Axioms for Modelling Cubical Type Theory in a Topos

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1 Axioms for Modelling Cubical Type Theory in a Topos Ian Orton 1 and Andrew M. Pitts 1 1 University of Cambridge Computer Laboratory, Cambridge CB3 0FD, UK Ian.Orton@cl.cam.ac.uk, Andrew.Pitts@cl.cam.ac.uk Abstract The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an interval-like object I in a topos to give a model of type theory in which elements of identity types are morphisms from I. Cohen, Coquand, Huber and Mörtberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of Kan filling that lies at the heart of their constructive interpretation of Voevodsky s univalence axiom. Furthermore, since our axioms can be satisfied in a number of different ways, we show that there is a range of topos-theoretic models of homotopy type theory in this style ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases models of dependent type theory, homotopy type theory, cubical sets, cubical type theory, topos, univalence Digital Object Identifier /LIPIcs Introduction Cubical type theory [8] provides a constructive justification of Voevodsky s univalence axiom, an axiom that has important consequences for the formalisation of mathematics within Martin-Löf type theory [28]. Working informally in constructive set theory, the authors of [8] give a model of their type theory using the category Ĉ of set-valued contravariant functors on a small category C which is the Lawvere theory for de Morgan algebra [4, Chapter XI]; see [26]. The representable functor on the generic de Morgan algebra it contains is used as an interval object I in Ĉ, with proofs of equality modelled by the corresponding notion of path, that is, by morphisms with domain I. The authors call the objects of Ĉ cubical sets. They have a richer structure compared with previous, synonymous notions [5, 18]. For one thing they allow path types to be modelled simply by exponentials X I, rather than by name abstractions [25, Chapter 4]. More importantly, the de Morgan algebra operations endow I with path-connection algebra structure that considerably simplifies the definition and properties of the constructive notion of Kan filling that lies at the heart of [8]. In particular, the filling operation is obtained from a simple special case that composes a filling at one end of the interval to a filling at the other end. Coquand [9] has suggested that this distinctive composition operation can be understood in terms of the properties of partial elements within the internal higher-order logic of toposes [21]. In this paper we show that that is indeed the case and usefully so; in particular something less than de Morgan algebra structure is sufficient and other models emerge. To accomplish this, we find it helpful to work not in higher-order predicate logic, but in extensional type theory equipped with an impredicative universe of propositions Ω, standing for the subobject classifier of the topos. Ian Orton and Andrew M. Pitts; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 0:2 Axioms for Modelling Cubical Type Theory in a Topos Working in that language, our axiomatisation concerns two structures that a topos E may possess: an object I that is endowed with some elementary characteristics of the unit interval; and a subobject of propositions Cof Ω whose elements we call cofibrant propositions and which determine the subobjects that are relevant for Kan filling (for example, unions of faces of hypercubes in the case of [8]). With respect to these two structures we define what it means for a family of types to be a fibration in the sense of Cohen et al. The relation of extending partial elements to total elements expressed within the internal type theory leads to a rather appealing and simple notion of fibration. In order that such fibrations model univalent type theory, we make a series of postulates about the interval and cofibrant subobjects that are true of the presheaf model in [8], as follows. We postulate that the interval is a non-trivial path-connection algebra (pos 2 pos 4 ), that the end points of the interval determine cofibrant subobjects (pos 5 ) and that cofibrant subobjects are closed under binary union (pos 6 ). These assumptions are enough to prove that fibrations are (stably) closed under taking dependent products and dependent functions; furthermore path types are fibrations and satisfy the rules for identity types propositionally. To get true Martin-Löf style identity types it suffices to assume that cofibrant subobjects are closed under composition (pos 7 ), which allows us to carry out a version of Swan s construction [27]. To show that a natural number object is fibrant and that fibrations are closed under binary coproducts, it suffices to further assume that the interval enjoys a form of connectedness (pos 1 ). In Section 5 we consider a weak form of univalence [28, Section 2.10], the correspondence between type-valued paths and functions that are equivalences modulo path-based equality. To do so we give the glueing construct of [8] in the internal type theory of the topos and use a postulate that ensures it has the necessary strictness properties (pos 8 ). However, we do not yet have an internalisation of the construction used in [8] to obtain a fibrant universe satisfying the univalence axiom. In Section 6 we indicate why the model in [8] satisfies our axioms and more generally which other presheaf toposes satisfy them. There is some freedom in choosing the subobject of cofibrant propositions. The path-connection algebra structure we assume for the interval I (Figure 4) is much weaker than being a de Morgan algebra. In particular, we avoid the use of a de Morgan involution and as a result the topos of simplicial sets contains a model of the axioms. In Section 7 we consider some related work and indicate future directions. The definitions and constructions we carry out in the internal type theory of toposes are sufficiently involved to warrant machine-assisted formalisation. Our tool of choice is Agda [2]. We persuaded it to provide an impredicative universe of mere propositions [28, Section 3.3] using a method due to Escardo [12]. Our development can be found at ~rio22/agda/cubical-topos/root.html. 2 Internal Type Theory of a Topos We rely on the categorical semantics of dependent type theory in terms of categories with families (CwF) [11]. For each topos E (with subobject classifier : 1 Ω) one can find a CwF with the same objects, such that the category of families at each object X is equivalent to the slice category E/X. This can be done in a number of different ways; for example [24, Example 6.14], or the more recent references [20, Section 1.3], [23] and [3], which cater for categories more general than a topos (and for contextual/comprehension categories rather than CwFs in the first two cases). Using the objects, families and elements of this CwF as a signature, we get an internal type theory, canonically interpreted in the CwF

