Notes from the HoTT Workshop in Oxford

Transcription

1 Notes from the HoTT Workshop in Oxford 7th 10th November 2014 Clive Newstead Contents 1 Benedikt Ahrens: Univalent FOLDS 2 2 Egbert Rijke: An algebraic formulation of dependent type theory 4 3 Peter Lumsdaine: Coherence constructions for dependent type theory 4 4 Chris Kapulkin: Type theory and locally cartesian closed quasicategories 6 5 Thorsten Altenkirch and Ambrus Kaposi: A syntax for cubical type theory 6 6 Andrej Bauer: Type theory with a reflection rule 7 7 Thierry Coquand: Variation on cubical sets 7 8 Dan Licata: A cubical type theory 8 9 Guillaume Brunerie: The fourth homotopy group of the three-sphere 9 10 Benno van den Berg: Path object categories Matthijs Vákár: Splitting the atom of dependent types James Cranch: How (not) to define categories in HoTT Evan Cavallo: The Mayer Vietoris sequence in HoTT Neil Strickland: Proof assistants as a routine tool? 16 1

2 All the Oxford talks were recorded and have been uploaded to YouTube: 1 Benedikt Ahrens: Univalent FOLDS The equivalence principle of higher category theory says that meaningful statements should be invariant under equivalence. First-Order Logic with Dependent Sorts (FOLDS) was introduced by Makkai as a language for describing higher-categorical structures in which this would always be true, because there is no equality that can distinguish equivalent structures. More recently, Homotopy Type Theory (HoTT) is a foundation for mathematics, in which Voevodsky s Univalence Axiom (UA) enforces the equivalence principle for infinity-groupoids by essentially defining equal to mean equivalent. In previous work, by relativizing UA, we defined a notion of univalent or saturated (1- )category in HoTT that satisfies the principle of equivalence. We now extend this to other higher-categorical structures by defining them á la FOLDS inside HoTT. Any FOLDS-signature comes with a canonical notion of univalence for its structures, and such univalent structures satisfy the principle of equivalence. Examples include n-categories, dagger-categories, and doubly weak double categories. Notes The motivation for this work comes from the equivalence principle, which says that reasoning should be invariant under equivalence. This is not always the case. For example, let the statement ϕ(x) be x is a category with a unique terminal object ; then ϕ(set) is true, but there are categories C Set for which ϕ(c) is false. Problems arise when ϕ mentions equality-on-objects. Theorem [Freyd and Blanc]. A statement about categories is invariant under categorical equivalence if and only if it can be posed in the language of dependent types. The goal of this research is to extend this result of Freyd and Blanc to higher categorical structures, using FOLDS: first-order logic with dependent sorts. As an example, consider the following possible definition of a category. A category consists of... ˆ A collection O of objects and, for each x, y : O, a collection A(x, y) of arrows x y; ˆ For each x, y, z : O, f : A(x, y), g : A(y, z) and h : A(x, z) there is a collection T x,y,z:a (f, g, h) of assertions that h = g f; 2

3 ˆ For each x : O and f : A(x, x) there is a collection I x (f) of assertions that f = id X ; ˆ For each x, y : O and f, g : A(x, y) there is a collection E x,y (f, g) of assertions that f = g; ˆ If t 1 : T (f, g, h 1 ) and t 2 : T (f, g, h 2 ) then E(h 1, h 2 ); ˆ E is a congruence with respect to T, I, and is an equivalence relation. Theorem. Categories C and D are equivalent if and only if there is a span C P D of surjective equivalences. The above definition and theorem motivate what is to follow. A category is a one-way category if there are no infinite ascending chains of morphisms and, for all objects A, the number of morphisms with domain A is finite. A FOLDS structure is a covariant presheaf X : Σ Set on a one-way category Σ. The elements of Σ are called kinds. The matching object of k Σ is M k (X) = lim (k k) id k X k This encodes dependencies of x X(k). The corresponding matching map is the induced map X k M k (X). The fibres of the matching map are sorts. Given Σ-structures X, Y : Σ Set, a natural transformation f : X Y is very surjective if all the maps are surjective. A FOLDS equivalence from X to Y is a span X P Y of very surjective maps. Theorem. A statement about FOLDS structures is invariant under FOLDS-equivalence if and only if it can be posed in dependent type theory with only E on the top level. We seek an analogous result for higher groupoids. For a category C, the saturation of C is an internal category groupoids. Ĉ = A O in the category of Lemma. A functor F : C D is an equivalence if and only if F : Ĉ D is a levelwise equivalence of groupoids. Ĉ satisfies: ˆ A O O is a discrete fibration; ˆ Hom(a, b) Iso(a, b) is an isomorphism for all a, b, where Hom(a, b) is the collection of morphisms a b in O and Iso(a, b) is the collection of objects of A that are morphisms a b in the category structure A O. We call such a category univalent (or saturated). Theorem. A statement or construction about univalent categories is invariant under equivalence in univalent foundations. 3

