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87 88 89 90</head> 91 92<body> 93 94<div id="first-bar" class="sticky-top inner-top"> 95 <div class="container clearfix"> 96 <header class="row float-left"> 97 <h1><a href="/">Kerodon</a></h1> 98 </header> 99 100 $\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ 101 $\newcommand\empty{}$ 102 103 <nav id="navigation" class="d-none d-lg-block float-right"> 104 <ul id="quicklinks"> 105 <li><a href="/about">about</a> 106 <li><a href="/changes">changes</a> 107 <li><a href="/recent-comments">recent comments</a> 108 <li><a href="/bibliography">bibliography</a> 109 </ul> 110 111 <form id="quicksearch" action="/search"> 112 <input type="search" placeholder="keywords or a tag" name="query"><input type="submit" value=""> 113 </form> 114 </nav> 115 116 <nav id="burger" class="d-lg-none float-right"> 117 <input type="checkbox" id="burger-checkbox"> 118 <label id="burger-box" for="burger-checkbox"></label> 119 120 <div class="d-none" id="burger-content"> 121 122 123 124 <ul class="neighbours clearfix"> 125 126 127 <li class="right"><a href="/tag/02GZ">next</a> 128 129 </ul> 130 131 <span id="hamburger-toggle"></span> 132 133 134 <hr> 135 136 <ul class="tag-info"> 137 138 <li class="structure-link"><a href="#structure">go to structure</a> 139 140 141 <li class="tag-statistics"><a href="/tag/0000/statistics">statistics</a> 142 143 144 </ul> 145 146 <a class="btn btn-success" href="/tag/0000/comments">comments (0)</a> 147 148 149 150 <hr> 151 152 <form id="burger-quicksearch" action="/search"> 153 <input type="search" placeholder="keywords or a tag" name="query"><input type="submit" value=""> 154 </form> 155 156 </div> 157 </nav> 158 </div> 159</div> 160 161<nav class="row" id="second-bar"> 162 163 <div id="breadcrumb"> 164 <div class="container"> 165 <ul id="crumbs"> 166 167 168 169 <li class="active">Part <a href="/tag/0000" data-tag="0000">1</a>: Foundations 170 <span id="citation">(<a href="/tag/0000/cite">cite</a>)</span> 171 172 173 174 </ul> 175 <br style="clear: both"> 176 </div> 177 </div> 178 179 180 181 <div id="sidebar-parent" class="container d-none d-lg-block"> 182 <div class="row"> 183 <div class="col-lg-9"></div> 184 185 <div class="col-lg-3"> 186 <div id="sidebar-container"> 187 <section id="sidebar"> 188 189 <div id="sidebar-interaction"> 190 191 <ul class="neighbours clearfix"> 192 193 194 <li class="right"><a href="/tag/02GZ">next</a> 195 196 </ul> 197 198 <span id="sidebar-toggle"></span> 199 </div> 200 201 <hr> 202 203 204 <div id="sidebar-structure"> 205 206 <input type="checkbox" id="quicknav-toggle" checked> 207 <label for="quicknav-toggle"></label> 208 209 <ul id="quicknav"> 210 211 <li><a class="sidebar-link" href="/tag/0001">Chapter <span data-tag="0001">1</span></a>: The Language of $\infty $-Categories 212 213 <li><a class="sidebar-link" href="/tag/007J">Chapter <span data-tag="007J">2</span></a>: Examples of $\infty $-Categories 214 215 <li><a class="sidebar-link" href="/tag/00SY">Chapter <span data-tag="00SY">3</span></a>: Kan Complexes 216 217 <li><a class="sidebar-link" href="/tag/01GP">Chapter <span data-tag="01GP">4</span></a>: The Homotopy Theory of $\infty $-Categories 218 219 <li><a class="sidebar-link" href="/tag/01J2">Chapter <span data-tag="01J2">5</span></a>: Fibrations of $\infty $-Categories 220 221 <li><a class="sidebar-link" href="/tag/04QH">Chapter <span data-tag="04QH">10</span></a>: Exactness and Animation 222 223 <li><a class="sidebar-link" href="/tag/00FN">Chapter <span data-tag="00FN">11</span></a>: Retired Tags 224 225 </ul> 226 227 228 <ul class="tag-info"> 229 <li class="structure-link"><a href="#structure">go to structure</a> 230 </ul> 231 </div> 232 233 <hr> 234 235 236 <div id="sidebar-related"> 237 <a class="btn btn-success" href="/kerodon.pdf#nameddest=0000">view as pdf</a> 238 239 240 241 <ul class="tag-info"> 242 243 <li class="tag-statistics"><a href="/tag/0000/statistics">statistics</a> 244 245 246 </ul> 247 </div> 248 249 </section> 250 </div> 251 </div> 252 </div> 253 </div> 254 255</nav> 256
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262 263 264<div class="container" id="content"> 265 <div class="d-none">$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$</div> 266 267 268<section id="main" class="col-xs-12 col-sm-12 col-md-12 col-lg-9 col-xl-9"> 269 270<div class="html" id="tag"> 271 <h1 id="Part:Foundations" class="tex-part"><span data-tag="0000">1</span> Foundations</h1> 272 273 274 275 276
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285 286 287 288 289 290 <h2 id="structure">Structure</h2> 291 <ul class="tree" id="tree"> 292 293 <li id="0001"> 294 Chapter <a href="/tag/0001" data-tag="0001">1</a>: The Language of $\infty $-Categories 295 296 297 298 <ul> 299 <li id="0004"> 300 Section <a href="/tag/0004" data-tag="0004">1.1</a>: Simplicial Sets 301 302 303 304 <ul> 305 <li id="04ZB"> 306 Subsection <a href="/tag/04ZB" data-tag="04ZB">1.1.1</a>: Face Operators 307 308 309 310 <li id="04ZJ"> 311 Subsection <a href="/tag/04ZJ" data-tag="04ZJ">1.1.2</a>: Degeneracy Operators 312 313 314 315 <li id="04ZS"> 316 Subsection <a href="/tag/04ZS" data-tag="04ZS">1.1.3</a>: Dimensions of Simplicial Sets 317 318 319 320 <li id="0010"> 321 Subsection <a href="/tag/0010" data-tag="0010">1.1.4</a>: The Skeletal Filtration 322 323 324 325 <li id="00FQ"> 326 Subsection <a href="/tag/00FQ" data-tag="00FQ">1.1.5</a>: Discrete Simplicial