3 I. Orton and A. M. Pitts 0:3 postulate pos 1 : [ (ϕ : I Ω). ( (i : I). ϕ i ϕ i) ( (i : I). ϕ i) ( (i : I). ϕ i)] pos 2 : [ (0 = 1)] pos 3 : [ (i : I)(e : {0,1}). cnx e e i = e = cnx e i e] pos 4 : [ (i : I)(e : {0,1}). cnx e e i = i = cnx e i e] where {0,1} {i : I i = 0 i = 1} and : {0,1} {0,1} satisfies 0 = 1 and 1 = 0 Figure 1 Postulates about the interval object in the standard fashion [16]. We make definitions and postulates in this internal language for E using a concrete syntax inspired by Agda [2]. Dependent functions are written as (x : A) B and dependent products as (x : A) B. The subobject classifier Ω becomes an impredicative universe of propositions in the internal type theory with logical connectives, equality and quantifiers,,,,,, =, (x : A), (x : A). Its universal property gives rise to comprehension subtypes: given Γ, x : A ϕ(x) : Ω, then Γ {x : A ϕ(x)} is a type whose terms are those t : A for which ϕ(t) is provable, with the proof being treated irrelevantly. 1 Taking A = 1 to be terminal, for each ϕ : Ω we have a type whose inhabitation corresponds to provability of ϕ: [ϕ] {_ : 1 ϕ} (1) We make extensive use of these types in connection with the partial elements of a type A, which are often taken to be sub-singletons, that is, terms s : A Ω satisfying (x x : A). s x s x x = x. However, it will be more convenient to work with an extensionally equivalent representation as dependent pairs ϕ : Ω and f : [ϕ] A. The proposition ϕ is the extent of the partial element; in terms of sub-singletons it is equal to (x : A). s x. Instead of quantifying externally over the objects, families and elements of the CwF associated with E, we will assume E comes with an internal full subtopos U. In the internal language we use U as a Russell-style universe (that is, if A : U, then A itself denotes a type) containing Ω and closed under forming products, exponentials and comprehension subtypes. 3 Path Types The homotopical approach to type theory [28] views elements of identity types as paths between the two elements being equated. We try to take this literally, using paths in a topos E that are morphisms out of a distinguished object I, called the interval. We assume it is equipped with morphisms 0, 1 : 1 I and cnx 0, cnx 1 : I I I satisfying the postulates in Figure 1. Postulate pos 1 amounts to a categorical connectedness property of the interval (see Section 6.1). Postulate pos 2 says that the interval is non-trivial and postulates pos 3 and pos 4 endow it with a form of path-connection algebra structure [7]. In the model of [8] this is given by the lattice structure of the interval, taking cnx 0 i j to be meet i j, cnx 1 i j to be join i j and using the fact that 0 and 1 are respectively least and greatest elements (and are distinct). 1 Our Agda development is proof relevant, so that terms of comprehension types contain a proof of membership as a component.

4 0:4 Axioms for Modelling Cubical Type Theory in a Topos Given A : U, we call terms of type I A paths in A. The path type associated with A is _ _ : A A U where a 0 a 1 {p : I A p 0 = a 0 p 1 = a 1 } (2) Can these types be used to model the rules for Martin-Löf identity types? We can certainly interpret the identity introduction rule (reflexivity), since degenerate paths given by constant functions k a i a satisfy k : {A : U}(a : A) a a. 2 However, we need further assumptions to interpret the identity elimination rule, otherwise known as path induction [28, Section ]. Coquand and Danielsson [10] give an alternative (propositionally equivalent) formulation of identity elimination in terms of substitution functions a 0 a 1 P a 0 P a 1 and contractibility of singleton types (a 1 : A) (a 0 a 1 ); see [5, Figure 2]. The path-connection algebra structure gives the latter, since using pos 3 and pos 4 we have ctr : {A : U}{a 0 a 1 : A}(p : a 0 a 1 ) (a 0, k a 0 ) (a 1, p) (3) ctr p i (p i, λj p(cnx 0 i j)) However, to get suitably behaved substitution functions we have to consider families of types endowed with some extra structure; and that structure has to lift through the type-forming operations (products, functions, identity types, etc). This is what the definitions in the next section achieve. 4 Cohen-Coquand-Huber-Mörtberg (CCHM) Fibrations In this section we show how to generalise the notion of fibration introduced in [8, Definition 13] from the particular presheaf model considered there to any topos with an interval object as in the previous section. To do so we introduce the notion of cofibrant proposition and use it to internalise the composition and filling operations described in [8]. 4.1 Cofibrant propositions Kan filling and other cofibrancy conditions on collections of subspaces have to do with extending maps from a subspace to the whole space. Here we consider subspaces as subobjects in toposes. Since subobjects are classified by morphisms to Ω, it is possible to consider collections of subobjects that are specified generically by giving a property of propositions, in other words by giving a subobject Cof Ω. A monomorphism m : A B is in the corresponding collection if its classifier λ(y : B) (x : A). m x = y : B Ω factors through Cof Ω. We reduce extension properties of morphisms out of the domains of such monomorphisms to extension properties of partial elements f : [ϕ] A whose extents ϕ are in Cof. Definition 4.1 (Cofibrant partial elements, A). We call terms of type Cof cofibrant propositions. Given a type A : U, we define the type of cofibrant partial elements of A to be A (ϕ : Cof) ([ϕ] A). Such a partial element (ϕ, f) : A extends to a : A if the following relation holds (ϕ, f) a (u : [ϕ]). f u = a (4) 2 Here and elsewhere we use the Agda convention that braces {} indicate implicit arguments.

5 I. Orton and A. M. Pitts 0:5 postulate pos 5 : [ (i : I)(e : {0,1}). cof(i = e)] pos 6 : [ (ϕ ψ : Ω). cof ϕ cof ψ cof(ϕ ψ)] pos 7 : [ (ϕ ψ : Ω). cof ϕ (ϕ cof ψ) cof(ϕ ψ)] Figure 2 Postulates about cofibrant propositions Figure 2 gives the simple properties of cofibrant propositions we need in order to define an internal notion of fibration generalising Definition 13 of [8] and show that it is closed under forming Σ-, Π- and Id-types, as well as basic datatypes. In the figure cof : Ω Ω is the classifying morphism of the subobject Cof Ω, so that Cof = {ϕ : Ω cof ϕ}. Note that postulate pos 5 implies that cof holds. Thus : 1 Ω factors through Cof Ω to give a monomorphism : 1 Cof which is generic in the sense that any cofibrant monomorphism is obtainable from it via pullback. Postulates pos 2 and pos 5 together imply that cof holds, so that A is always a cofibrant monomorphism, where is the initial object. One can show that postulate pos 7 is equivalent to requiring that the collection of cofibrant monomorphisms is closed under composition (see Appendix A). Postulate pos 6 says that the union of two cofibrant subobjects is again cofibrant and this allows us to take the join of compatible cofibrant partial elements: Definition 4.2 (Join of compatible cofibrant partial elements). Say that two cofibrant partial elements (ϕ, f), (ψ, g) : A are compatible if they agree wherever they are defined: (ϕ, f) (ψ, g) (u : [ϕ])(v : [ψ]). f u = g v (5) In that case we can form their join f g : [ϕ ψ] A, since in the topos there is a pushout square as shown below. We get (ϕ ψ, f g) in A because of postulate pos 6. [ϕ ψ] [ϕ] f [ψ] [ϕ ψ] f g g 4.2 Composition and filling structures A The outer square commutes because (ϕ, f) (ψ, g) holds and then f g is the unique induced morphism out of the pushout. If A : I U is an interval-indexed family of types, then we think of terms of dependent function type Π I A (i : I) A i as dependently typed paths. We call terms of type (Π I A) cofibrant-partial paths. Given (ϕ, f) : (Π I A), we can evaluate it at a point i : I of the interval to get a cofibrant partial element (ϕ, i : (A i): (ϕ, i (ϕ, λ(u : [ϕ]) f u i) (6) An operation for A : I U for filling from 0 takes any (ϕ, f) : (Π I A) and any a 0 : A 0 with (ϕ, 0 a 0 and extends (ϕ, f) to a dependently typed path g : Π I A with g 0 = a 0. Since we are not assuming any structure on the interval for reversing paths (such as a de Morgan involution), we also need to consider the symmetric notion of filling from 1:

