Notes on homotopical algebra Z.L. Low 20th September 2013 P These notes are intended as a kind of annotated index to the various standard references in homotopical algebra: the focus is on definitions and statements of results, not proofs. i C Foundations .. Set theory ............................. .. Accessibility and ind-completions . .. Change of universe ......................... .. Small object arguments ...................... Simplicial sets .. Basics ............................... .. Nerves, skeletons, and coskeletons . .. The Kan–Quillen model structure . .. Intrinsic homotopy ......................... .. Bisimplicial sets and cosimplicial simplicial sets . .. Bar and cobar complexes ..................... .. Homotopy limits and colimits . Simplicial categories .. Basics ............................... .. Homotopical aspects . .. Simplicial and cosimplicial objects . .. Homotopy-coherent diagrams . iii C Homotopical categories .. Basics ............................... .. Homotopical Kan extensions . .. Quillen–Verdier derived functors . .. DHKS derived functors . .. Three-arrow calculi . Model categories .. Basics ............................... .. Left and right homotopy . .. The homotopy category . .. Quillen functors . .. Reedy diagrams . .. Virtual cofibrancy and fibrancy . .. Framings and resolutions . .. Derived hom-spaces . Topics in model categories .. Combinatorial model categories . .. Cisinski model categories . .. Monoidal model categories . Quasicategories .. Basics ............................... .. The Joyal model structure . Derivators .. Basics ............................... .. Homotopy limits and colimits . .. Basic localisers . .. The minimal basic localiser . iv Contents Generalities .. Adjoints and mates . .. Cartesian closed categories . .. Factorisation systems . .. Relative categories . .. Kan extensions . .. Ends and coends . Higher generalities .. Monoidal categories . .. Categories with actions . Bibliography Index v 0 F . Set theory In category theory it is often convenient to invoke a certain set-theoretic device commonly known as a ‘Grothendieck universe’, but we shall say simply ‘uni- verse’, so as to simplify exposition and proofs by eliminating various circumlo- cutions involving cardinal bounds, proper classes etc. Definition ... A pre-universe is a set 퐔 satisfying these axioms: . If 푥 ∈ 푦 and 푦 ∈ 퐔, then 푥 ∈ 퐔. If 푥 ∈ 퐔 and 푦 ∈ 퐔 (but not necessarily distinct), then {푥, 푦} ∈ 퐔. If 푥 ∈ 퐔, then 풫 (푥) ∈ 퐔, where 풫 (푥) denotes the set of all subsets of 푥. If 푥 ∈ 퐔 and 푓 : 푥 → 퐔 is a map, then ⋃푖∈푥 푓(푖) ∈ 퐔. A universe is a pre-universe 퐔 with this additional property: . 휔 ∈ 퐔, where 휔 is the set of all finite (von Neumann) ordinals. Example ... The empty set is a pre-universe, and with very mild assumptions, so is the set 퐇퐅 of all hereditarily finite sets. ¶ ... The notion of universe makes sense in any material set theory, but their existence must be postulated. We adopt the following: • Grothendieck–Verdier universe axiom. For each set 푥, there exists a universe 퐔 with 푥 ∈ 퐔. 0. F For definiteness, we may take our base theory to be Mac Lane set theory, which is a weak subsystem of Zermelo–Fraenkel set theory with choice (ZFC). Readers interested in the details of Mac Lane set theory are referred to [Mathias, 2001], but in practice, as long as one is working at all times inside some universe, one may as well be working in ZFC. Indeed: Proposition ... With the assumptions of Mac Lane set theory, any universe is a transitive model of ZFC. Proof. Let 퐔 be a universe. By definition, 퐔 is a transitive set containing pairs, power sets, unions, and 휔, so the axioms of extensionality, empty set, pairs, power sets, unions, choice, and infinity are all automatically satisfied. Wemust show that the axiom schemas of separation and replacement are also satisfied, and in fact it is enough to check that replacement is valid; but this is straightfor- ward using axioms and . ■ Definition ... Let 퐔 be a pre-universe. A 퐔-set is a member of 퐔, a 퐔-class is a subset of 퐔, and a proper 퐔-class is a 퐔-class that is not a 퐔-set. Lemma ... A 퐔-class 푋 is a 퐔-set if and only if there exists a 퐔-class 푌 such that 푋 ∈ 푌 . ■ Proposition ... If 퐔 is a universe in Mac Lane set theory, then the collection of all 퐔-classes is a transitive model of Morse–Kelley class–set theory (MK), and so is a transitive model of von Neumann–Bernays–Gödel class–set theory (NBG) in particular. ■ Definition ... A 퐔-small category is a category ℂ such that ob ℂ and mor ℂ are 퐔-sets. A locally 퐔-small category is a category 풟 satisfying these condi- tions: • ob 풟 and mor 풟 are 퐔-classes, and • for all objects 푥 and 푦 