Sets 327 328 329 330 <li id="001D"> 331 Subsection <a href="/tag/001D" data-tag="001D">1.1.6</a>: Directed Graphs as Simplicial Sets 332 333 334 </ul> 335 336 337 <li id="0508"> 338 Section <a href="/tag/0508" data-tag="0508">1.2</a>: From Topological Spaces to Simplicial Sets 339 340 341 342 <ul> 343 <li id="00G5"> 344 Subsection <a href="/tag/00G5" data-tag="00G5">1.2.1</a>: Connected Components of Simplicial Sets 345 346 347 348 <li id="001Q"> 349 Subsection <a href="/tag/001Q" data-tag="001Q">1.2.2</a>: The Singular Simplicial Set of a Topological Space 350 351 352 353 <li id="001X"> 354 Subsection <a href="/tag/001X" data-tag="001X">1.2.3</a>: The Geometric Realization of a Simplicial Set 355 356 357 358 <li id="000K"> 359 Subsection <a href="/tag/000K" data-tag="000K">1.2.4</a>: Horns 360 361 362 363 <li id="002G"> 364 Subsection <a href="/tag/002G" data-tag="002G">1.2.5</a>: Kan Complexes 365 366 367 </ul> 368 369 370 <li id="002L"> 371 Section <a href="/tag/002L" data-tag="002L">1.3</a>: From Categories to Simplicial Sets 372 373 374 375 <ul> 376 <li id="002M"> 377 Subsection <a href="/tag/002M" data-tag="002M">1.3.1</a>: The Nerve of a Category 378 379 380 381 <li id="04FY"> 382 Subsection <a href="/tag/04FY" data-tag="04FY">1.3.2</a>: Example: Monoids as Simplicial Sets 383 384 385 386 <li id="002Y"> 387 Subsection <a href="/tag/002Y" data-tag="002Y">1.3.3</a>: Recovering a Category from its Nerve 388 389 390 391 <li id="0030"> 392 Subsection <a href="/tag/0030" data-tag="0030">1.3.4</a>: Characterization of Nerves 393 394 395 396 <li id="0035"> 397 Subsection <a href="/tag/0035" data-tag="0035">1.3.5</a>: The Nerve of a Groupoid 398 399 400 401 <li id="00HE"> 402 Subsection <a href="/tag/00HE" data-tag="00HE">1.3.6</a>: The Homotopy Category of a Simplicial Set 403 404 405 406 <li id="00HJ"> 407 Subsection <a href="/tag/00HJ" data-tag="00HJ">1.3.7</a>: Example: The Path Category of a Directed Graph 408 409 410 </ul> 411 412 413 <li id="0039"> 414 Section <a href="/tag/0039" data-tag="0039">1.4</a>: $\infty $-Categories 415 416 417 418 <ul> 419 <li id="003F"> 420 Subsection <a href="/tag/003F" data-tag="003F">1.4.1</a>: Objects and Morphisms 421 422 423 424 <li id="003L"> 425 Subsection <a href="/tag/003L" data-tag="003L">1.4.2</a>: The Opposite of an $\infty $-Category 426 427 428 429 <li id="003U"> 430 Subsection <a href="/tag/003U" data-tag="003U">1.4.3</a>: Homotopies of Morphisms 431 432 433 434 <li id="0041"> 435 Subsection <a href="/tag/0041" data-tag="0041">1.4.4</a>: Composition of Morphisms 436 437 438 439 <li id="0049"> 440 Subsection <a href="/tag/0049" data-tag="0049">1.4.5</a>: The Homotopy Category of an $\infty $-Category 441 442 443 444 <li id="004Q"> 445 Subsection <a href="/tag/004Q" data-tag="004Q">1.4.6</a>: Isomorphisms 446 447 448 </ul> 449 450 451 <li id="0055"> 452 Section <a href="/tag/0055" data-tag="0055">1.5</a>: Functors of $\infty $-Categories 453 454 455 456 <ul> 457 <li id="0057">
458 Subsection <a href="/tag/0057" data-tag="0057">1.5.1</a>: Examples of Functors 459 460 461 462 <li id="005H"> 463 Subsection <a href="/tag/005H" data-tag="005H">1.5.2</a>: Commutative Diagrams 464 465 466 467 <li id="005Z"> 468 Subsection <a href="/tag/005Z" data-tag="005Z">1.5.3</a>: The $\infty $-Category of Functors 469 470 471 472 <li id="006B"> 473 Subsection <a href="/tag/006B" data-tag="006B">1.5.4</a>: Digression: Lifting Properties 474 475 476 477 <li id="006V"> 478 Subsection <a href="/tag/006V" data-tag="006V">1.5.5</a>: Trivial Kan Fibrations 479 480 481 482 <li id="0078"> 483 Subsection <a href="/tag/0078" data-tag="0078">1.5.6</a>: Uniqueness of Composition 484 485 486 487 <li id="00J3"> 488 Subsection <a href="/tag/00J3" data-tag="00J3">1.5.7</a>: Universality of Path Categories 489 490 491 </ul> 492 493 </ul> 494 495 496 <li id="007J"> 497 Chapter <a href="/tag/007J" data-tag="007J">2</a>: Examples of $\infty $-Categories 498 499 500 501 <ul> 502 <li id="00BL"> 503 Section <a href="/tag/00BL" data-tag="00BL">2.1</a>: Monoidal Categories 504 505 506 507 <ul> 508 <li id="00BT"> 509 Subsection <a href="/tag/00BT" data-tag="00BT">2.1.1</a>: Nonunital Monoidal Categories 510 511 512 513 <li id="00C2"> 514 Subsection <a href="/tag/00C2" data-tag="00C2">2.1.2</a>: Monoidal Categories 515 516 517 518 <li id="00CP"> 519 Subsection <a href="/tag/00CP" data-tag="00CP">2.1.3</a>: Examples of Monoidal Categories 520 521 522 523 <li id="00CV"> 524 Subsection <a href="/tag/00CV" data-tag="00CV">2.1.4</a>: Nonunital Monoidal Functors 525 526 527 528 <li id="00DD"> 529 Subsection <a href="/tag/00DD" data-tag="00DD">2.1.5</a>: Lax Monoidal Functors 530 531 532 533 <li id="00DX"> 534 Subsection <a href="/tag/00DX" data-tag="00DX">2.1.6</a>: Monoidal Functors 535 536 537 538 <li id="00ED"> 539 Subsection <a href="/tag/00ED" data-tag="00ED">2.1.7</a>: Enriched Category Theory 540 541 542 </ul> 543 544 545 <li id="007K"> 546 Section <a href="/tag/007K" data-tag="007K">2.2</a>: The Theory of Bicategories 547 548 549 550 <ul> 551 <li id="007P"> 552 Subsection <a href="/tag/007P" data-tag="007P">2.2.1</a>: Bicategories 553 554 555 556 <li id="0083"> 557 Subsection <a href="/tag/0083" data-tag="0083">2.2.2</a>: Examples of Bicategories 558 559 560 561 <li id="008A"> 562 Subsection <a href="/tag/008A" data-tag="008A">2.2.3</a>: Opposite and Conjugate Bicategories 563 564 565 566 <li id="008G"> 567 Subsection <a