6 0:6 Axioms for Modelling Cubical Type Theory in a Topos Definition 4.3 (Filling structures). The type of filling structures for I-indexed families of types, Fill : (e : {0,1})(A : I U) U, is defined by: Fill e A (ϕ : Cof)(f : [ϕ] Π I A)(a : {a : A e (ϕ, e a }) {g : Π I A (ϕ, f) g g e = a} (7) A notable feature of [8] compared with preceding work [5] is that such filling structure can be constructed from a simpler composition structure that just produces an extension at one end of a cofibrant-partial path from an extension at the other end. We will deduce this using postulates pos 3 pos 6 after defining the main notion of this paper. Definition 4.4 (The CwF of CCHM fibrations). A CCHM fibration (A, α) over a type Γ : U is a family A : Γ U equipped with a fibration structure α : Fib A, where Fib : {Γ : U}(A : Γ U) U is defined by Fib {Γ} A (e : {0,1})(p : I Γ) Comp e (A p) (8) Here Comp : (e : {0,1})(A : I U) U is the type of composition structures for I-indexed families: Comp e A (ϕ : Cof)(f : [ϕ] Π I A) {a 0 : A e (ϕ, e a 0 } {a 1 : A e (ϕ, e a 1 } (9) CCHM fibrations are closed under re-indexing: given γ : Γ and A : Γ U, we get a function [_]γ : Fib A Fib(A γ) defined by [α]γ e p α e (γ p). So we get the structure of a Category with Families by taking families to be CCHM fibrations (A, α) over each Γ : U and elements of such a family to be terms of type (x : Γ) A x. Remark 4.5 (Fibrant objects). We say A : U is a fibrant object if we have a fibration structure for the constant family λ(_ : 1) A over the terminal object 1. Note that if A : Γ U has a fibration structure, then for each x : Γ the type A x : U is fibrant. However the converse is not true: having a family of fibration structures, that is, an element of (x : Γ) Fib(λ(_ : 1) A x), is weaker than having a fibration structure for A : Γ U. For example, having a fibration structure allows one to transport elements along paths in Γ (see the subst functions defined below in (15)), whereas clearly a family of fibrant objects may not possess such transport operations. If α : Fill e A, then λϕ f a α ϕ f a e : Comp e A and so every filling structure gives rise to a composition structure. Conversely, the composition structure of a CCHM fibration gives rise to filling structure: Lemma 4.6. Given Γ : U, A : Γ U, e : {0,1}, α : Fib A and p : I Γ, there is a filling structure fill e α p : Fill e (A p) that agrees with α at e, that is: (ϕ : Cof)(f : [ϕ] Π I A)(a : A(p e)). (ϕ, f)@e a fill e α p ϕ f a e = α e p ϕ f a (10) Proof. The construction follows [8, Section 4.4], but just using the path-connection algebra structure on I, rather than a de Morgan algebra structure, making use of postulates pos 2 pos 6 and Definition 4.2. See Appendix B for the details. Compared with [5], the fact that filling can be defined from composition considerably simplifies the process of lifting fibration structure through the usual type-forming constructs.

7 I. Orton and A. M. Pitts 0:7 Theorem 4.7 (Fibrant Σ- and Π-types). There are functions Fib Σ : {Γ : U}{A 1 : Γ U}{A 2 : (x : Γ) A 1 x U} Fib A 1 Fib A 2 Fib(Σ A 1 A 2 ) Fib Π : {Γ : U}{A 1 : Γ U}{A 2 : (x : Γ) A 1 x U} Fib A 1 Fib A 2 Fib(Π A 1 A 2 ) (11) (12) where Σ A 1 A 2 x (a 1 : A 1 x) A 2 (x, a 1 ) and Π A 1 A 2 x (a 1 : A 1 x) A 2 (x, a 1 ). These functions are stable under re-indexing. Hence the category with families given by CCHM fibrations has Σ- and Π-types. Proof. The proof is inspired by the constructions in [8] and is given in Appendix C. The following two theorems are the first time we need to use postulate pos 1. Their proofs are simpler than the ones in [8] and are given in Appendices D and E. Theorem 4.8 (N is fibrant). If the topos E has a natural number object 1 Z N S N, then there is a function Fib N : {Γ : U} Fib(λ(_ : Γ) N). Hence the category with families given by CCHM fibrations has a natural number object. inl inr Theorem 4.9 (Fibrant coproducts). Writing A 1 A 1 + A 2 A 2 for the coproduct of A 1 and A 2 in E, we lift this to families of types, _ _ : {Γ : U}(A 1 A 2 : Γ U) Γ U, by defining (A 1 A 2 ) x A 1 x + A 2 x. Then there is a function Fib : {Γ : U}{A 1 A 2 : Γ U} Fib A 1 Fib A 2 Fib(A 1 A 2 ) (13) and this fibration structure on coproducts is stable under re-indexing. Hence the category with families given by CCHM fibrations has coproducts. 4.3 Identity types Theorem 4.10 (Fibrant path types). There is a function Fib Path : {Γ : U}{A : Γ U} Fib A Fib(Path A), where Path A : (x : Γ) (A x A x) U is given by Path A (x, (a 0, a 1 )) a 0 a 1 (14) where is as in (2). This fibration structure on path types is stable under re-indexing. Proof. The proof uses postulates pos 2 pos 6 ; see Appendix F. These path types in the CwF of CCHM fibrations (Definition 4.4) satisfy the Coquand- Danielsson formulation of identity types with propositional computation properties [5, Figure 2]. Thus in addition to the contractibility of singleton types (3), we get substitution functions for transporting elements of a fibration along a path subst : {Γ : U}{A : Γ U}{α : Fib A}{x 0 x 1 : Γ} (x 0 x 1 ) A x 0 A x 1 (15) subst p a α 0 p elim a where for any B : U, elim : [ ] B denotes the unique function given by initiality of [ ]. Note that (, elim ) : B because : Cof by pos 2 and pos 5 ; furthermore (, elim ) b holds for any b : B. Using Lemma 4.6 we have that these substitution functions satisfy a propositional computation rule: H : {Γ : U}{A : Γ U}{α : Fib A}{x : Γ}(a : A x) (a subst(k x) a) (16) H a fill 0 α (k x) elim a

