Notes on Higher Categories and Categorical Logic Lectures by Mike Shulman and Peter LeFanu Lumsdaine; Notes by Jacob Alexander Gross August 12, 2016 Contents 1Preface 1 2Preliminaries 2 3SyntaxandFreeObjects 7 4TypeTheoryforCategorieswithProducts 11 5MotivatingExample(MonoidObjects) 15 6Multicategories 16 7Type-TheoreticProofofUniquenessofInverses 20 8HeytingAlgebras 23 9First-OrderLogic 26 10 First-Order Theories 28 11 Quantifiers as Adjoints 29 12 First-Order Hyperdoctrines 33 13 Bi-Initiality 36 14 More on Bi-Initiality 42 15 The Classifying Category of an Algebraic Theory 45 16 From Regular Categories to Regular Hyperdoctrines 49 17 The Logic of Presheaves 51 1 18 Cartesian Closed Multicategories 56 19 On Completeness 61 20 Intuitionistic Higher-Order Logic 66 21 The Free Topos 70 22 Some Dependent Types 74 23 References 77 1 Preface The “Category Theory 2016” conference was held during August 2016 at Dal- housie University, located in the Canadian provence of Nova Scotia. During the month prior to this conference Mike Shulman and Peter LeFanu Lumsdaine, as part of the AARMS summer school, gave a series of lectures each morning comprising a course called “Higher Categories and Categorical Logic.” These are one of the participants’ notes. The lectures could be summarized as saying that the goal is to prove things in (1-/higher/multi-)categories with the syntax of type theory by using type theories to present free structures. The first 10 lectures were given by Shulman and the next 10 by Lumsdaine. The first lecture begins at Section 3. The section after this one, “Preliminaries” breezes over a couple of defini- tions/facts about monads, monoidal categories, equivalences and graphs which are essential to digesting the material. A solid grounding in categories, functors, natural transformations, (co)limits, adjoints and Yoneda’s lemma is assumed. Multicategories, especially cartesian multicategories, are used often. All the multicategorical theory is introduced as needed throughout the notes. That aside, the reader may still find it helpful (as the note-taker did) to give the occasional glance at Tom Leinster’s book [Lei03] when more details are desired. Lastly, at the end of some sections there are exercises. There were three homeworks and one final exam given by the instructors. The exercises are all taken from the collection of these four documents and put at the end of whatever lecture the problem seemed most appropriate to. This will allow the reader to, if desired, their hands dirty with pens, paper, chalk, coffee, etc. And of course, any inaccuracies are entirely my own. Conventions That two objects A, B are isomorphic will be written A = B. • ∼ That two categories C , D are equivalent will be written C D. • ≃ The natural numbers start at 0. • 2 2 Preliminaries We try to bring to the forefront of the readers mind, who may be somewhat fresh to category theory, some of the key definitions and theorems from elementary category theory relevant to the notes. Adjoints Definition 2.1. A cartesian closed category C is category with finite prod- ucts such that for every X C , the functor X : C C has a right adjoint. ∈ −× → Theorem 2.2. (Freyd’s Adjoint Functor Theorem) Let G : D C be a continuous functor. The following conditions are sufficient for G to admit→ a left adjoint 1. (General) D is complete and locally small and G satisfies the solution set condition: For each object X C there is a small set I and an I-indexed ∈ family of arrows fi : X GAi such that every arrow h : X GA can be written as a composite h→= Gt f , some some i I and some→t : A A. ◦ i ∈ i → 2. (Special) C is locally small and D is locally small, complete and well- powered with a cogenerating set: asetS of objects such that the D( ,S ) − α are jointly faithful (where Sα ranges over all elements of S). Equivalence Definition 2.3. The skeleton of a category C ,writtenskC ,isthefull isomorphism-dense subcategory of C for which no two distinct objects are iso- morphic. Theorem 2.4. The following are equivalent there is an equivalence of categories C D • ≃ there is a functor F : C D that is full, faithful and essentially surjective • → C and D have isomorphic skeleta skC = skD • ∼ Definition 2.5. An adjoint equivalence is an adjunction F G for which the unit and counit are natural isomorphisms. ⊣ Proposition 2.6. An adjoint equivalence is an equivalence of categories. Proposition 2.7. If two categories C and D are equivalent, then there exists an adjoint equivalence between them. 3 Graphs Definition 2.8. A (directed) graph G consists