href="/tag/008G" data-tag="008G">2.2.4</a>: Functors of Bicategories 568 569 570 571 <li id="008V"> 572 Subsection <a href="/tag/008V" data-tag="008V">2.2.5</a>: The Category of Bicategories 573 574 575 576 <li id="0097"> 577 Subsection <a href="/tag/0097" data-tag="0097">2.2.6</a>: Isomorphisms of Bicategories 578 579 580 581 <li id="00FC"> 582 Subsection <a href="/tag/00FC" data-tag="00FC">2.2.7</a>: Strictly Unital Bicategories 583 584 585 586 <li id="02GD"> 587 Subsection <a href="/tag/02GD" data-tag="02GD">2.2.8</a>: The Homotopy Category of a Bicategory 588 589 590 </ul> 591 592 593 <li id="009P"> 594 Section <a href="/tag/009P" data-tag="009P">2.3</a>: The Duskin Nerve of a Bicategory 595 596 597 598 <ul> 599 <li id="009T"> 600 Subsection <a href="/tag/009T" data-tag="009T">2.3.1</a>: The Duskin Nerve 601 602 603 604 <li id="00AB"> 605 Subsection <a href="/tag/00AB" data-tag="00AB">2.3.2</a>: From Bicategories to $\infty $-Categories 606 607 608 609 <li id="00AQ"> 610 Subsection <a href="/tag/00AQ" data-tag="00AQ">2.3.3</a>: Thin $2$-Simplices of a Duskin Nerve 611 612 613 614 <li id="00AT"> 615 Subsection <a href="/tag/00AT" data-tag="00AT">2.3.4</a>: Recovering a Bicategory from its Duskin Nerve 616 617 618 619 <li id="00B9"> 620 Subsection <a href="/tag/00B9" data-tag="00B9">2.3.5</a>: The Duskin Nerve of a Strict Bicategory 621 622 623 </ul> 624 625 626 <li id="00JN">
627 Section <a href="/tag/00JN" data-tag="00JN">2.4</a>: Simplicial Categories 628 629 630 631 <ul> 632 <li id="00JQ"> 633 Subsection <a href="/tag/00JQ" data-tag="00JQ">2.4.1</a>: Simplicial Enrichment 634 635 636 637 <li id="00K4"> 638 Subsection <a href="/tag/00K4" data-tag="00K4">2.4.2</a>: Examples of Simplicial Categories 639 640 641 642 <li id="00KM"> 643 Subsection <a href="/tag/00KM" data-tag="00KM">2.4.3</a>: The Homotopy Coherent Nerve 644 645 646 647 <li id="00L0"> 648 Subsection <a href="/tag/00L0" data-tag="00L0">2.4.4</a>: The Path Category of a Simplicial Set 649 650 651 652 <li id="00LH"> 653 Subsection <a href="/tag/00LH" data-tag="00LH">2.4.5</a>: From Simplicial Categories to $\infty $-Categories 654 655 656 657 <li id="00LW"> 658 Subsection <a href="/tag/00LW" data-tag="00LW">2.4.6</a>: The Homotopy Category of a Simplicial Category 659 660 661 662 <li id="00MZ"> 663 Subsection <a href="/tag/00MZ" data-tag="00MZ">2.4.7</a>: Example: Braid Monoids 664 665 666 </ul> 667 668 669 <li id="00ND"> 670 Section <a href="/tag/00ND" data-tag="00ND">2.5</a>: Differential Graded Categories 671 672 673 674 <ul> 675 <li id="00NQ"> 676 Subsection <a href="/tag/00NQ" data-tag="00NQ">2.5.1</a>: Generalities on Chain Complexes 677 678 679 680 <li id="00P9"> 681 Subsection <a href="/tag/00P9" data-tag="00P9">2.5.2</a>: Differential Graded Categories 682 683 684 685 <li id="00PK"> 686 Subsection <a href="/tag/00PK" data-tag="00PK">2.5.3</a>: The Differential Graded Nerve 687 688 689 690 <li id="00SW"> 691 Subsection <a href="/tag/00SW" data-tag="00SW">2.5.4</a>: The Homotopy Category of a Differential Graded Category 692 693 694 695 <li id="00Q9"> 696 Subsection <a href="/tag/00Q9" data-tag="00Q9">2.5.5</a>: Digression: The Homology of Simplicial Sets 697 698 699 700 <li id="00QQ"> 701 Subsection <a href="/tag/00QQ" data-tag="00QQ">2.5.6</a>: The Dold-Kan Correspondence 702 703 704 705 <li id="00RF"> 706 Subsection <a href="/tag/00RF" data-tag="00RF">2.5.7</a>: The Shuffle Product 707 708 709 710 <li id="00S0"> 711 Subsection <a href="/tag/00S0" data-tag="00S0">2.5.8</a>: The Alexander-Whitney Construction 712 713 714 715 <li id="00SC"> 716 Subsection <a href="/tag/00SC" data-tag="00SC">2.5.9</a>: Comparison with the Homotopy Coherent Nerve 717 718 719 </ul> 720 721 </ul> 722 723 724 <li id="00SY"> 725 Chapter <a href="/tag/00SY" data-tag="00SY">3</a>: Kan Complexes 726 727 728 729 <ul> 730 <li id="00SZ"> 731 Section <a href="/tag/00SZ" data-tag="00SZ">3.1</a>: The Homotopy Theory of Kan Complexes 732 733 734 735 <ul> 736 <li id="00T0"> 737 Subsection <a href="/tag/00T0" data-tag="00T0">3.1.1</a>: Kan Fibrations 738 739 740 741 <li id="00UG"> 742 Subsection <a href="/tag/00UG" data-tag="00UG">3.1.2</a>: Anodyne Morphisms 743 744 745 746 <li id="00TJ"> 747 Subsection <a href="/tag/00TJ" data-tag="00TJ">3.1.3</a>: Exponentiation for Kan Fibrations 748 749 750 751 <li id="0219"> 752 Subsection <a href="/tag/0219" data-tag="0219">3.1.4</a>: Covering Maps 753 754 755 756 <li id="00TR"> 757 Subsection <a href="/tag/00TR" data-tag="00TR">3.1.5</a>: The Homotopy Category of Kan Complexes 758 759 760 761 <li id="01YX"> 762 Subsection <a href="/tag/01YX" data-tag="01YX">3.1.6</a>: The $\infty $-Category of Spaces 763 764 765 766 <li id="00U1"> 767 Subsection <a href="/tag/00U1" data-tag="00U1">3.1.7</a>: Homotopy Equivalences and Weak Homotopy Equivalences 768 769 770 771 <li id="00UU"> 772 Subsection <a href="/tag/00UU" data-tag="00UU">3.1.8</a>: Fibrant Replacement 773 774 775 </ul> 776 777 778 <li id="00V2"> 779 Section <a href="/tag/00V2" data-tag="00V2">3.2</a>: Homotopy Groups 780 781 782 783 <ul> 784 <li id="00V4"> 785 Subsection <a href="/tag/00V4" data-tag="00V4">3.2.1</a>: Pointed Kan Complexes 786 787 788 789 <li id="00VJ"> 790 Subsection <a href="/tag/00VJ" data-tag="00VJ">3.2.2</a>: The Homotopy Groups of a Kan Complex 791 792 793 794 <li id="00W3">