8 0:8 Axioms for Modelling Cubical Type Theory in a Topos To get Martin-Löf identity types with standard judgemental rather than propositional computation properties from these path types, we use a version of Swan s construction [27] like the one in Section 9.1 of [8], but only using the path-connection algebra structure on I, rather than a de Morgan algebra structure. This is the only place that postulate pos 7 is used; we need the fact that the universe given by Cof and [_] : Cof U is closed under dependent products: Lemma The following term of type Ω is provable: (ϕ : Ω)(f : [ϕ] Ω). cof ϕ ( (u : [ϕ]). cof(f u)) cof( (u : [ϕ]). f u). Proof. Note that if u : [ϕ] then ( (v : [ϕ]). f v) = f u and hence cof( (v : [ϕ]). f v) = cof(f u). So (u : [ϕ]). cof(f u) equals ϕ cof( (v : [ϕ]). f v). Therefore from cof ϕ and (u : [ϕ]). cof(f u) by postulate pos 7 we get cof(ϕ (v : [ϕ]). f v) and hence cof( (v : [ϕ]). f v), since ( (v : [ϕ]). f v) ϕ. Using this lemma we get the following result, whose proof is given in Appendix G. Theorem 4.12 (Fibrant identity types). Define identity types by: Id : {Γ : U}(A : Γ U) (x : Γ) (A x A x) U (17) Id A (x, (a 0, a 1 )) (p : Path A (x, (a 0, a 1 )) {ϕ : Cof ϕ (i : I). p i = a 0 } Then there is a function Fib Id : {Γ : U}{A : Γ U} Fib A Fib(Id A) and the fibrations (Id A, Fib Id A) can be given the structure of Martin-Löf identity types in the category with families of CCHM fibrations. 5 Towards Univalence In this section we present a weak form of univalence [28, Section 2.10], using a glueing construction as in [8]. Specifically we show how to transform equivalences into paths and vice-versa just for fibrant objects, rather than for fibrant families of objects (cf. Remark 4.5). This is because we currently have no method, expressed within the internal type theory of a topos, for constructing a universe in the Category with Families of Definition 4.4 that satisfies the univalence axiom. We represent a proof of equality between types A, B : U as a fibrant family P : I U such that P 0 = A and P 1 = B. Note that since P has a fibration structure, A and B are necessarily fibrant objects. We show that given such a family P it is always possible to construct an equivalence f : A B. Conversely, given an equivalence f : A B between fibrant objects, it is always possible to construct such a P, provided cofibrant propositions satisfy a certain strictness property (Figure 3). We begin by defining a path type for elements of the universe U, in the style of (2). To do this we assume a second universe U 1 with U : U 1. We sometimes refer to terms of type U as small types and terms of type U 1 as large types. Definition 5.1 (A path type for U). Define _ U _ : U U U 1 by A U B {P : I U P 0 = A P 1 = B} (18) Next we recall the notion of equivalence using Voevodsky s definition in terms of having contractible homotopy fibres [28, Section 4.4]. We also need the notion of quasi-inverse [28, Section 4.1]. We have seen that for CCHM fibrations, path types have the properties of

9 I. Orton and A. M. Pitts 0:9 Martin-Löf equality types, up to propositional equality. It is a standard result of homotopy type theory that the type of equivalences Equiv f is a mere proposition, that is, up to path equality there is at most one equivalence structure for f; whereas Qinv f is not necessarily a mere proposition (see Chapter 4 of [28]). Nevertheless, these two types are logically equivalent in the sense that there are functions between them in each direction. Definition 5.2 (Contractibility, equivalence and quasi-inverse structure). Contr : U U Contr A (a 0 : A) ((a : A) a 0 a) Equiv : {A B : U}(f : A B) U (20) Equiv f (b : B) Contr((a : A) f a b) Qinv : {A B : U}(f : A B) U (21) Qinv f (g : B A) ((b : B) f(g b) b) ((a : A) g(f a) a) Theorem 5.3. There is a function pathtoequiv : {A B : U}(P : A U B)(ρ : Fib P ) (f : A B) Equiv f (22) Proof. (Sketch, see Appendix H for details.) Define maps f : A B and g : B A like so: (19) f a ρ 0 id elim a g b ρ 1 id elim b This definition is well-typed since P 0 = A and P 1 = B. Since both functions are defined using composition structure, for every a : A and b : B we can use filling (Lemma 4.6) to find dependently typed paths p, q : Π I P with p 0 = a, p 1 = f a, q 0 = b and q 1 = g b (23) Since P : I U has a fibration structure, A and B are fibrant objects. Therefore as in Section 4.3, we have path types for them satisfying the properties of identity types propositionally. So it is possible to combine the paths (23) to get a g (f a) and b f (g b) and hence f and g are quasi-inverses; from which we can construct an equivalence structure [28, Chapter 4]. We now wish to construct a map going the other way, from equivalences to paths in the universe. To do so we use the notion of cofibrant-partial families of types: given a type Γ : U and a cofibrant property Φ : Γ Cof, we define (Γ, Φ) : U by Γ, Φ (x : Γ) [Φ x] (24) and say that a term of type A : (Γ, Φ) U is a cofibrant-partial family of types over Γ. Next we introduce the glueing construction which allows one to extend such a cofibrantpartial family A : (Γ, Φ) U along a function f : (x : Γ)(v : [Φ x]) A(x, v) B x to a total family over Γ. Definition 5.4 (Glueing). Given Γ : U, Φ : Γ Cof, A : Γ, Φ U, B : Γ U and

