Emily Riehl Johns Hopkins University The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories joint with Dominic Verity and Michael Shulman Homotopy Type Theory Electronic Seminar Talks Abstract Homotopy type theory provides a “synthetic” framework that is suitable for developing the theory of mathematical objects with natively homotopical content. A famous example is given by (∞, 1)-categories — aka ∞-categories — which are categories given by a collection of objects, a homotopy type of arrows between each pair, and a weak composition law. This talk will compare two “synthetic” developments of the theory of ∞-categories • the first (with Verity) using 2-category theory and • the second (with Shulman) using a simplicial augmentation of homotopy type theory due to Shulman by considering in parallel their treatment of the theory of adjunctions between ∞-categories. The hope is to spark a discussion about the merits and drawbacks of various approaches to synthetic mathematics. Homotopical trinitarianism? homotopical mathematics formalize internalize ↺ homotopy type theory higher categories interpret homotopical mathematics synthetic math cat of structures ↻ homotopy type theory higher categories internal logic Synthetic homotopy theory homotopical mathematics synthetic math cat of structures ↻ homotopy type theory higher categories internal logic ∞-groupoids synthetic math cat of structures ↻ HoTT sSet internal logic Synthetic ∞-category theory homotopical mathematics synthetic math cat of structures ↻ homotopy type theory higher categories internal logic ∞-categories synthetic math cat of structures ↻ simplicial HoTT Rezk internal logic Synthetic ∞-category theory homotopical mathematics synthetic math cat of structures ↻ homotopy type theory higher categories internal logic ∞-categories synthetic math cat of structures ↻ ∞-cosmos qCat internal logic Plan 1. The synthetic theory of ∞-categories 2. The synthetic theory of ∞-categories Plan 1. The semantic theory of ∞-categories 2. The synthetic theory of ∞-categories in an ∞-cosmos 3. The synthetic theory of ∞-categories in homotopy type theory 4. Discussion 1 The semantic theory of ∞-categories The idea of an ∞-category ∞-categories are the nickname that Jacob Lurie gave to (∞, 1)-categories: categories weakly enriched over homotopy types. The schematic idea is that an ∞-category should have • objects • 1-arrows between these objects • with composites of these 1-arrows witnessed by invertible 2-arrows • with composition associative (and unital) up to invertible 3-arrows • with these witnesses coherent up to invertible arrows all the way up The problem is that this definition is not very precise. Models of ∞-categories Rezk Segal RelCat Top-Cat 1-Comp qCat • topological categories and relative categories are strict objects but the correct maps between them are tricky to understand • quasi-categories (originally weak Kan complexes) are the basis for the R-Verity synthetic theory of ∞-categories • Rezk spaces (originally complete Segal spaces) are the basis for the R-Shulman synthetic theory of ∞-categories 2 The synthetic theory of ∞-categories in an ∞-cosmos ∞-cosmoi An ∞-cosmos is an axiomatization of the properties of