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
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).
Theorem(R-Shulman). Given functors 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴 between Segal types the type of quasi-transposing adjunctions 푓 ⊣ 푢 is equivalent to the type of quasi-diagrammatic adjunctions 푓 ⊣ 푢. Coherent adjunction data A half-adjoint diagrammatic adjunction consists of:
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
Theorem (R-Shulman). Given functors 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴 between Segal types the type of transposing adjunctions 푓 ⊣ 푢 is equivalent to the type of half-adjoint diagrammatic adjunctions 푓 ⊣ 푢. Uniqueness of coherent adjunction data
If 휂∶ hom(id퐵, 푢푓) is a unit, then that adjunction is uniquely determined: 퐵→퐵 Theorem (R-Shulman). Given Segal types 퐴 and 퐵, functors 푢∶ 퐴 → 퐵 and 푓∶ 퐵 → 퐴, and a natural transformation 휂∶ hom(id퐵, 푢푓) the 퐵→퐵 following are equivalent propositions: • The type of (휖, 훼, 훽, 휇, 휔, 휏) extending (푓, 푢, 휂) to a half-adjoint diagrammatic adjunction. • The propositional truncation of the type of (휖, 훼, 훽) extending (푓, 푢, 휂) to a quasi-diagrammatic adjunction.
Theorem (R-Shulman). Given the data (푓, 푢, 휂, 휖, 훼) as in a quasi-diagrammtic adjunction, the following are equivalent propositions: • The type of (훽, 휇, 휔, 휏) extending this data to a half-adjoint diagrammatic adjunction. • The propositional truncation of the type of 훽 extending this data to a quasi-diagrammatic adjunction. Where does Rezk-completeness come in?
For Rezk types — the synthetic ∞-categories — adjoints are literally unique, not just “unique up to isomorphism”:
Theorem (R-Shulman). Given a Segal type 퐵, a Rezk type 퐴, and a functor 푢∶ 퐴 → 퐵, the following types are equivalent propositions: • The type of transposing left adjoints of 푢. • The type of half-adjoint diagrammatic left adjoints of 푢. • The propositional truncation of the type of quasi-diagrammatic left adjoints of 푢. 4
Discussion Closing thoughts
• In an ∞-cosmos, we prove that there exists a quasi-diagrammatic adjunction if and only if there exists a quasi-transposing adjunction. In simplicial HoTT, we prove the types of such are equivalent, which conveys more information (though I’m not exactly sure what). • The ∞-cosmos Rezk does not see Segal or ordinary types — because we’ve axiomatized the fibrant objects, rather than the full model category. • It seems to be much easier to produce an ∞-cosmos than to define a model of simplicial HoTT. • But overall the experiences of working with either approach to the synthetic theory of ∞-categories are strikingly similar — and I’m not sure I entirely understand why that is. References For more on homotopical trinitarianism, see:
Michael Shulman • Homotopical trinitarianism: a perspective on homotopy type theory, home.sandiego.edu/∼shulman/papers/trinity.pdf
For more on the synthetic theory of ∞-categories, see:
Emily Riehl and Dominic Verity • ∞-category theory from scratch, arXiv:1608.05314 • ∞-Categories for the Working Mathematician, www.math.jhu.edu/∼eriehl/ICWM.pdf Emily Riehl and Michael Shulman • A type theory for synthetic ∞-categories, Higher Structures 1(1):116–193, 2017; arXiv:1705.07442
Thank you!