10 0:10 Axioms for Modelling Cubical Type Theory in a Topos f : (x : Γ)(v : [Φ x]) A(x, v) B x), define: Glue Φ A B f : Γ U (25) Glue Φ A B f x (g : (v : [Φ x]) A(x, v)) {b : B x (v : [Φ x]). f x v (g v) = b} glue f : ((x, u) : Γ, Φ) A(x, u) Glue Φ A B f x (26) glue f (x, u) a (λ(_ : [Φ x]) a, f x u a) unglue f : (x : Γ) Glue Φ A B f x B x (27) unglue f x snd We show that this construction has a fibration structure in the case that f is an equivalence (fiber-wise), so that we can model the glueing construction from cubical type theory as a CCHM fibration. Theorem 5.5. Glue Φ A B f has a fibration structure if A and B have one and if f has the structure of an equivalence. In other words there is a function Fib Glue : {Γ : U}{Φ : Γ Cof}{A : Γ, Φ U}{B : Γ U} (f : (x : Γ)(u : [Φ x]) A(x, u) B x) ((x : Γ)(v : [Φ x]) Equiv(f x v)) Fib A Fib B Fib(Glue Φ A B f) Proof. The proof we give differs from that in [8], in that we do not need cofibrant propositions to be closed under I-indexed conjunction (cf. the quantifier defined in Section 4.1 of [8]). 3 See Appendix I. We now have a way to interpret the glueing operation from [8] that meets some of the necessary requirements; see [8, Figure 4]. However, the current construction does not have certain strictness properties. In particular cubical type theory requires that, when restricted to a context where Φ holds, glueing should be equal to A on the nose, that is that for any x : Γ with u : [Φ x], we should have Glue Φ A B f x = A(x, u). To satisfy such a requirement we make a further postulate that allows us to extend a partial type along an isomorphism (Definition 5.6) to get a total type. Note that this is weaker than directly postulating a strict glueing construction that extends along equivalences. Isomorphisms have inverses up to the extensional equality of the internal type theory, in contrast to equivalences which only have inverses up to path equality. Definition 5.6 (Isomorphism). Define _ = _ : U U U by A = B {f : A B ( g : B A) (g f = id) (f g = id)} (29) We postulate that any partial type A that is isomorphic to a total type B everywhere that A is defined, can be extended to a total type B that is isomorphic to B; see Figure 3. Lemma 5.7. For all (x, u) : Γ, Φ we have glue f (x, u) : A(x, u) = (Glue Φ A B f) x. Proof. Take an arbitrary (x, u) : Γ, Φ. In this case we have [Φ x] = 1 and hence: (Glue Φ A B f) x = (g : 1 A(x, u)) {b : B x f x u (g u) = b} = (g : A(x, u)) {b : B x f x u g = b} = A(x, u) (28) with glue f (x, u) being one side of this isomorphism. 3 However, unlike in [8], our construction does not yield an element Fib Glue f ε α β that restricts to α on Φ, which seems to be needed to satisfy the full univalence axiom.

11 I. Orton and A. M. Pitts 0:11 postulate pos 8 : {ϕ : Cof}(A : [ϕ] U)(B : U)(s : (u : [ϕ]) A u = B) (B : U) {s : B = B (u : [ϕ]). A u = B s u = s } Figure 3 The strictness postulate Using postulate pos 8 we obtain a version of glueing satisfying the strictness requirement discussed above: Definition 5.8 (Strict Glueing). Given Γ : U, Φ : Γ Cof, A : Γ, Φ U, B : Γ U and f : (x : Γ)(v : [Φ x]) A(x, v) B x, define SGlue Φ A B f : Γ U by SGlue Φ A B f x fst(pos 8 (λu : [Φ x] A(x, u)) (Glue Φ A B f x) (λu : [Φ x] glue f (x, u))) (30) Note that SGlue has the desired strictness property: given any (x, u) : Γ, Φ, by pos 8 we have A(x, u) = fst(pos 8 (λu : [Φ x] A(x, u)) (Glue Φ A B f x) (λu : [Φ x] glue f (x, u))) and hence (x : Γ)(u : [Φ x]). SGlue Φ A B f x = A(x, u) (31) Theorem 5.9. SGlue has a fibration structure if A and B have one and f has the structure of an equivalence. Proof. It is easy to show that (fibrewise) isomorphisms preserve fibration structures. Hence we obtain a fibration structure on SGlue by transporting the structure obtained from Fib Glue (Theorem 5.5) along the isomorphism from pos 8. We are now able to construct a map from equivalences to paths in the universe: Theorem There is a function equivtopath : {A B : U}{α : Fib(λ_ : 1 A)}{β : Fib(λ_ : 1 B)} (f : A B) (Equiv f) (P : A U B) (Fib P ) (32) Proof. (Sketch, see Appendix J for details.) Define the following: Φ : I Cof Φ i (i = 0) (i = 1) C : (I, Φ) U C (i, u) ((λ_ : [i = 0] A) (λ_ : [i = 1] B)) u f : (i : I)(u : [Φ i]) C(i, u) B f i (λ_ : [i = 0] f) (λ_ : [i = 1] id) Now let P SGlue Φ C (λ_ : I B) f and observe that P 0 = A and P 1 = B by the strictness property of SGlue. Further, we can show that f is an equivalence since f and the identity are both equivalences; and using α, β and pos 1, we can define a fibration structure on C. Hence, by Theorem 5.9, we get a fibration structure on P.

12 0:12 Axioms for Modelling Cubical Type Theory in a Topos The univalence axiom for a universe V in a CwF (with Σ-, Π- and Id-types) states that for every A, B : V the canonical function from Id V A B to (f : A B) Equiv f is an equivalence. As mentioned previously, we currently have no method for constructing a univalent universe V in the CwF of Definition 4.4 using the internal type theory. So we cannot even state, let alone prove the univalence axiom. However, for any two fibrant objects A, B : U (see Remark 4.5), we can prove related results regarding the following functions: pathtopath : (P : A U B)(ρ : Fib P ) (P : A U B) (ρ : Fib P ) pathtopath P ρ equivtopath (fst(pathtoequiv P ρ)) (snd(pathtoequiv P ρ)) equivtoequiv : (f : A B)(e : Equiv f) (f : A B) (e : Equiv f ) equivtoequiv f e pathtoequiv (fst(equivtopath f e)) (snd(equivtopath f e)) The following results are proved in Appendices K and L. Theorem Given fibrant objects A, B : U, there is a function: pathinv : (P : A U B)(ρ : Fib P )(i : I) P i U fst(pathtopath P ρ) i (33) Theorem Given fibrant objects A, B : U, there is a function: equivinv : (f : A B)(e : Equiv f) f fst(equivtoequiv f e) (34) As a corollary, since for a function f between fibrant objects Equiv f is a mere proposition [28, Section 3.3], we can deduce that there exists a function: equivinv : (f : A B)(e : Equiv f) (f, e) (equivtoequiv f e). (35) 6 Satisfying the Axioms Working informally in constructive set theory, the authors of [8] give a model of their type theory using the topos Ĉ of contravariant set-valued functors on a particular small category C that they call the category of cubes. Its objects are all finite subsets of a countably infinite set with decidable equality (whose elements can be thought of as names of cartesian directions). Given two such subsets X and Y, the C-morphisms X Y are all de Morgan algebra [4, Chapter XI] homomorphisms from the free de Morgan algebra on the finite set Y to that on X; composition and identities are as in the (opposite of the) category of sets. Thus C is in fact a presentation of de Morgan algebra as a Lawvere theory [22, 1]: C is universal among categories with finite products containing an internal de Morgan algebra. Let us replace C by an arbitrary small category C and see what is required of C for the topos Ĉ to have an interval object and subobject of cofibrant propositions satisfying the postulates in Figures 1 3. We do not aim for complete generality, just enough to encompass some examples of independent interest such as C = the category of inhabited finite linearly ordered sets [0 < 1 < < n], for which Ĉ = sset, the category of simplicial sets, widely used in homotopy theory [15]. 6.1 The interval object We take the interval object I Ĉ to be the representable functor y i C(_, i) on some object i C. The following theorem gives a useful criterion for such an interval object to satisfy postulate pos 1. Its proof is given in Appendix M.