of a set of objects G0, together with for each object pair of objects X, Y, a set of arrows G (X, Y ). There is a forgetful functor U : Cat Gr taking a category to its underlying graph. This functor admits a left adjoint→ F Gr Cat. U Definition 2.9. A multigraph G consists of a set of objects G0, together with for every object X a finite list of objects A1,...,An a set of arrows G (A1,...,An; X) Monads Definition 2.10. A functor U : D C is monadic if it has a left adjoint → F : C D and the canonical comparison functor D C T is an equivalence of → → categories (where C T denotes the Eilenberg-Moore category). Definition 2.11. A split coequalizer is a collection of objects and arrows t t f A B h C g such that f s =1 • ◦ B g s = t h • ◦ ◦ h t =1 , and • ◦ C h f = h g • ◦ ◦ Definition 2.12. Let U : C D be a functor. Then, a U-split pair is a → parallel pair of morphisms f,g in C for which there exists a split coequalizer t t Uf UA UB h C Ug in D. Definition 2.13. (Beck’s Monadicity Theorem) A functor U : C D if monadic if and only if → 1. U admits a left adjoint 2. U reflects isomorphisms 3. C has coequalizers of U-split pairs and U preserves them 4 Monoidal Categories Definition 2.14. A strict monoidal category is a category C along with a tensor product functor : C C C and a unit object I such that ⊗ × → A (B C)=(A B) C, ⊗ ⊗ ⊗ ⊗ A I = A, ⊗ I A = A, ⊗ for all A, B, C C . ∈ Definition 2.15. A monoidal category is a category C along with a tensor product functor : C C C ,aunit object I along with natural isomorphisms ⊗ × → associators αA,B,C : A (B C) (A B) C, left unitors λA : I A A and right unitors ρ : A⊗ I ⊗A satisfying→ ⊗ the⊗pentagon ⊗ → ⊗ → αA,B C,D (A (B C)) D ⊗ A ((B C) D) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ α 1 1 α A,B,C ⊗ D A ⊗ B,C,D ((A B) C) D A (B (C D)) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ αA B,C,D ⊗ αA,B,C D ⊗ (A B) (C D) ⊗ ⊗ ⊗ and triangle α (A I) B A,I,B A (I B) ⊗ ⊗ ⊗ ⊗ ρA 1B 1 λ ⊗ A ⊗ A A B ⊗ identities. Theorem 2.16. (Coherence for Monoidal Categories) Given a monoidal category (C , ,I,α,ρ,λ), any well-typed equation constructed only from α,λ,ρ and their inverses⊗ holds. Definition 2.17. Let C and D be monoidal categories. A lax monoidal functor (F,φ):C D is a functor F : C D along with coherence morphisms → → φ : FA FB F (A B),φ: I FI A,B ⊗ → ⊗ I → such that, for all A, B, C C the following diagrams commute ∈ 5 φA,B 1FC φA B,C (FA FB) FC ⊗ F (A B) FC ⊗ F ((A B) C) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ αFA,FB,FC FαA,B,C 1FA φB,C φA,B C FA (FB FC) ⊗ FA F (B C) ⊗ F (A (B C)) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 1 φ φA,I FA I FA ⊗ FA FI F (A I) ⊗ ⊗ ⊗ ρFA FρA FA φ 1 φI,A I FA ⊗ FA FI FA F (I A) ⊗ ⊗ ⊗ λFA FλA FA Definition 2.18. A colax monoidal functor (F,φ):C D is a functor → F : C D together with coherence morphisms → φ : FA FB F (A B),φ: FI I A,B ⊗ → ⊗ I → satisfying conditions dual to those required of coherence morphisms for lax monoidal functors. Definition. 2.19. A weak (resp. strict) monoidal functor is a lax monoidal functor (F,φ) for which all coherence morphisms are isomorphisms (resp. iden- tities). Theorem 2.20. (Strictification) For every monoidal category C there is a weak monoidal functor out of C into a strict monoidal category, which is an equivalence of categories. Definition 2.21. A braided monoidal category is a monoidal category along with a natural isomorphism σ : A B B A, A,B ⊗ → ⊗ called braiding, such that the hexagon identities hold: 6 σA,B C A (B C) ⊗ (B C) A ⊗ ⊗ ⊗ ⊗ 1 1 αA,B,C− αB,C,A− (A B) C B (C A) ⊗ ⊗ ⊗ ⊗ σ 1 1 σ A,B ⊗ C B ⊗ A,C α (B A) C B,A,C B (A C) ⊗ ⊗ ⊗ ⊗ σA B,C (A B) C ⊗ C (A B) ⊗ ⊗ ⊗ ⊗ αA,B,C αC,B,A A (B C) (C A) B ⊗ ⊗ ⊗ ⊗ 1 σ σ 1 A ⊗ B,C A,C ⊗ B 1 α− A (C B) A,C,B (A C) B. ⊗ ⊗ ⊗ ⊗ Definition 2.22. A symmetric monoidal category is a bradied monoidal category (C ,σ) such that σB,A σA,B =1A B ◦ ⊗ for all objects A, B C . In this case we call σ the symmetry. ∈ Definition 2.23. Let V be a symmetric monoidal category. A V-category is the following data : A class C of objects • 0 for all objects A, B C , an hom object C (A, B) V • ∈ 0 ∈ for all objects A, B, C C a composition morphism in V • ∈ 0 c : C (A, B) C (B,C) C (A, C) A,B,C ⊗ → for every object A C ,aV-morphism u : I C (A, A) • ∈ A → such that the diagrams 7 (C (A, B) C (B,C)) C (C, D) ∼= C (A, B) (C (B,C) C (C, D)) ⊗ ⊗ ⊗ ⊗ c 1 1 c A,B,C ⊗ ⊗ B,C,D C (A, C) C (C, D) C (A, B) C (B,D) ⊗ ⊗ cA,C,D cA,B,D C (A, D) I C (A, B) ∼= C (A, B) ∼= C (A, B) I ⊗ ⊗ u 1 1 u A ⊗ ⊗ B cA,B,C C (A, A) C (A, B) C (A, B) C (A, B) C (B,B) ⊗ cA,B,C ⊗ commute.