795 Subsection <a href="/tag/00W3" data-tag="00W3">3.2.3</a>: The Group Structure on $\pi _{n}(X,x)$ 796 797 798 799 <li id="02L4"> 800 Subsection <a href="/tag/02L4" data-tag="02L4">3.2.4</a>: Contractibility 801 802 803 804 <li id="00WD"> 805 Subsection <a href="/tag/00WD" data-tag="00WD">3.2.5</a>: The Connecting Homomorphism 806 807 808 809 <li id="00WL"> 810 Subsection <a href="/tag/00WL" data-tag="00WL">3.2.6</a>: The Long Exact Sequence of a Fibration 811 812 813 814 <li id="00WU"> 815 Subsection <a href="/tag/00WU" data-tag="00WU">3.2.7</a>: Whitehead's Theorem for Kan Complexes 816 817 818 819 <li id="00XB"> 820 Subsection <a href="/tag/00XB" data-tag="00XB">3.2.8</a>: Closure Properties of Homotopy Equivalences 821 822 823 </ul> 824 825 826 <li id="00XF"> 827 Section <a href="/tag/00XF" data-tag="00XF">3.3</a>: The $\operatorname{Ex}^{\infty }$ Functor 828 829 830 831 <ul> 832 <li id="00XT"> 833 Subsection <a href="/tag/00XT" data-tag="00XT">3.3.1</a>: Digression: Braced Simplicial Sets 834 835 836 837 <li id="00XH"> 838 Subsection <a href="/tag/00XH" data-tag="00XH">3.3.2</a>: The Subdivision of a Simplex 839 840 841 842 <li id="00Y5"> 843 Subsection <a href="/tag/00Y5" data-tag="00Y5">3.3.3</a>: The Subdivision of a Simplicial Set 844 845 846 847 <li id="00YR"> 848 Subsection <a href="/tag/00YR" data-tag="00YR">3.3.4</a>: The Last Vertex Map 849 850 851 852 <li id="00Z1"> 853 Subsection <a href="/tag/00Z1" data-tag="00Z1">3.3.5</a>: Comparison of $X$ with $\operatorname{Ex}(X)$ 854 855 856 857 <li id="00ZA"> 858 Subsection <a href="/tag/00ZA" data-tag="00ZA">3.3.6</a>: The $\operatorname{Ex}^{\infty }$ Functor 859 860 861 862 <li id="00ZM"> 863 Subsection <a href="/tag/00ZM" data-tag="00ZM">3.3.7</a>: Application: Characterizations of Weak Homotopy Equivalences 864 865 866 867 <li id="00ZR"> 868 Subsection <a href="/tag/00ZR" data-tag="00ZR">3.3.8</a>: Application: Extending Kan Fibrations 869 870 871 </ul> 872 873 874 <li id="0106"> 875 Section <a href="/tag/0106" data-tag="0106">3.4</a>: Homotopy Pullback and Homotopy Pushout Squares 876 877 878 879 <ul> 880 <li id="010K"> 881 Subsection <a href="/tag/010K" data-tag="010K">3.4.1</a>: Homotopy Pullback Squares 882 883 884 885 <li id="0112"> 886 Subsection <a href="/tag/0112" data-tag="0112">3.4.2</a>: Homotopy Pushout Squares 887 888 889 890 <li id="011H"> 891 Subsection <a href="/tag/011H" data-tag="011H">3.4.3</a>: Mather's Second Cube Theorem 892 893 894 895 <li id="011Q">
896 Subsection <a href="/tag/011Q" data-tag="011Q">3.4.4</a>: Mather's First Cube Theorem 897 898 899 900 <li id="0120"> 901 Subsection <a href="/tag/0120" data-tag="0120">3.4.5</a>: Digression: Weak Homotopy Equivalences of Semisimplicial Sets 902 903 904 905 <li id="012B"> 906 Subsection <a href="/tag/012B" data-tag="012B">3.4.6</a>: Excision 907 908 909 910 <li id="012K"> 911 Subsection <a href="/tag/012K" data-tag="012K">3.4.7</a>: The Seifert van-Kampen Theorem 912 913 914 </ul> 915 916 917 <li id="0513"> 918 Section <a href="/tag/0513" data-tag="0513">3.5</a>: Truncations and Postnikov Towers 919 920 921 922 <ul> 923 <li id="0517"> 924 Subsection <a href="/tag/0517" data-tag="0517">3.5.1</a>: Connectivity 925 926 927 928 <li id="051J"> 929 Subsection <a href="/tag/051J" data-tag="051J">3.5.2</a>: Connectivity as a Lifting Property 930 931 932 933 <li id="051Z"> 934 Subsection <a href="/tag/051Z" data-tag="051Z">3.5.3</a>: Coskeletal Simplicial Sets 935 936 937 938 <li id="052Q"> 939 Subsection <a href="/tag/052Q" data-tag="052Q">3.5.4</a>: Weakly Coskeletal Simplicial Sets 940 941 942 943 <li id="053L"> 944 Subsection <a href="/tag/053L" data-tag="053L">3.5.5</a>: Higher Groupoids 945 946 947 948 <li id="0543"> 949 Subsection <a href="/tag/0543" data-tag="0543">3.5.6</a>: Higher Fundamental Groupoids 950 951 952 953 <li id="054N"> 954 Subsection <a href="/tag/054N" data-tag="054N">3.5.7</a>: Truncated Kan Complexes 955 956 957 958 <li id="055K"> 959 Subsection <a href="/tag/055K" data-tag="055K">3.5.8</a>: The Postnikov Tower of a Kan Complex 960 961 962 963 <li id="055Y"> 964 Subsection <a href="/tag/055Y" data-tag="055Y">3.5.9</a>: Truncated Morphisms 965 966 967 </ul> 968 969 970 <li id="012Y"> 971 Section <a href="/tag/012Y" data-tag="012Y">3.6</a>: Comparison with Topological Spaces 972 973 974 975 <ul> 976 <li id="0130"> 977 Subsection <a href="/tag/0130" data-tag="0130">3.6.1</a>: Digression: Finite Simplicial Sets 978 979 980 981 <li id="013C"> 982 Subsection <a href="/tag/013C" data-tag="013C">3.6.2</a>: Exactness of Geometric Realization 983 984 985 986 <li id="013P"> 987 Subsection <a href="/tag/013P" data-tag="013P">3.6.3</a>: Weak Homotopy Equivalences in Topology 988 989 990 991 <li id="0141"> 992 Subsection <a href="/tag/0141" data-tag="0141">3.6.4</a>: The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$ 993 994 995 996 <li id="0147"> 997 Subsection <a href="/tag/0147" data-tag="0147">3.6.5</a>: Comparison of Homotopy Categories 998 999 1000 1001 <li id="021Q"> 1002 Subsection <a href="/tag/021Q" data-tag="021Q">3.6.6</a>: Serre Fibrations 1003 1004 1005 </ul> 1006 1007 </ul> 1008 1009 1010 <li id="01GP"> 1011 Chapter <a href="/tag/01GP" data-tag="01GP">4</a>: The Homotopy Theory of $\infty $-Categories 1012 1013 1014 1015 <ul> 1016 <li id="01B8"> 1017 Section <a href="/tag/01B8" data-tag="01B8">4.1</a>: Inner Fibrations 1018 1019 1020 1021 <ul> 1022 <li id="01B9"> 1023 Subsection <a href="/tag/01B9" data-tag="01B9">4.1.1</a>: Inner Fibrations of Simplicial Sets 1024 1025 1026 1027 <li id="01CD"> 1028 Subsection <a