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169    <li class="active">Part <a href="/tag/0000" data-tag="0000">1</a>: Foundations
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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
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271  <h1 id="Part:Foundations" class="tex-part"><span data-tag="0000">1</span> Foundations</h1>
272
273
274
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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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