13 I. Orton and A. M. Pitts 0:13 constants: 0 1 binary operations: equations: 0 x = 0 = x 0 1 x = x = x 1 0 x = x = x 0 1 x = 1 = x 1 Figure 4 Path-connection algebra Theorem 6.1. In a presheaf topos Ĉ, a representable functor I = y i satisfies postulate pos 1 (Figure 1) if C is a cosifted category, that is, if finite products in Set commute with colimits over C op [13]. A more elementary characterisation of cosiftedness is that C is inhabited and for every pair of objects c, c C the category of spans c c is a connected category [1, Theorem 2.15]. The category has this property and hence the natural candidate for an interval in sset, namely y i when i is the 1-simplex [0 < 1], satisfies pos 1. Any category with finite products trivially has the property. This is the case for C and thus the interval in the model of [8], where C = C and i is any one-element set (the underlying object of the internal de Morgan algebra), satisfies pos 1. In addition to pos 1, the other postulates in Figure 1 say that I is a non-trivial model of the algebraic theory in Figure 4, that we call path connection algebra. (See also Definition 1.7 of [14], which considers a similar notion in a more abstract setting.) The 1-simplex in sset is a path-connection algebra, the constants being its two end points and the binary operations being induced by the order-preserving binary operations of minimum and maximum on [0 < 1]. The generic de Morgan algebra in Ĉ is a path-connection algebra: the constants are the least and greatest elements and the binary operations are meet and join. An obvious variation on the theme of [8] would be to replace C by the Lawvere theory for path-connection algebra. 6.2 Cofibrant propositions and the strictness postulate In a topos with an interval object, there are many candidates for a subobject Cof Ω satisfying the postulates in Figure 2. For example, relying on the impredicative nature of the subobject classifier, one can take an internal intersection of all subobjects of Ω containing {0} and {1} and closed under binary union and dependent intersection to obtain a smallest Cof satisfying pos 5 pos 7. Although it is not described that way, this is the notion of cofibrant proposition in Ĉ used for the model in [8]. Next we discuss satisfaction of the strictness postulate pos 8 in presheaf toposes. To do so, recall that in a topos of presheaves Ĉ, the subobject classifier Ω maps each c C to the set of sieves on c, that is, pre-composition closed subsets S obj(c/c). Let Ω dec Ω be the subpresheaf consisting of those S that are decidable subsets of obj(c/c). Of course if the ambient set theory satisfies the Law of Excluded Middle, then Ω dec = Ω. In general Ω dec classifies monomorphisms α : F G in Ĉ for which each injective function α c : F (c) G(c) has decidable image. The proof of the following result is sketched in Appendix N. Theorem 6.2. Interpreting the universe U as the Hofmann-Streicher lifting [17] of a Grothendieck universe in Set, a subobject Cof Ω in a presheaf topos Ĉ satisfies the strictness postulate pos 8 (Figure 3) if it is contained in Ω dec Ω. It is not hard to see that Ω dec also satisfies pos 6 and pos 7. It also satisfies pos 5 if for example equality of C-morphisms is decidable. Therefore we get a model of our axioms in Ĉ by taking Cof = Ω dec, for any cosifted C that has decidable equality of morphisms and a

14 0:14 Axioms for Modelling Cubical Type Theory in a Topos representable with the structure of a path-connection algebra; C and are both examples, as is the Lawvere theory of path-connection algebra. In the case of Ĉ this is a different choice of cofibrant proposition to the one in [8]. In the case of ˆ = sset, it remains to be investigated what is the relationship between this constructive model based on CCHM fibrations and Voevodsky s non-constructive model of univalent type theory using classical simplicial sets [20]. 7 Related and Future Work The work reported here was inspired by [8]. We have shown how to express Cohen, Coquand, Huber and Mörtberg s notion of fibration in the internal type theory of a topos (Definition 4.4). We found that quite a simple collection of axioms (Figures 1 3) suffices for this to model Martin-Löf type theory with path types satisfying a weak form of univalence. The construction in Section 8.2 of [8] of a fibrant universe satisfying the full univalence axiom uses a modified form of Hofmann-Streicher lifting [17] within the ambient constructive set theory. We plan to investigate whether that, or indeed some other universe construction, can be axiomatised within the internal type theory of a topos. We found that only a simple path-connection algebra, rather than a de Morgan algebra structure, is needed on the interval. Furthermore, the collection of propositions suitable for Kan filling can be chosen in various ways. This allows a model of our axioms in constructive simplicial sets as one example, besides variations on the notion of cubical set. In Section 6 we only considered how presheaf categories can satisfy our axioms. It would be interesting to consider models in sheaf toposes, particularly gros toposes such as [19] that allow the interval object to be the usual topological interval. Our concerns (modelling intensional type theory with path-like identity types, getting cubical and simplicial sets as instances) are similar to those of Gambino and Sattler [14]; although our methods differ, it seems we arrive at the same notion of fibration, although this remains to be investigated. Birkedal et al [6] are developing guarded cubical type theory with a semantics based on an axiomatic version of [8] within the internal logic of a presheaf topos. We believe that our approach to modelling cubical type theory is more general than theirs, but that there will be interesting synergies. Acknowledgements. We thank Thierry Coquand for many conversations about the work reported in [8] and the possibility of an internal characterisation of its constructions. We are grateful to him, Christian Sattler, Bas Spitters and Andrew Swan for comments on the work reported here. References 1 J. Adámek, J. Rosický, and E. M. Vitale. Algebraic Theories: a Categorical Introduction to General Algebra, volume 184 of Cambridge Tracts in Mathematics. Cambridge University Press, Agda Project. wiki.portal.chalmers.se/agda. 3 S. Awodey. Natural models of homotopy type theory. ArXiv e-prints, arxiv: v2 [math.ct], March R. Balbes and P. Dwinger. Distributive Lattices. University of Missouri Press, M. Bezem, T. Coquand, and S. Huber. A model of type theory in cubical sets. In R. Matthes and A. Schubert, editors, Proc. TYPES 2013, volume 26 of Leibniz International Proceedings in Informatics (LIPIcs), pages Leibniz-Zentrum fuer Informatik, 2014.

