Notes on homotopical algebra

Updates

• 10 Nov 2015: Minor corrections and clarifications, typographical fixes.
• 19 Sep 2015: Corrections to the definition of ‘homotopical calculus of right/left fractions’ and related proofs; correction to the construction of free algebras for pointed endofunctors; other minor corrections.
• 20 Jul 2015: Minor corrections and additions; new formatting.
• 20 Feb 2015: Correction to the statement and proof of the unbounded acyclic assembly lemma, added the fact that the realisation of a simplicial simplicial object is its diagonal.
• 6 Feb 2015: Added some information about the tensor product of chain complexes; correction to the proof of the main theorem of calculus of correspondences; other minor fixes.
• 15 Jan 2015: Added a section about double complexes; added a section about simplicial objects in abelian categories; other minor fixes.
• 16 Dec 2014: Started a section about the calculus of correspondences.
• 12 Dec 2014: Added a section about inverse diagrams in Cisinski fibration categories; fixed the definition of basic fibration category; added Cisinski's asphericity criterion; other additions and corrections.
• 8 Dec 2014: Started a new chapter on categories with fibrations; reorganised other material; other additions and corrections.
• 29 Nov 2014: Added a section about filtered complexes; other corrections.
• 21 Nov 2014: Added a section about the derived category of an abelian category.
• 15 Nov 2014: Started a new chapter on homological algebra; added a section about abelian categories; other minor fixes.
• 8 Nov 2014: Added a new section on the rectification of homotopy-coherent diagrams; removed the old section on homotopy-coherent diagrams; other additions and corrections.
• 31 Oct 2014: Added some information about simplicially enriched Bousfield–Kan extensions.
• 24 Oct 2014: Rewrote the section on homotopy (co)limits in Kan-enriched categories; inserted a section on homotopy (co)ends in Kan-enriched categories; reordered the sections in the chapter on simplicial categories; other additions and corrections.
• 8 Oct 2014: Added more details about the construction of the Bergner model structure; inserted a section on Kan-enriched categories; other additions and corrections.
• 29 Aug 2014: Improved the proof of the Dwyer–Kan theorem; other additions and corrections.
• 27 Aug 2014: Introduced calculi of cocycles; fixed the proof of the fundamental theorem of homotopical three-arrow calculi; other minor corrections.
• 22 Aug 2014: Split off a section on homotopical calculi of fractions; other additions.
• 19 Aug 2014: Added a proof of Quillen's Theorem B; other additions.
• 18 Aug 2014: Inserted a new section on the Dwyer–Kan theorem comparing homotopy function complexes and the hammock localisation; added Thomason's homotopy colimit theorem and more information about Grothendieck fibrations; other minor fixes.
• 8 Aug 2014: Inserted a new section on Bousfield–Kan extensions; added some results on maximal deformation retracts; other minor corrections.
• 5 Aug 2014: Changed the definitions of resolvable category and (co)simplicial resolution model structure; fixed the definition of morphism of zigzag types.
• 22 Jul 2014: Rewrote the sections on Reedy model structures and derived hom-spaces; other minor fixes.
• 9 Jul 2014: Added some justification for the standard simplicial localisation; added a section on proper model categories; other minor changes.
• 2 Jul 2014: Added the Bousfield–Kan comparison theorem; added a proof of the product–descent formula for homotopy limits; added some remarks on the fundamental category of a graph; corrections to the proof concerning Quillen's subdivision functor; other minor changes.
• 26 Jun 2014: Added a new section on the Dwyer–Kan and Bergner model structures; added a proof that the category of simplices functor is the left half of an adjoint homotopical equivalence; replaced Hirschhorn's left Bousfield localisation theorem with Smith's; other minor changes.
• 17 Jun 2014: Split off a section on Kan's Sd and Ex functors; added Cisinski's proof that Kan's Ex functor is a homotopical equivalence; added the fact that the enriched simplicial homotopy category of a homotopy cocomplete Kan-enriched category is cotensored.
• 14 Jun 2014: Added a new section on the homotopy theory of nerves; added a new section on homotopy limits and colimits in Kan-enriched categories; added more on the Dwyer–Kan theory of simplicially enriched categories; other minor changes.
• 26 May 2014: Added new sections on enriched categories; added some information about fibrant replacement of simplicially enriched categories; reordered sections in the chapter on simplicial categories; other minor changes.
• 15 May 2014: Replaced the section on homotopy (co)limits of simplicial sets with a section on Bousfield–Kan (co)limits; other minor changes.
• 9 May 2014: Added information about the calculus of right fractions for categories of fibrant objects; added the fact that the Kan–Quillen model structure is a left Bousfield localisation of the Joyal model structure; other minor changes.
• 7 May 2014: Added another version of Verdier's hypercovering theorem; other minor changes.
• 2 May 2014: Started a new section on Verdier's hypercovering theorem; added a proof that the cofibations in the injective model structure for strong stacks are the monomorphisms; other minor fixes.
• 25 Apr 2014: Added some information about stacks of ∞-groupoids; added some transfer theorems for model structures; other minor fixes.
• 24 Apr 2014: Added a section about hypercovers; added a section about descent and hypersheaves; added information about the Blander model structure on simplicial presheaves; other minor fixes.