4 See also ˆ TeX source for Benedikt s talk (along similar lines) at Halifax: benediktahrens/folds/tree/master/tex 2 Egbert Rijke: An algebraic formulation of dependent type theory Egbert introduced his E-systems, formulating dependent type theory algebraically by piecing together the theories of extension, weakening, projection, substitution, and the empty context and families, as shown in the following diagram: extension weakening substitution empty/families projection DTT See also ˆ Slides: 3 Peter Lumsdaine: Coherence constructions for dependent type theory Coherence constructions are a vexing technical hurdle which most models of dependent type theory, especially homotopical ones, have to tackle in some way. I will survey the main known approaches, including the constructions of Hofmann, Voevodsky/Hofmann Streicher, Lumsdaine Warren, and Curien Garner Hofmann. I will also discuss why we need to worry about coherence in the first place. 4

5 Notes In order to give a model of a theory T, we specify how particular objects are modelled (e.g. in sset, we model types as fibrations, terms as sections, Π-types as Π-functors, identity types as path spaces,... ) and extend automatically to an interpretation of the whole syntax. This interpretation is a universal property of the syntax, saying the syntax is an initial object C T in some category StrMod T of (strict) models of T. Usually this is done by first obtaining a weak model C, the category of which can be denoted by WkMod T, and strictifying somehow to obtain a strong model C. We would like to have C C. To this end, we can define a comprehension category is a triple (C, Ty, χ), which can be thought of as follows: ˆ C is a category of contexts. ˆ For Γ C there is a fibre Ty(Γ) of types over Γ. ˆ For Γ f Ty(f) Γ there is a substitution Ty(Γ) Ty(Γ ), which is functorial in f up to coherent isomorphism. ˆ The image of A Ty(Γ) under χ is a morphism χ(a) : Γ.A Γ; we can think of terms a : A as sections Γ Γ.A of χ(a). The category of comprehension categories will be denoted CompCat. A comprehension category (C, Ty, χ) is split if Ty is strictly functorial in f up to isomorphism; denote the category of split comprehension categories by StrCompCat. Proposition. If T is a theory given from just structural rules then C T is initial in StrCompCat. If T is a purely algebraic extension of such a theory then C T is initial in a particular subcategory of StrCompCat. Theorem. Let C be a comprehension category. Then there exist split comprehension categories C and C! which are (in some sense) equivalent to C. The following results summarise why these are useful for resolving the coherence issues at question. Theorem [Hofmann]. If C is a comprehension category with pseudostable structure, this lifts to strictly stable structure on C. Theorem [Lumsdaine & Warren]. If C is a comprehension category with strictly stable structure and (certain) products and exponentials, this lifts to strictly stable structure on C!. Proposition. If C has identity types and satisfies a reflection rule then weakly stable structure is automatically pseudostable. See also ˆ arxiv: : 5