href="/tag/01CD" data-tag="01CD">4.1.2</a>: Subcategories of $\infty $-Categories 1029 1030 1031 1032 <li id="01BM"> 1033 Subsection <a href="/tag/01BM" data-tag="01BM">4.1.3</a>: Inner Anodyne Morphisms 1034 1035 1036 1037 <li id="01BS"> 1038 Subsection <a href="/tag/01BS" data-tag="01BS">4.1.4</a>: Exponentiation for Inner Fibrations 1039 1040 1041 1042 <li id="0226"> 1043 Subsection <a href="/tag/0226" data-tag="0226">4.1.5</a>: Inner Covering Maps 1044 1045 1046 </ul> 1047 1048 1049 <li id="014H"> 1050 Section <a href="/tag/014H" data-tag="014H">4.2</a>: Left and Right Fibrations 1051 1052 1053 1054 <ul> 1055 <li id="01GT"> 1056 Subsection <a href="/tag/01GT" data-tag="01GT">4.2.1</a>: Left and Right Fibrations of Simplicial Sets 1057 1058 1059 1060 <li id="0156"> 1061 Subsection <a href="/tag/0156" data-tag="0156">4.2.2</a>: Fibrations in Groupoids 1062 1063 1064 1065 <li id="022K">
1066 Subsection <a href="/tag/022K" data-tag="022K">4.2.3</a>: Left and Right Covering Maps 1067 1068 1069 1070 <li id="014S"> 1071 Subsection <a href="/tag/014S" data-tag="014S">4.2.4</a>: Left Anodyne and Right Anodyne Morphisms 1072 1073 1074 1075 <li id="0152"> 1076 Subsection <a href="/tag/0152" data-tag="0152">4.2.5</a>: Exponentiation for Left and Right Fibrations 1077 1078 1079 1080 <li id="00T8"> 1081 Subsection <a href="/tag/00T8" data-tag="00T8">4.2.6</a>: The Homotopy Extension Lifting Property 1082 1083 1084 </ul> 1085 1086 1087 <li id="015M"> 1088 Section <a href="/tag/015M" data-tag="015M">4.3</a>: The Slice and Join Constructions 1089 1090 1091 1092 <ul> 1093 <li id="015N"> 1094 Subsection <a href="/tag/015N" data-tag="015N">4.3.1</a>: Slices of Categories 1095 1096 1097 1098 <li id="0160"> 1099 Subsection <a href="/tag/0160" data-tag="0160">4.3.2</a>: Joins of Categories 1100 1101 1102 1103 <li id="016K"> 1104 Subsection <a href="/tag/016K" data-tag="016K">4.3.3</a>: Joins of Simplicial Sets 1105 1106 1107 1108 <li id="017E"> 1109 Subsection <a href="/tag/017E" data-tag="017E">4.3.4</a>: Joins of Topological Spaces 1110 1111 1112 1113 <li id="017V"> 1114 Subsection <a href="/tag/017V" data-tag="017V">4.3.5</a>: Slices of Simplicial Sets 1115 1116 1117 1118 <li id="018E"> 1119 Subsection <a href="/tag/018E" data-tag="018E">4.3.6</a>: Slices of $\infty $-Categories 1120 1121 1122 1123 <li id="018P"> 1124 Subsection <a href="/tag/018P" data-tag="018P">4.3.7</a>: Slices of Left and Right Fibrations 1125 1126 1127 </ul> 1128 1129 1130 <li id="01EM"> 1131 Section <a href="/tag/01EM" data-tag="01EM">4.4</a>: Isomorphisms and Isofibrations 1132 1133 1134 1135 <ul> 1136 <li id="01ER"> 1137 Subsection <a href="/tag/01ER" data-tag="01ER">4.4.1</a>: Isofibrations of $\infty $-Categories 1138 1139 1140 1141 <li id="019C"> 1142 Subsection <a href="/tag/019C" data-tag="019C">4.4.2</a>: Isomorphisms and Lifting Properties 1143 1144 1145 1146 <li id="01CZ"> 1147 Subsection <a href="/tag/01CZ" data-tag="01CZ">4.4.3</a>: The Core of an $\infty $-Category 1148 1149 1150 1151 <li id="01DF"> 1152 Subsection <a href="/tag/01DF" data-tag="01DF">4.4.4</a>: Natural Isomorphisms 1153 1154 1155 1156 <li id="01F0"> 1157 Subsection <a href="/tag/01F0" data-tag="01F0">4.4.5</a>: Exponentiation for Isofibrations 1158 1159 1160 </ul> 1161 1162 1163 <li id="01DE"> 1164 Section <a href="/tag/01DE" data-tag="01DE">4.5</a>: Equivalence 1165 1166 1167 1168 <ul> 1169 <li id="01DQ"> 1170 Subsection <a href="/tag/01DQ" data-tag="01DQ">4.5.1</a>: Equivalences of $\infty $-Categories 1171 1172 1173 1174 <li id="03T1"> 1175 Subsection <a href="/tag/03T1" data-tag="03T1">4.5.2</a>: Minimal $\infty $-Categories 1176 1177 1178 1179 <li id="032Y"> 1180 Subsection <a href="/tag/032Y" data-tag="032Y">4.5.3</a>: Categorical Pullback Squares 1181 1182 1183 1184 <li id="01E7"> 1185 Subsection <a href="/tag/01E7" data-tag="01E7">4.5.4</a>: Categorical Equivalence 1186 1187 1188 1189 <li id="01F6"> 1190 Subsection <a href="/tag/01F6" data-tag="01F6">4.5.5</a>: Categorical Pushout Squares 1191 1192 1193 1194 <li id="01FU"> 1195 Subsection <a href="/tag/01FU" data-tag="01FU">4.5.6</a>: Isofibrations of Simplicial Sets 1196 1197 1198 1199 <li id="0346"> 1200 Subsection <a href="/tag/0346" data-tag="0346">4.5.7</a>: Isofibrant Diagrams 1201 1202 1203 1204 <li id="02TM"> 1205 Subsection <a href="/tag/02TM" data-tag="02TM">4.5.8</a>: Relative Exponentiation 1206 1207 1208 </ul> 1209 1210 1211 <li id="01J3"> 1212 Section <a href="/tag/01J3" data-tag="01J3">4.6</a>: Morphism Spaces 1213 1214 1215 1216 <ul> 1217 <li id="01J4"> 1218 Subsection <a href="/tag/01J4" data-tag="01J4">4.6.1</a>: Morphism Spaces 1219 1220 1221 1222 <li id="01HM"> 1223 Subsection <a href="/tag/01HM" data-tag="01HM">4.6.2</a>: Digression: Universal Property of the Join 1224 1225 1226 1227 <li id="01K2"> 1228 Subsection <a href="/tag/01K2" data-tag="01K2">4.6.3</a>: Digression: Categorical Mapping Cylinders 1229 1230 1231 1232 <li id="01KE">