in 풟, the hom-set 풟(푥, 푦) is a 퐔-set. An essentially 퐔-small category is a category 풟 for which there exist a 퐔-small category ℂ and a functor ℂ → 풟 that is fully faithful and essentially surjective on objects. Proposition ... If 픻 is a 퐔-small category and 풞 is a locally 퐔-small cat- egory, then the functor category [픻, 풞] is locally 퐔-small. .. Set theory Proof. Strictly speaking, this depends on the set-theoretic implementation of ordered pairs, categories, functors, etc., but at the very least [픻, 풞] should be isomorphic to a locally 퐔-small category. In the context of [픻, 풞], we may regard functors 픻 → 풞 as being the pair consisting of the graph of the object map ob 픻 → ob 풞 and the graph of the morphism map mor 픻 → mor 풞, and these are 퐔-sets by the 퐔-replacement axiom. Similarly, if 퐹 and 퐺 are objects in [픻, 풞], then we may regard a natural transformation 훼 : 퐹 ⇒ 퐺 as being the triple (퐹 , 퐺, 퐴), where 퐴 is the set of all pairs (푐, 훼푐). ■ One complication introduced by having multiple universes concerns the ex- istence of (co)limits. Theorem .. (Freyd). Let 풞 be a category and let 휅 be a cardinal such that |mor 풞| ≤ 휅. If 풞 has products for families of size 휅, then any two parallel morphisms in 풞 must be equal. Proof. Suppose, for a contradiction, that 푓, 푔 : 푋 → 푌 are distinct morphisms in 풞. Let 푍 be the product of 휅-many copies of 푌 in 풞. The universal property of products implies there are at least 2휅-many distinct morphisms 푋 → 푍; but 풞(푋, 푍) ⊆ mor 풞, so this is an absurdity. ■ Definition ... Let 퐔 be a pre-universe. A 퐔-complete (resp. 퐔-cocomplete) category is a category 풞 with the following property: • For all 퐔-small categories 픻 and all diagrams 퐴 : 픻 → 풞, a limit (resp. colimit) of 퐴 exists in 풞. We may instead say 풞 has all finite limits (resp. finite colimits) in the special case 퐔 = 퐇퐅. Proposition ... Let 풞 be a category and let 퐔 be a non-empty pre-universe. The following are equivalent: (i) 풞 is 퐔-complete. (ii) 풞 has all finite limits and products for all families of objects indexed bya 퐔-set. 0. F (iii) For each 퐔-small category 픻, there exists an adjunction Δ ⊣ lim : [픻, 풞] → 풞 ←−픻 where Δ푋 is the constant functor with value 푋. Dually, the following are equivalent: (i′) 풞 is 퐔-cocomplete. (ii′) 풞 has all finite colimits and coproducts for all families of objects indexed by a 퐔-set. (iii′) For each 퐔-small category 픻, there exists an adjunction lim ⊣ Δ : 풞 → [픻, 풞] −→픻 where Δ푋 is the constant functor with value 푋. Proof. This is a standard result; but we remark that we do require a sufficiently powerful form of the axiom of choice to pass from (ii) to (iii). □ ¶ ... In the explicit universe convention, the words ‘set’, ‘class’, etc. have their usual meanings, and in the one-universe convention, these instead abbreviate ‘퐔-set’, ‘퐔-class’, etc. for a fixed (but arbitrary) universe 퐔. However, the word ‘category’ always refers to a category that is contained in some universe, which may or may not be locally 퐔-small, and we shall use the word ‘ensemble’ to refer to sets which may or may not be in 퐔. In subsequent chapters, the implicit universe convention should be assumed unless otherwise stated. We now recall some definitions and results about ordinal and cardinal num- bers. Readers familiar with axiomatic set theory may wish to skip ahead. Definition ... A von Neumann ordinal is a set 훼 with the following prop- erties: • If 푥 ∈ 푦 and 푦 ∈ 훼, then 푥 ∈ 훼. • The binary relation ∈ is strict total ordering of 훼. • If 푆 is a subset of 훼 such that – ∅ ∈ 푆, .. Set theory – If 훽 ∈ 푆 and 훽 ∪ {훽} ∈ 훼, then 훽 ∪ {훽} ∈ 푆. – If 푇 ⊆ 푆, then ⋃ 푇 ∈ 푆. then 푆 = 훼. We identify 0 with the von Neumann ordinal ∅, and by induction, we identify the natural number 푛 + 1 with the von Neumann ordinal {0, … , 푛}. Proposition ... (i) If 훼 is a von Neumann ordinal, then every member of 훼 is an initial segment of 훼 and is in particular a von Neumann ordinal. (ii) If 훼 is a von Neumann ordinal, so is 훼 ∪ {훼}. (This is usually denoted by 훼 + 1 and called the successor of 훼.) (iii) The union of a set 푆 of von Neumann ordinals is another von Neumann ordinal. (This is usually denoted by sup 푆 and called the supremum of 푆.) (iv) If 퐔 is a pre-universe and 휅(퐔) is the set of von Neumann ordinals in 퐔, then 휅(퐔) a von Neumann ordinal, but 휅(퐔) ∉ 퐔. Proof. Claims (i) – (iii) are all easy, and claim (iv) is Burali-Forti’s paradox. ⧫ Theorem .. (Classification of well-orderings). (i) In Zermelo–Fraenkel set theory, every well-ordered set is isomorphic to a unique von Neumann ordinal. (ii) In Mac Lane set theory, if 퐔 is a pre-universe and 푋 is a well-ordered set in 퐔, then 푋 is isomorphic to a unique von Neumann ordinal in 퐔.