15 I. Orton and A. M. Pitts 0:15 6 L. Birkedal, A. Bizjak, R. Clouston, H. B. Grathwohl, B. Spitters, and A. Verozzi. Guarded cubical type theory. Abstract of a talk at TYPES 2016, Novi Sad, Serbia, R. Brown and G. H. Mosa. Double categories, 2-categories, thin structures and connections. Theory and Applications of Categories, 5(7): , C. Cohen, T. Coquand, S. Huber, and A. Mörtberg. Cubical type theory: a constructive interpretation of the univalence axiom. Preprint, December T. Coquand. A cubical type theory, August Slides of an invited talk at Domains XII, University of Cork, Ireland, 10 T. Coquand and N. A. Danielsson. Isomorphism is equality. Indagationes Mathematicae, 24(4): , P. Dybjer. Internal type theory. In S. Berardi and M. Coppo, editors, Types for Proofs and Programs, volume 1158 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, M. Escardo. A small type of propositions à la Voevodsky in Agda, August bham.ac.uk/~mhe/impredicativity/. 13 P. Gabriel and F. Ulmer. Lokal Präsentierbare Kategorien, volume 221 of Lecture Notes in Mathematics. Springer-Verlag, N. Gambino and C. Sattler. Uniform Fibrations and the Frobenius Condition. ArXiv e-prints, arxiv: , October P. G. Goerss and J. F. Jardine. Simplicial Homotopy Theory. Modern Birkhäuser Classics. Birkhäuser Basel, M. Hofmann. Syntax and semantics of dependent types. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, Publications of the Newton Institute, pages Cambridge University Press, M. Hofmann and T. Streicher. Lifting Grothendieck universes. Unpublished note, S. Huber. A Model of Type Theory in Cubical Sets. Licentate thesis, University of Gothenburg, May P. T. Johnstone. On a topological topos. Proceedings of the London Mathematical Society, s3-38(2): , C. Kapulkin, P. L. Lumsdaine, and V. Voevodsky. The simplicial model of univalent foundations. arxiv preprint, arxiv: v2 [math.lo], April J. Lambek and P. J. Scott. Introduction to Higher Order Categorical Logic. Cambridge University Press, F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings of the National Academy of Sciences of the United States of America, 50(1): , P. L. Lumsdaine and M. A. Warren. The local universes model: An overlooked coherence construction for dependent type theories. ACM Trans. Comput. Logic, 16(3):23:1 23:31, July A. M. Pitts. Categorical logic. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures, pages Oxford University Press, A. M. Pitts. Nominal Sets: Names and Symmetry in Computer Science, volume 57 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, B. Spitters. Cubical sets as a classifying topos. Abstract of a talk at TYPES 2015, Tallinn, Estonia, A. Swan. An algebraic weak factorisation system on 01-substitution sets: A constructive proof. ArXiv e-prints, arxiv: , September URL: arxiv.org/abs/ The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations for Mathematics. homotopytypetheory.org/book, Institute for Advanced Study, 2013.

16 0:16 Axioms for Modelling Cubical Type Theory in a Topos A Dependent Conjunction of Cofibrant Propositions A monomorphism f : A B in E is cofibrant if its classifying morphism B Ω factors through Cof Ω, or equivalently, if (b : B). cof( (a : A). f a = b) holds in the internal language. Lemma. Postulate pos 7 is equivalent to requiring the class of cofibrant monomorphisms to be closed under composition. Proof. First suppose that pos 7 holds and that f : A B and g : B C are cofibrant. So both (b : B). cof( (a : A). f a = b) and (c : C). cof( (b : B). g b = c) hold and we wish to prove (c : C). cof( (a : A). g(f a) = c). Note that for b : B and c : C g b = c ( (a : A). g(f a) = c) = ( (a : A). f a = b) cof( (a : A). g(f a) = c) = cof( (a : A). f a = b) = since g is a mono So for ϕ (b : B). g b = c and ψ (a : A). g(f a) = c, we have cof ϕ and ϕ cof ψ. Therefore by pos 7 we get cof(φ ψ), which is equal to cof(ψ) since ψ ϕ. So we do indeed have (c : C). cof( (a : A). g(f a) = c). Conversely, suppose cofibrant monomorphisms are closed under composition and that ϕ, ψ : Ω satisfy cof ϕ and ϕ cof ψ. That cof ϕ holds is equivalent to the monomorphism [ϕ] 1 being cofibrant; and since ϕ (ψ = ϕ ψ) (cof ψ = cof(ϕ ψ)) from ϕ cof ψ we get ϕ cof(ϕ ψ) and hence the monomorphism [ϕ ψ] [ϕ] is cofibrant. Composing these monomorphisms, we have that [ϕ ψ] 1 is cofibrant, that is, cof(ϕ ψ) holds. B Proof of Lemma 4.6 Suppose Γ : U, A : Γ U, e : {0,1}, α : Fib A, p : I Γ, ϕ : Cof, f : [ϕ] Π I (A p), a : A(p e) with (ϕ, e a, and i : I. Using Definition 4.2 we can define fill e α p ϕ f a i α e (p i) (ϕ i = e) (f i g i) a, where (36) p : I I Γ is defined by p i j p(cnx e i j) f : (i : I) [ϕ] Π I (A (p i)) is defined by f i u j f u (cnx e i j) g : (i : I) {g : [i = e] Π I (A (p i)) (ϕ, f i) (i = e, g )} is defined by g i v j a. Note that ϕ i = e is in Cof by postulates pos 5 and pos 6. Furthermore g is well-defined, because if v : [i = e], then by pos 3 we have a : A(p e) = A(p(cnx e e j) = A(p e j) = (A (p e)) j = (A (p i)) j; so g : [i = e] Π I (A (p i)). Similarly, if u : [ϕ] and v : [i = e], then f i u j = f u (cnx e i j) = f u (cnx e e j) = f u e = a = g i v j; so (ϕ, f i) (i = e, g ). For each i : I we have fill e α p ϕ f a i : A(p i e) = A(p(cnx e i e)) = A(p i) by pos 4 ; and hence fill e α p ϕ f a : Π I (A p). Furthermore, since α e (p i) : Comp e (A (p i)) we have (u : [ϕ i = e]). (f i g i) u e = α e (p i) (ϕ i = e) (f i g i) a = fill e α p ϕ f a i. Hence for all i : I, by pos 4 we have (u : [ϕ]). f u i = f u (cnx e i e) = f i u e = fill e α p ϕ f a i, so that (ϕ, f) fill e α p ϕ f a; and (v : [i = e]). a = g i v e = fill e α p ϕ f a i, so that a = fill e α p ϕ f a e. So fill e α p : Fill e(a p). Note also that in the definition of fill,

