280:adding citations to reliable sources 185:, the following are all equivalent: 25: 631:((2nd ed.) ed.). Springer. 627:. Graduate Texts in Mathematics 597:Abstract and Concrete Categories 415:category of all small categories 252: 478:is complete but not cocomplete. 332:category of topological spaces 127:if all finite colimits exist. 1: 139:that a category is complete 137:existence theorem for limits 560:Categorical Homotopy Theory 46:exist. That is, a category 672: 350:category of abelian groups 85:is one in which all small 603:. John Wiley & Sons. 483:category of metric spaces 371:category of vector spaces 123:). Dually, a category is 656:Limits (category theory) 406:compact Hausdorff spaces 244:Examples and nonexamples 516:partially ordered class 404:, the category of all 218:and a terminal object. 450:finite abelian groups 192:is finitely complete, 50:is complete if every 276:improve this section 174:, or, equivalently, 135:It follows from the 390:category of modules 125:finitely cocomplete 91:bicomplete category 83:cocomplete category 42:in which all small 619:Mac Lane, Saunders 494:category of fields 456:finite-dimensional 431:, the category of 422:, the category of 341:category of groups 100:limits (even when 476:complete lattices 359:category of rings 312: 311: 304: 117:finitely complete 96:The existence of 73:) has a limit in 36:complete category 16:(Redirected from 663: 642: 614: 602: 582: 581: 552: 546: 543: 509:complete lattice 474:The category of 469:abelian category 454:The category of 448:The category of 442:The category of 394:commutative ring 323:category of sets 307: 300: 296: 293: 287: 256: 248: 237:posetal category 178:and coproducts. 170:and all (small) 21: 671: 670: 666: 665: 664: 662: 661: 660: 646: 645: 639: 617: 611: 600: 593: 590: 588:Further reading 585: 570: 554: 553: 549: 544: 540: 536: 520:ordinal numbers 433:simplicial sets 308: 297: 291: 288: 273: 257: 246: 206:terminal object 133: 28: 23: 22: 15: 12: 11: 5: 669: 667: 659: 658: 648: 647: 644: 643: 637: 615: 609: 589: 586: 584: 583: 568: 547: 537: 535: 532: 531: 530: 523: 512: 501: 490: 479: 472: 461: 460: 459: 452: 446: 437: 436: 435: 426: 417: 408: 399: 380: 361: 352: 343: 334: 325: 310: 309: 260: 258: 251: 245: 242: 227:small category 220: 219: 209: 199: 193: 141:if and only if 132: 129: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 668: 657: 654: 653: 651: 640: 638:0-387-98403-8 634: 630: 626: 625: 620: 616: 612: 610:0-471-60922-6 606: 599: 598: 592: 591: 587: 579: 575: 571: 569:9781139960083 565: 561: 557: 551: 548: 542: 539: 533: 528: 524: 521: 517: 513: 510: 506: 502: 499: 495: 491: 488: 484: 480: 477: 473: 470: 466: 462: 458:vector spaces 457: 453: 451: 447: 445: 441: 440: 438: 434: 430: 427: 425: 421: 418: 416: 412: 409: 407: 403: 400: 398: 395: 391: 387: 385: 381: 379: 376: 372: 368: 366: 362: 360: 356: 353: 351: 347: 344: 342: 338: 335: 333: 329: 326: 324: 320: 317: 316: 314: 313: 306: 303: 295: 285: 281: 277: 271: 270: 266: 261:This section 259: 255: 250: 249: 243: 241: 238: 233: 231: 228: 223: 217: 213: 210: 207: 203: 200: 197: 194: 191: 188: 187: 186: 184: 179: 177: 173: 169: 164: 162: 158: 154: 150: 146: 142: 138: 130: 128: 126: 122: 118: 113: 111: 110:thin category 107: 103: 99: 94: 92: 88: 84: 80: 76: 72: 68: 64: 60: 56: 53: 49: 45: 41: 37: 33: 19: 628: 623: 596: 559: 556:Riehl, Emily 550: 541: 497: 486: 428: 419: 410: 401: 396: 383: 382: 377: 364: 363: 354: 345: 336: 327: 318: 298: 289: 274:Please help 262: 234: 229: 224: 221: 211: 201: 195: 189: 182: 180: 168:coequalizers 165: 160: 156: 134: 124: 120: 116: 114: 106:proper class 101: 97: 95: 90: 82: 74: 66: 62: 58: 54: 47: 35: 29: 444:finite sets 292:August 2012 32:mathematics 534:References 240:lattices. 172:coproducts 145:equalizers 18:Cocomplete 578:881162803 263:does not 216:pullbacks 153:pullbacks 89:exist. A 650:Category 621:(1998). 558:(2014). 176:pushouts 149:products 131:Theorems 87:colimits 57: : 40:category 527:trivial 518:of all 392:over a 373:over a 284:removed 269:sources 143:it has 65:(where 52:diagram 635:  607:  576:  566:  424:wheels 413:, the 388:, the 369:, the 357:, the 348:, the 339:, the 330:, the 321:, the 79:Dually 44:limits 601:(PDF) 505:poset 498:Field 463:Any ( 402:CmptH 375:field 367:-Vect 104:is a 71:small 38:is a 633:ISBN 605:ISBN 574:OCLC 564:ISBN 514:The 492:The 481:The 429:sSet 386:-Mod 355:Ring 267:any 265:cite 214:has 81:, a 34:, a 487:Met 465:pre 420:Whl 411:Cat 337:Grp 328:Top 319:Set 278:by 98:all 69:is 30:In 652:: 572:. 503:A 496:, 485:, 346:Ab 235:A 225:A 159:, 77:. 61:→ 641:. 629:5 613:. 580:. 511:. 467:) 397:R 384:R 378:K 365:K 305:) 299:( 294:) 290:( 286:. 272:. 230:C 212:C 208:, 202:C 196:C 190:C 183:C 161:g 157:f 121:J 102:J 75:C 67:J 63:C 59:J 55:F 48:C 20:)

Index

Cocomplete
mathematics
category
limits
diagram
small
Dually
colimits
proper class
thin category
existence theorem for limits
if and only if
equalizers
products
pullbacks
coequalizers
coproducts
pushouts
terminal object
pullbacks
small category
posetal category

cite
sources
improve this section
adding citations to reliable sources
removed
Learn how and when to remove this message
category of sets

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