6 4 Chris Kapulkin: Type theory and locally cartesian closed quasicategories Suppose C is a contextual category with Π-, Σ- and Id- structures (i.e. a categorical model of a type theory admitting rules for Π-, Σ- and Id-types). Then (the underlying category of) C can be equipped with a class of weak equivalences defined syntactically. We may therefore consider the simplicial localization of C, which is the quasicategory obtained by inverting in the suitable higher-categorical sense the weak equivalences in C. In this talk, I will outline a proof that the resulting quasicategory is locally cartesian closed. Notes Recall that a category with weak equivalences is a pair (C, W), where C is a category and W is a subcategory of C containing all identity morphisms. Denote the category of categories with weak equivalences by wecat. Let Th Π,Σ,Id denote the category of contextual categories and contextual functors. There is a functor Cl : Th Π,Σ,Id wecat defined by Cl(T) = (C T, W), where W is the subcategory with all syntactically defined weak equivalences. There is a functor Ho : wecat Cat sset to the category Cat of quasicategories, which are simplicial sets with the unique horn filling property. The question we seek to address is: what do we know about Ho (Cl(T))? Some motivation comes from correspondences arising from internal languages. For example, λ-theories correspond with cartesian closed categories, and extensional Martin-Löf type theory corresponds with locally cartesian closed categories. Intensional Martin-Löf type theory must correspond with something quasicategorical. Theorem. Ho (Cl(T)) is a locally cartesian closed quasicategory. This theorem combines results of Avigad, Kapulkin, Lumsdaine and Szumi lo. 5 Thorsten Altenkirch and Ambrus Kaposi: A syntax for cubical type theory Vladimir Voevodsky put forward the challenge to define semisimplicial types in HoTT - it seems that this is impossible due to the coherence problems that arise. Voevodsky put forward a 6

7 new type theory HTS based on extensional type theory which distinguishes between types and pretypes. As an alternative we propose a type theory based on intensional type theory which can be easily modelled in existing systems such as Agda. In this type theory as in HTS we can avoid the coherence problem by using a strict equality. See also ˆ Slides: 6 Andrej Bauer: Type theory with a reflection rule I will report on the progress in implementation of a type system HTS0 proposed by Vladimir Voevodsky. The type system is tricky to implement because it has a (strict) equality type which reflects into judgmental equality. This gives us ample opportunities for making mistakes and for derailing the usual algorithms, but it also makes the system extremely expressive. For instance, we can define in it the notion of propositional truncation, or any other higher-inductive type, with the desired computational rules. The plan is to create a system that is useful for experimenting with various concepts in homotopy type theory. See also ˆ Slides: ˆ Andromeda (GitHub): 7 Thierry Coquand: Variation on cubical sets We present a variation of the cubical set model, which adds connections and diagonal operations, together with a prototype implementation. Adding connections allows us to interpret the computation rule for the identity elimination as a judgmental equality. By adding diagonals, we also can interpret the two computation rules for circle elimination (on points and on paths) as judgmental equalities. 7

8 Notes The motivation for this work is that we seek a computational model of type theory, and it will give a universal nominal extension of λ-calculus and a justification of function extensionality from function extensionality in the metalogic. The model of type theory in Kan simplicial sets and Kan cubical sets use classical logic. For example, the law of excluded middle must be used to prove that B Kan implies B A Kan, or that if E B is a Kan fibration and b 0 b 1 in B then E(b 0 ) is homotopy equivalent to E(b 1 ). Cubical sets. Given a finite set I, we will think of elements of I as symbols, names or dimension labels. The cube category C will have finite sets as objects. Morphisms are functions f : I J together with decompositions I = I 0, I 1, I such that f is injective on I. We will think of morphisms f : I J as being substitutions, so we will write them on the right and composition will go forwards, i.e. fg = g f and Af = f(a). A cubical set is a covariant presheaf X : C Set. We think of elements of X(I) as a set of I-cubes, i.e. I -dimensional cubes living in the dimensions labelled by the elements of I. Closed types A will be interpreted as covariant presheaves on C. Closed types I A which depend on a dimension in a finite set I will be interpreted as covariant presheaves on I/ C. [My notes on the rest of the talk aren t very good because it was mostly syntax and the slides went very fast.] See also ˆ Slides: 8 Dan Licata: A cubical type theory In this talk, I will describe work in progress (joint with Guillaume Brunerie) on a cubical syntax for type theory. The goal of the work is to provide a syntactic type theory where the computational aspects of the cubical sets model by Bezem, Coquand, and Huber can be expressed. I will describe a boundaries-as-terms cubical type theory, where the basic judgement is the term u is an n- cube in the type A, together with its boundary, and the cubical operations (faces, degeneracies, symmetries, and diagonals) can be applied to any term. This syntax permits a clean formulation of the computation rules for Kan fillings in Π, Σ, and identity types. Moreover, diagonals support the computation rules for higher inductive types. The most salient open issue is how to define the Kan fillings for the universe and for univalence; I would welcome collaboration on this after the talk. 8