1233 Subsection <a href="/tag/01KE" data-tag="01KE">4.6.4</a>: Oriented Fiber Products 1234 1235 1236 1237 <li id="01KW"> 1238 Subsection <a href="/tag/01KW" data-tag="01KW">4.6.5</a>: Pinched Morphism Spaces 1239 1240 1241 1242 <li id="04HY"> 1243 Subsection <a href="/tag/04HY" data-tag="04HY">4.6.6</a>: Digression: Diagrams in Slice $\infty $-Categories 1244 1245 1246 1247 <li id="01LA"> 1248 Subsection <a href="/tag/01LA" data-tag="01LA">4.6.7</a>: Morphism Spaces in the Homotopy Coherent Nerve 1249 1250 1251 1252 <li id="01PF"> 1253 Subsection <a href="/tag/01PF" data-tag="01PF">4.6.8</a>: Composition of Morphisms 1254 1255 1256 </ul> 1257 1258 1259 <li id="05EA"> 1260 Section <a href="/tag/05EA" data-tag="05EA">4.7</a>: Truncated Objects and Higher Homotopy Categories 1261 1262 1263 1264 <ul> 1265 <li id="05EB"> 1266 Subsection <a href="/tag/05EB" data-tag="05EB">4.7.1</a>: Truncated Objects 1267 1268 1269 1270 <li id="05F8"> 1271 Subsection <a href="/tag/05F8" data-tag="05F8">4.7.2</a>: Truncated Morphisms 1272 1273 1274 1275 <li id="02H6"> 1276 Subsection <a href="/tag/02H6" data-tag="02H6">4.7.3</a>: Initial and Final Objects 1277 1278 1279 1280 <li id="04UX"> 1281 Subsection <a href="/tag/04UX" data-tag="04UX">4.7.4</a>: Discrete and Subterminal Objects 1282 1283 1284 1285 <li id="04VD"> 1286 Subsection <a href="/tag/04VD" data-tag="04VD">4.7.5</a>: Monomorphisms 1287 1288 1289 1290 <li id="0583"> 1291 Subsection <a href="/tag/0583" data-tag="0583">4.7.6</a>: Locally Truncated $\infty $-Categories 1292 1293 1294 1295 <li id="057D"> 1296 Subsection <a href="/tag/057D" data-tag="057D">4.7.7</a>: $n$-Categories 1297 1298 1299 1300 <li id="058P"> 1301 Subsection <a href="/tag/058P" data-tag="058P">4.7.8</a>: Minimality Conditions 1302 1303 1304 1305 <li id="0596"> 1306 Subsection <a href="/tag/0596" data-tag="0596">4.7.9</a>: Higher Homotopy Categories 1307 1308 1309 </ul> 1310 1311 1312 <li id="0578"> 1313 Section <a href="/tag/0578" data-tag="0578">4.8</a>: Full, Faithful, and Essentially Surjective Functors 1314 1315 1316 1317 <ul> 1318 <li id="01JG"> 1319 Subsection <a href="/tag/01JG" data-tag="01JG">4.8.1</a>: Essentially Surjective Functors 1320 1321 1322 1323 <li id="059U"> 1324 Subsection <a href="/tag/059U" data-tag="059U">4.8.2</a>: Full Functors 1325 1326 1327 1328 <li id="05AW"> 1329 Subsection <a href="/tag/05AW" data-tag="05AW">4.8.3</a>: Faithful Functors 1330 1331 1332 1333 <li id="06SQ"> 1334 Subsection <a href="/tag/06SQ" data-tag="06SQ">4.8.4</a>: Detecting Equivalences of $\infty $-Categories 1335 1336 1337 1338 <li id="05BY"> 1339 Subsection <a href="/tag/05BY" data-tag="05BY">4.8.5</a>: Categorically Connective Functors 1340 1341 1342 1343 <li id="06SY"> 1344 Subsection <a href="/tag/06SY" data-tag="06SY">4.8.6</a>: Digression: $n$-Categorical Fibrations 1345 1346 1347 1348 <li id="05CP"> 1349 Subsection <a href="/tag/05CP" data-tag="05CP">4.8.7</a>: Canonical Factorizations 1350 1351 1352 1353 <li id="05DG"> 1354 Subsection <a href="/tag/05DG" data-tag="05DG">4.8.8</a>: Categorically Connective Morphisms of Simplicial Sets 1355 1356 1357 </ul> 1358 1359 1360 <li id="03PP"> 1361 Section <a href="/tag/03PP" data-tag="03PP">4.9</a>: Size Conditions on $\infty $-Categories 1362 1363 1364 1365 <ul> 1366 <li id="03PV"> 1367 Subsection <a href="/tag/03PV" data-tag="03PV">4.9.1</a>: Ordinals and Well-Orderings 1368 1369 1370 1371 <li id="03QW"> 1372 Subsection <a href="/tag/03QW" data-tag="03QW">4.9.2</a>: Cardinals and Cardinality 1373 1374 1375 1376 <li id="03RB"> 1377 Subsection <a href="/tag/03RB" data-tag="03RB">4.9.3</a>: Small Sets 1378 1379 1380 1381 <li id="03S1"> 1382 Subsection <a href="/tag/03S1" data-tag="03S1">4.9.4</a>: Small Simplicial Sets 1383 1384 1385 1386 <li id="03SH"> 1387 Subsection <a href="/tag/03SH" data-tag="03SH">4.9.5</a>: Essential Smallness 1388 1389 1390 1391 <li id="03TJ"> 1392 Subsection <a href="/tag/03TJ" data-tag="03TJ">4.9.6</a>: Small Kan Complexes 1393 1394 1395 1396 <li id="03TN"> 1397 Subsection <a href="/tag/03TN" data-tag="03TN">4.9.7</a>: Local Smallness 1398 1399 1400 1401 <li id="047A">
1402 Subsection <a href="/tag/047A" data-tag="047A">4.9.8</a>: Small Fibrations 1403 1404 1405 </ul> 1406 1407 </ul> 1408 1409 1410 <li id="01J2"> 1411 Chapter <a href="/tag/01J2" data-tag="01J2">5</a>: Fibrations of $\infty $-Categories 1412 1413 1414 1415 <ul> 1416 <li id="01T1"> 1417 Section <a href="/tag/01T1" data-tag="01T1">5.1</a>: Cartesian Fibrations 1418 1419 1420 1421 <ul> 1422 <li id="01T4"> 1423 Subsection <a href="/tag/01T4" data-tag="01T4">5.1.1</a>: Cartesian Edges of Simplicial Sets 1424 1425 1426 1427 <li id="01TK"> 1428 Subsection <a href="/tag/01TK" data-tag="01TK">5.1.2</a>: Cartesian Morphisms of $\infty $-Categories 1429 1430 1431 1432 <li id="01TW"> 1433 Subsection <a href="/tag/01TW" data-tag="01TW">5.1.3</a>: Locally Cartesian Edges 1434 1435 1436 1437 <li id="01U9"> 1438 Subsection <a href="/tag/01U9" data-tag="01U9">5.1.4</a>: Cartesian Fibrations 1439 1440 1441 1442 <li id="01UW"> 1443 Subsection <a href="/tag/01UW" data-tag="01UW">5.1.5</a>: Locally Cartesian Fibrations 1444 1445 1446 1447 <li id="023L"> 1448 Subsection <a href="/tag/023L" data-tag="023L">5.1.6</a>: Fiberwise Equivalence 1449 1450 1451 1452 <li id="0280"> 1453 Subsection <a href="/tag/0280" data-tag="0280">5.1.7</a>: Equivalence of Inner Fibrations 1454 1455 1456 </ul> 1457 1458 1459 <li id="019A"> 1460 Section <a href="/tag/019A" data-tag="019A">5.2</a>: Covariant Transport 1461 1462 1463 1464 <ul> 1465 <li id="01VF"> 1466 Subsection <a href="/tag/01VF" data-tag="01VF">5.2.1</a>: Exponentiation for Cartesian Fibrations 1467 1468 1469 1470 <li id="019M"> 1471 Subsection <a href="/tag/019M" data-tag="019M">5.2.2</a>: Covariant Transport Functors 1472 1473 1474 1475 <li id="0241"> 1476 Subsection <a href="/tag/0241" data-tag="0241">5.2.3</a>: Example: The Relative Join 1477 1478 1479 1480 <li id="024M"> 