qCat. The category of quasi-categories has: • objects the quasi-categories 퐴, 퐵 • functors between quasi-categories 푓∶ 퐴 → 퐵, which define the points of a quasi-category Fun(퐴, 퐵) = 퐵퐴 • a class of isofibrations 퐸 ↠ 퐵 with familiar closure properties • so that (flexible weighted) limits of diagrams of quasi-categories and isofibrations are quasi-categories ∞-cosmoi An ∞-cosmos is an axiomatization of the properties of qCat. An ∞-cosmos has: • objects the ∞-categories 퐴, 퐵 • functors between ∞-categories 푓∶ 퐴 → 퐵, which define the points of a quasi-category Fun(퐴, 퐵) = 퐵퐴 • a class of isofibrations 퐸 ↠ 퐵 with familiar closure properties • so that (flexible weighted) limits of diagrams of ∞-categories and isofibrations are ∞-categories Theorem (R-Verity). qCat, Rezk, Segal, and 1-Comp define ∞-cosmoi. The homotopy 2-category The homotopy 2-category of an ∞-cosmos is a strict 2-category whose: • objects are the ∞-categories 퐴, 퐵 in the ∞-cosmos • 1-cells are the ∞-functors 푓∶ 퐴 → 퐵 in the ∞-cosmos 푓 • 2-cells we call ∞-natural transformations 퐴 ⇓훾 퐵 which are 푔 defined to be homotopy classes of 1-simplices in Fun(퐴, 퐵) Prop (R-Verity). Equivalences in the homotopy 2-category 푓 1퐴 1퐵 퐴 퐵 퐴 ⇓≅ 퐴 퐵 ⇓≅ 퐵 푔 푔푓 푓푔 coincide with equivalences in the ∞-cosmos. Adjunctions between ∞-categories An adjunction consists of: • ∞-categories 퐴 and 퐵 • ∞-functors 푢∶ 퐴 → 퐵, 푓∶ 퐵 → 퐴 • ∞-natural transformations 휂∶ id퐵 ⇒ 푢푓 and 휖∶ 푓푢 ⇒ id퐴 satisfying the triangle equalities 퐵 퐵 퐵 퐵 퐵 퐵 푓 푓 푢 푓 푓 ⇓휂 = = ⇓휂 푢 = = ⇓휖 푢 푢 푢 푓 ⇓휖 퐴 퐴 퐴 퐴 퐴 퐴 Write 푓 ⊣ 푢 to indicate that 푓 is the left adjoint and 푢 is the right adjoint. The 2-category theory of adjunctions Prop. Adjunctions compose: 푓′ 푓 푓푓′ 퐶 ⊥ 퐵 ⊥ 퐴 ⇝ 퐶 ⊥ 퐴 푢′ 푢 푢′푢 Prop. Adjoints to a given functor 푢∶ 퐴 → 퐵 are unique up to canonical isomorphism: if 푓 ⊣ 푢 and 푓′ ⊣ 푢 then 푓≅푓′. Prop. Any equivalence can be promoted to an adjoint equivalence: if 푢∶ 퐴 ∼ 퐵 then 푢 is left and right adjoint to its equivalence inverse. The universal property of adjunctions Any ∞-category 퐴 has an ∞-category of arrows hom퐴 ↠ 퐴 × 퐴 equipped with a generic arrow dom hom퐴 ⇓휅 퐴 cod 푓 Prop. 퐴 ⊥ 퐵 if and only if hom퐴(푓, 퐴) ≃퐴×퐵 hom퐵(퐵, 푢). 푢 Prop. If 푓 ⊣ 푢 with unit 휂 and counit 휖 then • 휂푏 is initial in hom퐵(푏, 푢) and • 휖푎 is terminal in hom퐴(푓, 푎). The free adjunction Theorem (Schanuel-Street). Adjunctions in a 2-category K correspond to 2-functors Adj → K, where Adj, the free adjunction, is a 2-category: op 횫−∞≅횫∞ op 횫+ + ⟂ − 횫+ op 횫∞≅횫−∞ 휂푢푓 id 휂 푢푓 푢휖푓 푢푓푢푓 푢푓푢푓푢푓 ⋯ 푢푓휂 휂푢푓 휂푢 푢 푢푓푢 푢휖푓 푢푓푢푓푢 푢푓푢푓푢푓푢 ⋯ 푢휖 푢푓휂 푢푓푢휖 Homotopy coherent adjunctions A homotopy coherent adjunction in an ∞-cosmos K is a simplicial functor Adj → K. Explicitly, it picks out: • a pair of objects 퐴, 퐵 ∈ K. • homotopy coherent diagrams op 횫+ → Fun(퐵, 퐵) 횫+ → Fun(퐴, 퐴) op 횫∞ → Fun(퐴, 퐵) 횫∞ → Fun(퐵, 퐴) that are functorial with respect to the composition action of Adj. Coherent adjunction data A homotopy coherent adjunction is a simplicial functor Adj → K. 