9 See also ˆ Slides: 9 Guillaume Brunerie: The fourth homotopy group of the threesphere In this talk we will give a proof that the fourth homotopy group of S 3 is Z/2Z. We will see that the proof is entirely constructive and homotopy-theoretic, hence can be carried inside homotopy type theory, and we will discuss the various advantages of having an HoTT-proof and the various challenges in formalizing it. Notes First we define the higher inductive type S n, which requires a few preliminary definitions. ˆ Given types A, B, C and functions f : C A and g : C B, the pushout A C B is a type with l : A A C B, r : B A C B, p : l(fc) = r(fc) c:c ˆ The suspension of a type A is the type ΣA 1 A 1. Let north, south : ΣA denote the corresponding poles. ˆ The circle S 1 is the higher inductive type with For n 1, we define S n+1 : ΣS n. base : S 1 and loop : base = S 1 base ˆ The loop space of a pointed type (A, a) is Ω a (A) : a = A a. The n th iterated loop space of (A, a), denoted Ω n a(a), is defined inductively by Ω 1 a(a) = Ω A a and Ω n+1 a (A) = Ω n refl a (Ω a (A)) ˆ The n th homotopy group of a pointed type (A, a) is π n (A, a) = Ω n a(a) 0 The goal is to be able to compute π k (S n ) for given k, n 1. 9

10 Hopf fibration. The Hopf fibration is a fibration over S 2 with fibre S 1 and total space S 3. We define it using the equivalence (fibres over S 2 ) S 2 Type To this end, is a multiplication µ : S 1 S 1 S 1 such that µ(x, ) and µ(, y) are equivalences, defined by µ(base, x) x and µ(loop, x) f(x) : x = S 1 x... where f : x:s 1(x = S1 x) is defined by f(base) loop and f(loop) : loop = loop... where is the filler of the square loop f(base) = f(base) loop. We can now define Hopf : S 2 Type by Hopf(north) Hopf(south) S 1 and, for y : S 1, Hopf(p(y)) µ(, y) : Hopf(north) = Type Hopf(south) The total space is the pushout of S 1 fst S 1 S 1 µ S 1, which is equivalent to the pushout of S 1 fst S 1 S 1 snd S 1, which is the join S 1 S 1, which is homotopy equivalent to S 3, as desired. The long exact sequence for homotopy groups gives that π 3 (S 2 ) = π 3 (S 3 ) = Z. James construction. A function f : A B is k-connected if π n (f) is an isomorphism for all n k and is surjective for n = k + 1. A type A is k-connected if A! 1 is k-connected. The James construction is as follows: if A is k-connected, then there exist types J n (A) for n N and maps i n : J n (A) J n+1 (A) such that ˆ J 0 (A) = 1 and J 1 (A) = A; ˆ i n is ((k 1) + m(k + 1))-connected; ˆ colim n J n (A) = ΩΣA. J 0 (A) (k 1) J 1 (A) (2k) J 2 (A) (3k+1) ΩΣA Now the map J 2 (S 2 ) ΩS 3 is 3-connected, so π 4 (S 3 ) = π 3 (Ω(S 3 )) = π 3 (J 2 (S 3 ) Moreover, we can define J 2 (S 3 ) = S 2 S2 S 2 (S 2 S 2 ) = S 2 S3 1 This latter pushout yields a map S 3 S 2 S3 1, and hence π 3 (α) : π 3 (S 3 ) π 3 (S 2 ) 10