1481 Subsection <a href="/tag/024M" data-tag="024M">5.2.4</a>: Fibrations over the $1$-Simplex 1482 1483 1484 1485 <li id="01VT"> 1486 Subsection <a href="/tag/01VT" data-tag="01VT">5.2.5</a>: The Homotopy Transport Representation 1487 1488 1489 1490 <li id="01Q7"> 1491 Subsection <a href="/tag/01Q7" data-tag="01Q7">5.2.6</a>: Elements of Set-Valued Functors 1492 1493 1494 1495 <li id="01QG"> 1496 Subsection <a href="/tag/01QG" data-tag="01QG">5.2.7</a>: Covering Space Theory 1497 1498 1499 1500 <li id="02RL"> 1501 Subsection <a href="/tag/02RL" data-tag="02RL">5.2.8</a>: Parametrized Covariant Transport 1502 1503 1504 </ul> 1505 1506 1507 <li id="035A"> 1508 Section <a href="/tag/035A" data-tag="035A">5.3</a>: Fibrations over Ordinary Categories 1509 1510 1511 1512 <ul> 1513 <li id="035D"> 1514 Subsection <a href="/tag/035D" data-tag="035D">5.3.1</a>: The Strict Transport Representation 1515 1516 1517 1518 <li id="035U"> 1519 Subsection <a href="/tag/035U" data-tag="035U">5.3.2</a>: Homotopy Colimits of Simplicial Sets 1520 1521 1522 1523 <li id="025W"> 1524 Subsection <a href="/tag/025W" data-tag="025W">5.3.3</a>: The Weighted Nerve 1525 1526 1527 1528 <li id="036V"> 1529 Subsection <a href="/tag/036V" data-tag="036V">5.3.4</a>: Scaffolds of Cocartesian Fibrations 1530 1531 1532 1533 <li id="037H"> 1534 Subsection <a href="/tag/037H" data-tag="037H">5.3.5</a>: Application: Classification of Cocartesian Fibrations 1535 1536 1537 1538 <li id="037W"> 1539 Subsection <a href="/tag/037W" data-tag="037W">5.3.6</a>: Application: Relative Exponentials 1540 1541 1542 1543 <li id="01US"> 1544 Subsection <a href="/tag/01US" data-tag="01US">5.3.7</a>: Application: Path Fibrations 1545 1546 1547 1548 <li id="05PM"> 1549 Subsection <a href="/tag/05PM" data-tag="05PM">5.3.8</a>: Digression: Minimal Fibrations 1550 1551 1552 </ul> 1553 1554 1555 <li id="01W4"> 1556 Section <a href="/tag/01W4" data-tag="01W4">5.4</a>: $\infty $-Bicategories 1557 1558 1559 1560 <ul> 1561 <li id="01W6"> 1562 Subsection <a href="/tag/01W6" data-tag="01W6">5.4.1</a>: Definitions 1563 1564 1565 1566 <li id="01WF"> 1567 Subsection <a href="/tag/01WF" data-tag="01WF">5.4.2</a>: Interior Fibrations 1568 1569 1570 1571 <li id="01WT">
1572 Subsection <a href="/tag/01WT" data-tag="01WT">5.4.3</a>: Slices of $\infty $-Bicategories 1573 1574 1575 1576 <li id="01X7"> 1577 Subsection <a href="/tag/01X7" data-tag="01X7">5.4.4</a>: The Local Thinness Criterion 1578 1579 1580 1581 <li id="01XA"> 1582 Subsection <a href="/tag/01XA" data-tag="01XA">5.4.5</a>: The Pith of an $\infty $-Bicategory 1583 1584 1585 1586 <li id="01XQ"> 1587 Subsection <a href="/tag/01XQ" data-tag="01XQ">5.4.6</a>: The Four-out-of-Five Property 1588 1589 1590 1591 <li id="01Y4"> 1592 Subsection <a href="/tag/01Y4" data-tag="01Y4">5.4.7</a>: Functors of $\infty $-Bicategories 1593 1594 1595 1596 <li id="01YK"> 1597 Subsection <a href="/tag/01YK" data-tag="01YK">5.4.8</a>: Strict $\infty $-Bicategories 1598 1599 1600 1601 <li id="02RX"> 1602 Subsection <a href="/tag/02RX" data-tag="02RX">5.4.9</a>: Comparison of Homotopy Transport Representations 1603 1604 1605 </ul> 1606 1607 1608 <li id="01YV"> 1609 Section <a href="/tag/01YV" data-tag="01YV">5.5</a>: The $\infty $-Category $\operatorname{\mathcal{QC}}$ 1610 1611 1612 1613 <ul> 1614 <li id="01Z5"> 1615 Subsection <a href="/tag/01Z5" data-tag="01Z5">5.5.1</a>: Digression: Slicing and the Homotopy Coherent Nerve 1616 1617 1618 1619 <li id="01ZV"> 1620 Subsection <a href="/tag/01ZV" data-tag="01ZV">5.5.2</a>: The $\infty $-Category of Pointed Spaces 1621 1622 1623 1624 <li id="0208"> 1625 Subsection <a href="/tag/0208" data-tag="0208">5.5.3</a>: The $\infty $-Category of $\infty $-Categories 1626 1627 1628 1629 <li id="020J"> 1630 Subsection <a href="/tag/020J" data-tag="020J">5.5.4</a>: The $\infty $-Bicategory of $\infty $-Categories 1631 1632 1633 1634 <li id="020S"> 1635 Subsection <a href="/tag/020S" data-tag="020S">5.5.5</a>: $\infty $-Categories with a Distinguished Object 1636 1637 1638 </ul> 1639 1640 1641 <li id="027M"> 1642 Section <a href="/tag/027M" data-tag="027M">5.6</a>: Classification of Cocartesian Fibrations 1643 1644 1645 1646 <ul> 1647 <li id="01R1"> 1648 Subsection <a href="/tag/01R1" data-tag="01R1">5.6.1</a>: Elements of Category-Valued Functors 1649 1650 1651 1652 <li id="026H"> 1653 Subsection <a href="/tag/026H" data-tag="026H">5.6.2</a>: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors 1654 1655 1656 1657 <li id="0273"> 1658 Subsection <a href="/tag/0273" data-tag="0273">5.6.3</a>: Comparison with the Category of Elements 1659 1660 1661 1662 <li id="0279"> 1663 Subsection <a href="/tag/0279" data-tag="0279">5.6.4</a>: Comparison with the Weighted Nerve 1664 1665 1666 1667 <li id="028U"> 1668 Subsection <a href="/tag/028U" data-tag="028U">5.6.5</a>: The Universality Theorem 1669 1670 1671 1672 <li id="02HU"> 1673 Subsection <a href="/tag/02HU" data-tag="02HU">5.6.6</a>: Application: Corepresentable Functors 1674 1675 1676 1677 <li id="029B"> 1678 Subsection <a href="/tag/029B" data-tag="029B">5.6.7</a>: Application: Extending Cocartesian Fibrations 1679 1680 1681 1682 <li id="02S9"> 1683 Subsection <a href="/tag/02S9" data-tag="02S9">5.6.8</a>: Transport Witnesses 1684 1685 1686 1687 <li id="02SM"> 1688 Subsection <a href="/tag/02SM" data-tag="02SM">5.6.9</a>: Proof of the Universality Theorem 1689 1690 1691 </ul> 1692 1693 </ul> 1694 1695 1696 <li id="04QH"> 1697 Chapter <a href="/tag/04QH" data-tag="04QH">10</a>: Exactness and Animation 1698 1699 1700 1701 <ul> 1702 <li id="04QJ"> 1703 Section <a href="/tag/04QJ" data-tag="04QJ">10.1</a>: Simplicial Objects of $\infty $-Categories 1704 1705 1706 1707 <ul> 1708 <li id="04QQ"> 1709 Subsection <a href="/tag/04QQ" data-tag="04QQ">10.1.1</a>: Geometric Realization 1710 1711 1712 1713 <li id="04R8"> 1714 Subsection <a href="/tag/04R8" data-tag="04R8">10.1.2</a>: Semisimplicial Objects 1715 1716 1717 1718 <li id="05G3"> 1719 Subsection <a href="/tag/05G3" data-tag="05G3">10.1.3</a>: Skeletal Simplicial Objects 1720 1721 1722 1723 <li id="05GU"> 1724 Subsection <a href="/tag/05GU" data-tag="05GU">10.1.4</a>: Coskeletal Simplicial Objects 1725 1726 1727 1728 <li id="04SV"> 1729 Subsection <a href="/tag/04SV" data-tag="04SV">10.1.5</a>: The Nerve of a Morphism 1730 1731 1732 1733 <li id="04S6">
1734 Subsection <a href="/tag/04S6" data-tag="04S6">10.1.6</a>: Split Simplicial Objects 1735 1736 1737 </ul> 1738 1739 1740 <li id="04TM"> 1741 Section <a href="/tag/04TM" data-tag="04TM">10.2</a>: Regular $\infty $-Categories 1742 1743 1744 1745 <ul> 1746 <li id="04TZ"> 1747 Subsection <a href="/tag/04TZ" data-tag="04TZ">10.2.1</a>: Sieves 1748 1749 1750 1751 <li id="04WG"> 1752 Subsection <a href="/tag/04WG" data-tag="04WG">10.2.2</a>: Quotient Morphisms 1753 1754 1755 1756 <li id="04X1"> 1757 Subsection <a href="/tag/04X1" data-tag="04X1">10.2.3</a>: Images 1758 1759 1760 1761 <li id="04XV"> 1762 Subsection <a href="/tag/04XV" data-tag="04XV">10.2.4</a>: Universal Quotient Morphisms 1763 1764 1765 1766 <li id="04YE"> 1767 Subsection <a href="/tag/04YE" data-tag="04YE">10.2.5</a>: Regular $\infty $-Categories 1768 1769 1770 </ul> 1771 1772 </ul> 1773 1774 1775 <li id="00FN"> 1776 Chapter <a href="/tag/00FN" data-tag="00FN">11</a>: Retired Tags 1777 1778 1779 1780 <ul> 1781 <li id="00MB"> 1782 Section <a href="/tag/00MB" data-tag="00MB">11.1</a>: Expanded Tags 1783 1784 1785 1786 <li id="03DT"> 1787 Section <a href="/tag/03DT" data-tag="03DT">11.2</a>: Discarded Diagrams 1788 1789 1790 1791 <li id="00MC"> 1792 Section <a href="/tag/00MC" data-tag="00MC">11.3</a>: Tags without Context 1793 1794 1795 1796 <li id="00MD"> 1797 Section <a href="/tag/00MD" data-tag="00MD">11.4</a>: Obsolete Constructions 1798 1799 1800 1801 <li id="01NH"> 1802 Section <a href="/tag/01NH" data-tag="01NH">11.5</a>: Tags Lost in Reorganization 1803 1804 1805 1806 <li id="00ME"> 1807 Section <a href="/tag/00ME" data-tag="00ME">11.6</a>: Reclassified Tags 1808 1809 1810 1811 <li id="00MF"> 1812 Section <a href="/tag/00MF" data-tag="00MF">11.7</a>: Retired Sections 1813 1814 1815 1816 <li id="01W2"> 1817 Section <a href="/tag/01W2" data-tag="01W2">11.8</a>: Temporarily Removed Tags 1818 1819 1820 1821 <li id="02QR"> 1822 Section <a href="/tag/02QR" data-tag="02QR">11.9</a>: Retired Subsections 1823 1824 1825 1826 <ul> 1827 <li id="01HF"> 1828 Subsection <a href="/tag/01HF" data-tag="01HF">11.9.1</a>: Detecting Equivalences of $\infty $-Categories 1829 1830 1831 1832 <li id="05NY"> 1833 Subsection <a href="/tag/05NY" data-tag="05NY">11.9.3</a>: Obsolete Discussion of Balanced Couplings 1834 1835 1836 1837 <li id="04GY"> 1838 Subsection <a href="/tag/04GY" data-tag="04GY">11.9.4</a>: Connectivity of Morphisms 1839 1840 1841 1842 <li id="03KC"> 1843 Subsection <a href="/tag/03KC" data-tag="03KC">11.9.5</a>: The Universal Property of Twisted Arrows 1844 1845 1846 1847 <li id="03P9"> 1848 Subsection <a href="/tag/03P9" data-tag="03P9">11.9.6</a>: Adjunctions as Profunctors 1849 1850 1851 1852 <li id="0194"> 1853 Subsection <a href="/tag/0194" data-tag="0194">11.9.7</a>: Towers of Kan Fibrations 1854 1855 1856 1857 <li id="024T"> 1858 Subsection <a href="/tag/024T" data-tag="024T">11.9.8</a>: Classification of Fibrations 1859 1860 1861 1862 <li id="029L"> 1863 Subsection <a href="/tag/029L" data-tag="029L">11.9.9</a>: Proof of the Universality Theorem 1864 1865 1866 1867 <li id="01N5"> 1868 Subsection <a href="/tag/01N5" data-tag="01N5">11.9.10</a>: Stability Properties of Localizations 1869 1870 1871 1872 <li id="01N8"> 1873 Subsection <a href="/tag/01N8" data-tag="01N8">11.9.11</a>: The Category of Simplices 1874 1875 1876 </ul> 1877 1878 1879 <li id="01W3"> 1880 Section <a href="/tag/01W3" data-tag="01W3">11.10</a>: Temporarily Retired Subsections 1881 1882 1883 1884 <ul> 1885 <li id="02GE"> 1886 Subsection <a href="/tag/02GE" data-tag="02GE">11.10.1</a>: Notational Junk 1887 1888 1889 1890 <li id="01S6"> 1891 Subsection <a href="/tag/01S6" data-tag="01S6">11.10.2</a>: The Transport Representation 1892 1893 1894 1895 <li id="02AJ"> 1896 Subsection <a href="/tag/02AJ" data-tag="02AJ">11.10.3</a>: Classical Stuff 1897 1898 1899 1900 <li id="02AK"> 1901 Subsection <a href="/tag/02AK" data-tag="02AK">11.10.4</a>: Classical Grothendieck Construction for Lax Functors 1902 1903 1904 1905 <li id="02AN">
1906 Subsection <a href="/tag/02AN" data-tag="02AN">11.10.5</a>: Explicit Transport for Set-Valued Functors 1907 1908 1909 1910 <li id="01AA"> 1911 Subsection <a href="/tag/01AA" data-tag="01AA">11.10.6</a>: Relative Homotopy Equivalences 1912 1913 1914 1915 <li id="01AP"> 1916 Subsection <a href="/tag/01AP" data-tag="01AP">11.10.7</a>: Covariant and Contravariant Equivalences 1917 1918 1919 </ul> 1920 1921 </ul> 1922 1923 1924 </ul> 1925
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