푢푓푢 푓푢푓 triangle equality witnesses 휂푢 푢휖 푓휂 휖푓 훼 훽 푢 푢 푓 푓 푓푢푓푢 푓푢푓푢 푓휂푢 휖∗휖 푓휂푢 휖∗휖 휔 휇 푓푢휖 1 휖 푓푢 푓훼 nat휖 id퐴 =⇒ 푓푢 id퐴 휖 휖 휖 푓푢 푓푢 푓푢푓푢 푓푢푓푢 푓휂푢 휖∗휖 푓휂푢 휖∗휖 휏 휇 휖푓푢 2 휖 푓푢 훽푢 nat휖 id퐴 =⇒ 푓푢 id퐴 휖 휖 휖 푓푢 푓푢 Existence of homotopy coherent adjunctions Theorem (R-Verity). Any adjunction in the homotopy 2-category of an ∞-cosmos extends to a homotopy coherent adjunction. Proof: Given adjunction data • 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴 • 휂∶ id퐵 ⇒ 푢푓 and 휖∶ 푓푢 ⇒ id퐴 • 훼 witnessing 푢휖 ∘ 휂푢 = id푢 and 훽 witnessing 휖푓 ∘ 푓휂 = id푓 forget to either • (푓, 푢, 휂) or • (푓, 푢, 휂, 휖, 훼) and use the universal property of the unit 휂 to extend all the way up. Theorem (R-Verity). Moreover, the spaces of extensions from the data (푓, 푢, 휂) or (푓, 푢, 휂, 휖, 훼) are contractible Kan complexes. 3 The synthetic theory of ∞-categories in homotopy type theory The intended model op op Set횫 ×횫 ⊃ Reedy ⊃ Segal ⊃ Rezk = = = = bisimplicial sets types types with types with composition composition & univalence Theorem (Shulman). Homotopy type theory is modeled by the category of Reedy fibrant bisimplicial sets. Theorem (Rezk). (∞, 1)-categories are modeled by Rezk spaces aka complete Segal spaces. The bisimplicial sets model of homotopy type theory has: • an interval type 퐼, parametrizing paths inside a general type • a directed interval type 2, parametrizing arrows inside a general type Paths and arrows • The identity type for 퐴 depends on two terms in 퐴: 푥, 푦 ∶ 퐴 ⊢ 푥 =퐴 푦 and a term 푝 ∶ 푥 =퐴 푦 defines a path in 퐴 from 푥 to 푦. • The hom type for 퐴 depends on two terms in 퐴: 푥, 푦 ∶ 퐴 ⊢ hom퐴(푥, 푦) and a term 푓 ∶ hom퐴(푥, 푦) defines an arrow in 퐴 from 푥 to 푦. Hom types are defined as instances of extension types axiomatized in a three-layered type theory with (simplicial) shapes due to Shulman [푥,푦] 1 + 1 퐴 hom퐴(푥, 푦) ≔ ⟨ ⟩ 2 ∑ (푥, 푦) ∞ Semantically, hom types 푥,푦∶퐴 hom퐴 recover the -category of arrows hom퐴 ↠ 퐴 × 퐴 in the ∞-cosmos Rezk. Segal, Rezk, and discrete types • A type 퐴 is Segal if every composable pair of arrows has a unique composite: if for every 푓 ∶ hom퐴(푥, 푦) and 푔 ∶ hom퐴(푦, 푧) 2 [푓,푔] Λ1 퐴 ⟨ ⟩ is contractible. Δ2 • A Segal type 퐴 is Rezk if every isomorphism is an identity: if id-to-iso∶ ∏ (푥 =퐴 푦) → (푥 ≅퐴 푦) is an equivalence. 푥,푦∶퐴 • A type 퐴 is discrete if every arrow is an identity: if id-to-arr∶ ∏ (푥 =퐴 푦) → hom퐴(푥, 푦) is an equivalence. 푥,푦∶퐴 Prop. A type is discrete if and only if it is Rezk and all of its arrows are isomorphisms — the discrete types are the ∞-groupoids. The 2-category of Segal types Prop (R-Shulman). • Any function 푓∶ 퐴 → 퐵 between Segal types preserves identities and composition. Morever, the type 퐴 → 퐵 of functors is again a Segal type. • Given functors 푓, 푔∶ 퐴 → 퐵 between Segal types there is an equivalence ∼ hom(푓, 푔) ∏ hom퐵(푓푎, 푔푎) 퐴→퐵 푎∶퐴 • Terms 훾 ∶ hom(푓, 푔), called natural transformations, are natural 퐴→퐵 and can be composed vertically and horizontally up to typal equality. Incoherent adjunction data A quasi-diagrammatic adjunction between types 퐴 and 퐵 consists of • functors 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴 • natural transformations 휂∶ hom(id퐵, 푢푓), 휖∶ hom(푓푢, id퐴) 퐵→퐵 퐴→퐴 • witnesses 훼 ∶ 푢휖 ∘ 휂푢 = id푢 and 훽 ∶ 휖푓 ∘ 푓휂 = id푓 A (quasi*-)transposing adjunction between types 퐴 and 퐵 consists of functors 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴 and a family of equivalences ∏ hom퐴(푓푏, 푎) ≃ hom퐵(푏, 푢푎) 푎∶퐴,푏∶퐵 (*together with their quasi-inverses and the witnessing homotopies).