11 But we already know what these two groups are: they re both Z. Some fiddling with the Blakers Massey theorem and long exact sequences yields π 4 (S 3 ) = π 3 (J 2 (S 2 )) = Z/Im(π 3 (α)) = Z/nZ for some n N. Classically we have n = 2, but proving this constructively appears to be trickier (if possible). 10 Benno van den Berg: Path object categories I will discuss categorical models of homotopy type theory. In the first part I will discuss an approach to such models using the notion of path object category : this provides a fairly simple approach and I will explain that both the category of simplicial sets and the category of cubical sets with connections carries such a structure (joint work with Simon Docherty). One drawback of this approach is that the syntactic category associated to type theory does not carry such a structure; so I will also discuss a more general approach based on categories with fibrations, which includes the syntactic category as an example. I will talk about a kind of homotopy exact completion for these categories and explain that to construct a model of Aczel s constructive set theory CZF in this exact completion only a very weak form of universe is needed in the original category (joint work with Ieke Moerdijk). Notes A tribe is a pair (C, F), where C is a category with a terminal object and and F is a class of C-morphisms (called fibrations ), such that ˆ F contains all isomorphisms and is closed under composition; ˆ Pullbacks of morphisms along fibrations exist and are fibrations; ˆ Every X 1 is a fibration. Given an object A, write C(A) for the full subcategory of C /A consisting of all fibrations with codomain A. A morphism is anodyne if it has the left lifting property with respect to all fibrations. A tribe (C, F) is an identity tribe if: ˆ If X A is a fibration then the diagonal X X A A in C(A) factors as an anodyne map followed by a fibration in C(A). Moreover this should be stable along any B A. 11

12 ˆ Anodyne maps are stable under pullbacks along fibrations. Some results about identity tribes include: ˆ Fibrations satisfy a path-lifting property, corresponding to transport. ˆ Anodyne maps are the same as strong deformation retracts. ˆ Any map can be factored as an anodyne map followed by a fibration, and this has the structure of a weak factorisation system. An identity tyibe (C, F) is closed if fibrations are precisely the maps with the right lifting property with respect to anodyne maps. A pseudoequivalence relation is a fibration R X X with reflexivity, symmetry and transitivity morphisms (X R, R R and R X R R, respectively), satisfying some conditions. A closed identity tribe gives (C, F) gives rise to a category whose objects are pseudoequivalence relations and whose morphisms (R X X) f (S Y Y ) are those f : X Y such that there is t : R S with the following diagram commuting: R t S (s, t) X X f f Y Y We say f g if there exists p : X S such that sp = f and tp = g. This category is called the homotopy exact completion of (C, F). There is a functor from C to its homotopy exact completion factoring through the homotopy category of C. A path object category is a category C with finite limits together with an endofunctor P : C C such that ˆ There are natural transformations r X : X P X, s X, t X : P X X and m X : P X X P X P X giving each X the structure of an internal category; ˆ P preserves finite limits; ˆ There is a natural transformation η : P P P such that This is called contraction structure. sη = id, tη = rt, (Ms)η = id, (Mt)η = rt, ηr = rr 12

13 11 Matthijs Vákár: Splitting the atom of dependent types Notes For propositional logic, the journey intuitionistic linear operational is well understood. For dependent type theory this is less clear. The motivations behind understanding linear and operational dependent type theory are: to depend understanding of HoTT as a foundation of mathematics, to develop a computational semantics of dependent types, and to do dependently-typed quantum computing. Linear logic is a weaker form of intuitionistic logic in which assumptions must be used exactly once; in particular, assumptions must be used, and upon use they must be thrown away. We can blend linear logic into dependent type theory by using contexts of the form ; Ξ. We think of as the intuitionistic region, from which variables can be used as many times as we like, and Ξ as the linear region, from which variables are used exactly once. The syntax has to be re-written, but mostly it carries over unchanged. (Some judgements about types require Ξ =.) Linear dependent types can then be interpreted in strict indexed symmetric multicategories. 12 James Cranch: How (not) to define categories in HoTT Work of Ahrens, Kapulkin and Shulman gives a convincing definition of 1-categories inside homotopy type theory. It s certainly natural to ask for much more: a full theory of (, 1)-categories. I ll describe a fragment of such a theory, where our categories are those which differ in a finitary way from the (, 1)-category of types. This fragment contains some handy examples (including all 1-categories), but also fails to capture some pretty crucial examples (such as most (, 1)- categories of structured types). I ll discuss the strengths and limitations, and hopefully describe some further aspirations. Notes HoTT is supposed to permit representations of interesting mathematical objects. represent categories in a fruitful way. We seek to Categories have previously been defined in HoTT by Ahrens, Kapulkin and Shulman; we indicate that we are using their definition by prefixing with AKS. 13