• 18 Apr 2014: Added some information about the Jardine model structure on simplicial presheaves.
• 16 Apr 2014: Added a section on the Heller–Joyal model structure on simplicial objects in a Grothendieck topos; added the definition of internal homotopy groups of internal Kan complexes.
• 11 Apr 2014: Reorganised and added new material on simplicial homotopy theory, including the definition of homotopy groups.
• 8 Apr 2014: Added some information about Kan's Ex^∞ functor, Dugger–Isaksen weak equivalences, and internal fibrant replacement.
• 11 Mar 2014: Clarified the proof of the universal property of cofibrant replacement.
• 15 Feb 2014: Fixed a gap in the proof of the theorem on accessible model structures; other minor corrections.
• 15 Jan 2014: Started a section about Bousfield localisation; added the definition of cellular combinatorial model category; added the definition of proper model category; other minor corrections.
• 30 Dec 2013: Started a chapter about homotopy toposes; added a section about familial regularity and exactness; added some material on simplicial objects.
• 20 Dec 2013: Started a section about categories of fibrant objects; added a proof of Joyal's determination principle for derivable categories; other minor corrections.
• 18 Dec 2013: Started a section about simplicial model categories; added a theorem on the existence of filtered colimits in accessible categories; other minor corrections.
• 26 Nov 2013: Inserted a new section about algebraic model categories; added more material about combinatorial model categories, including a proof that they are stable under universe enlargement.
• 15 Nov 2013: Added a new section about simplicial localisations; inserted a new section about accessible constructions; added some remarks about the accessibility of weak equivalences and fibrations in a combinatorial model category.
• 7 Oct 2013: Weakened the definition of ‘derivable category’; added the reflective localisation theorem; added some remarks on Cisinski model categories.
• 2 Oct 2013: Added a new section about two-arrow calculi; introduced the special 2-out-of-4 property; stated the universal property of cofibrant/fibrant replacements.
• 27 Sep 2013: Weakened the definition of ‘left/right deformation retract’, changed proofs for the non-functorial case; other minor fixes.
• 24 Sep 2013: Introduced derivable categories; strengthened the definition of ‘total derived functor’; added more details about derived hom-spaces.
• 20 Sep 2013: Added a new section about framings and resolutions and a section about derived hom-spaces; split off a chapter on topics on model categories.
• 12 Sep 2013: Started a new chapter on quasicategories; added a section about homotopy-coherent diagrams in simplicially enriched categories; fixed an error in the definition of ‘class of I-anodyne extensions’.
• 4 Sep 2013: Added some observations about cartesian model categories; other small fixes.
• 9 Aug 2013: Added some information about Lawvere cylinders.
• 8 Aug 2013: Added a new section about Cisinski model structures.
• 6 Aug 2013: Identified the minimal basic localiser as the weak homotopy equivalences; other small fixes.
• 5 Aug 2013: Added new sections about the bar construction and homotopy colimits in simplicial sets; started new chapter on simplicial categories.
• 29 Jul 2013: Added some information about the minimal basic localiser.
• 17 Jul 2013: Added more details about basic localisers; reversed the chirality in the terminology; other small fixes.
• 16 Jul 2013: Fixed a mistake in the definition of ‘basic localiser’; added a remark on the stability of the homotopy category under universe enlargement; added a comment on the accessibility of fibrations; other small fixes.
• 5 May 2013: Inserted a new section about virtually cofibrant and fibrant diagrams; added a new section about homotopy colimits and limits; added more material about deformation retracts and derived functors.
• 1 May 2013: Reorganised section about derivators, corrected some typos.
• 29 Apr 2013: Started a new chapter about derivators; added more material about conjugate pairs of natural transformations and exact squares.
• 27 Apr 2013: Split off a section about the Beck–Chevalley condition; added a section about three-arrow calculi.
• 24 Apr 2013: Fixed more typos in the Foundations chapter; added a section about Reedy diagrams and a section about monoidal model categories.
• 17 Apr 2013: Fixed some typos in the Foundations chapter; fixed the proof of the ternary factorisation system characterisation of model structures.
• 15 Apr 2013: Added more material on model categories, including a section on Quillen functors and combinatorial model categories.
• 12 Apr 2013: Fixed some typos; added material on homotopically initial/terminal objects and DHKS derived functors.
• 5 Apr 2013: Fixed the definition of ‘category with weak equivalences’; added material about Quillen–Verdier derived functors, plus some additional material on factorisation systems.
• 2 Apr 2013: Fixed a typo in the definition of ‘accessible extension’; added material about small object arguments and cofibrantly-generated factorisation systems.
• 31 Mar 2013: Fixed some typos and added more material about functorial factorisation systems.
• 29 Mar 2013: A correction regarding compact objects in functor categories, added material about cartesian closed categories and intrinsic homotopy.
• 18 Mar 2013: Added more material about change-of-universe, Kan extensions, weighted limits/colimits, ends/coends, and the Kan–Quillen model structure.
• 2 Mar 2013: Replaced a dangling reference to Makkai–Paré with a weaker lemma.
• 1 Mar 2013: Added some material about change-of-universe.
• 26 Feb 2013: Some corrections regarding relative categories; added some material about locally presentable categories.
• 14 Feb 2013: Completed up to the construction of the Quillen homotopy category of a model category.