14 Definition. An AKS precategory consists of: ˆ Obj : Type; ˆ Hom : Obj Obj Type; ˆ Identity and composition operations, axioms for unit and associativity; ˆ Requirement that all Hom-types be sets. This yields the problem that there are two notions of equivalence in a category, namely = Obj and categorical isomorphism (defined by composition). There is a morphism from = Obj to categorical isomorphisms, but this morphism may not be an equivalence. Definition. An AKS category is an AKS precategory in which the morphism = Obj isos is an equivalence. The added restriction that all Hom-types be sets means we are restricted to the discussion of 1-categories. We may seek more, but when we do we bump into the usual coherence issues. For instance, we need to specify: ˆ Composition gf; ˆ A path h(gf) = (hg)f; ˆ A 2-cell filling the following pentagon: (kh)(gf) ((kh)g)f k(h(gf)) (k(hg))f k((hg)f) The key observation is that Type is an (, 1)-category, which exists constructively with no added work to be done. The idea derived from the observation is that we can define some proper (, 1)- categories by specifying low-degree cells by hand and leaving the hard work for high-degree cells to Type. Definition. Let n N. An n-concrete category consists of: ˆ Obj : Type; ˆ Obj + : Obj Type; ˆ Hom : Obj Obj Type; 14

15 ˆ Hom + : X,Y :Obj Hom(X, Y ) Obj+ (X) Obj + (Y ); ˆ conf : X,Y :Obj is-(n 2)-truncated(Hom+ (X, Y )); ˆ Unit and associativity laws; ˆ Coherence laws for (n 1)-categories. For instance, a 0-concrete category is, in some sense, just a full subcategory of Type; in this case, the truncation requirement gives that, for all objects X and Y we have Hom(X, Y ) (Obj + (X) Obj(Y )). Examples. ˆ Any n-concrete category yields an (n + k)-concrete category for all k 1 by inserting refl for coherences at level higher than n. ˆ Types, sets, groupoids and finite sets are all 0-concrete categories. ˆ is a 1-concrete category, with Obj N, Hom defined recursively, Obj + : n (set with n elements), and so on. ˆ All AKS categories are 2-concrete, and those with trivial automorphisms are 1-concrete. ˆ (Non-example) Type is not n-concrete for any n. ˆ... but n-type is (n + 1) concrete for all n. See also ˆ Slides: ˆ arxiv: : 13 Evan Cavallo: The Mayer Vietoris sequence in HoTT We present an essentially topological proof that the Mayer Vietoris sequence is exact, using the Eilenberg Steenrod axioms for cohomology. The proof uses squares and cubes, defined as higher inductive types, to simplify computations dealing with higher paths. Adopting a technique used by Dan Licata to prove that the torus is equivalent to S 1 S 1, we use cube fillers to obtain certain necessary paths, thereby avoiding explicitly calculating these paths and obtaining a dramatically cleaner proof. 15

16 The cubical approach to higher paths has also proven generally useful for mechanization in Agda, as the cube type more precisely expresses common path types that arise in proving properties of HITs. [This research was sponsored in part by the National Science Foundation under grant numbers CCF and CCF (REU) as part of my senior thesis, and conducted with collaboration from Bob Harper, Dan Licata, Carlo Angiuli, and Ed Morehouse.] See also ˆ Slides: 14 Neil Strickland: Proof assistants as a routine tool? Proof assistants have existed for decades. The vision has always been that they should become a standard part of the tool kit of mathematical research, as is now the case for systems like Sage, Maple and Mathematica. Examples like the proof of the Odd Order Theorem show that when one has a dedicated team working for an extended period, it is possible to formalise highly complex proofs. However, despite renewed and broadened interest in formalisation stemming from the development of Homotopy Type Theory, proof assistants are not yet used routinely by many mathematicians. This talk will attempt to discuss the reasons for this, and what could be done about them. It will focus on issues of documentation, user interface design, software engineering and information management. I have some knowledge of Agda and Coq, but I am very far from being an expert on either system. However, I think that this is a context where an outsider s perspective is useful. Notes This talk highlighted many of the issues affecting mathematicians seeking to use proof assistants in particular, Coq to verify their proofs as a day-to-day task, particularly those who do not already do so. It was established that much needs to be done to make Coq more accessible to working mathematicians, but there is questionable availability of the resources needed to do this. See also ˆ Slides: 16