17 I. Orton and A. M. Pitts 0:17 if i = e, then (ϕ i = e) = ϕ by pos 2, p i = p and f i g i = f by pos 4 ; and hence we have (10). That fill is stable under re-indexing is the property (γ : Γ)(e : {0,1})(α : Fib A)(p : I ). fill{ } {A γ} e (α[γ]) p = fill{γ} {A} e α (γ p) and this is immediate from the definitions of α[γ] (Definition 4.4) and fill (36). C Proof of Theorem 4.7 For (11), given Γ : U, A 1 : Γ U, A 2 : (x : Γ) A 1 x U, α 1 : Fib A 1, α 2 : Fib A 2, e : {0,1}, p : I Γ, ϕ : Cof, f : [ϕ] Π I ((Σ A 1 A 2 ) p) and (a 1, a 2 ) : (Σ A 1 A 2 )(p e) with (ϕ, e (a 1, a 2 ), using Lemma 4.6 we define Fib Σ α 1 α 2 e p ϕ f (a 1, a 2 ) (α 1 e p ϕ f 1 a 1, α 2 e q ϕ f 2 a 2 ), where (37) f 1 : [ϕ] Π I (A 1 p) f 1 u i fst(f u i) q : I (x : Γ) A 1 x q p, fill e α 1 p ϕ f 1 a 1 f 2 : [ϕ] Π I (A 2 q) f 2 u i snd(f u i) and the definition of q using notation for pairing functions: _, _ : {Γ : U}{A 1 : Γ U}{A 2 : (x : Γ) A 1 x U} (f 1 : (x : Γ) A 1 x)(f 2 : (x : Γ) A 2 (x, f 1 x)) (x : Γ) (a 1 : A 1 x) A 2 (x, a 1 ) f 1, f 2 x (f 1 x, f 2 x) (38) Thus Fib Σ α 1 α 2 e p ϕ f (a 1, a 2 ) : (Σ A 1 A 2 )(p e); and since (u : [ϕ]). f 1 u e = α 1 e p ϕ f 1 a 1 = fill e α 1 p ϕ f 1 a 1 e (u : [ϕ]). f 2 u e = α 2 e q ϕ f 2 a 2 hold, it follows that (ϕ, f)@e Fib Σ α 1 α 2 e p ϕ f (a 1, a 2 ). Hence Fib Σ α 1 α 2 : Fib(Σ A 1 A 2 ). For (12), given Γ : U, A 1 : Γ U, A 2 : (x : Γ) A 1 x U, α 1 : Fib A 1, α 2 : Fib A 2, e : {0,1}, p : I Γ, ϕ : Cof, f : [ϕ] Π I ((Π A 1 A 2 ) p), g : (Π A 1 A 2 )(p e) with (ϕ, e g and a 1 : A 1 (p e), using Lemma 4.6 and the cofibrant partial elements (, elim ) from Sect. 4.3

18 0:18 Axioms for Modelling Cubical Type Theory in a Topos we define Fib Π α 1 α 2 e p ϕ f g a 1 α 2 e q ϕ f 2 a 2, where (39) f 1 : Π I (A 1 p) f 1 fill e α 1 p elim a 1 q : I (x : Γ) A 1 x q p, f 1 f 2 : [ϕ] Π I (A 2 q) f 2 u i f u i (f 1 i) a 2 : {a 2 : A 2 (q e) (ϕ, f 2 e a 2} a 2 g(f 1 e) Because fill e α 1 p elim a 1 e = a 1, we have Fib Π α 1 α 2 e p ϕ f g a 1 : A 2 (q b) = A 2 (p e, f 1 e) = A 2 (p e, a 1 ) (40) Furthermore, since (ϕ, f 2 e α 2 e q ϕ f 2 a 2, for any u : [ϕ] we have f u e a 1 = f u e (f 1 e) = f 2 u e = α 2 e q ϕ f 2 a 2 = Fib Π α 1 α 2 e p ϕ f g a 1 (41) Since (40) and (41) hold for all a 1 : A 1 (p e), from the first if follows that Fib Π α 1 α 2 e p ϕ f g : (Π A 1 A 2 )(p e) and from the second that (ϕ, e Fib Π α 1 α 2 e p ϕ f g. Therefore we have Fib Π α 1 α 2 : Fib(Π A 1 A 2 ). The stability of Fib Σ and Fib Π under re-indexing follows from that of fill (Lemma 4.6). D Proof of Theorem 4.8 Suppose Γ : U, e : {0,1}, p : I Γ, ϕ : Cof, f : [ϕ] Π I (λ_ N) and n : N with (ϕ, e n. Note that since in a topos equality on the natural number object is decidable, for each u : [ϕ] the property λ(i : I) (f u i = n) : I Ω is decidable; hence by postulate pos 1 and the fact that f u e = n, we also have f u e = n. Therefore we can get Fib N e p ϕ f n : {n : N (ϕ, e n } just by defining: Fib N e p ϕ f n n. E Proof of Theorem 4.9 The proof of (13) makes use of unique choice in the internal type theory of the topos: uc : (A : U)(ϕ : A Ω) [!(a : A). ϕ a] {a : A ϕ a} (42) where!(a : A). ϕ a (a : A). ϕ a (a : A). ϕ a a = a. Suppose we have Γ : U, A 1 A 2 : Γ U, α 1 : Fib A 1, α 2 : Fib A 2, e : {0,1}, p : I Γ, ϕ : Cof, g : [ϕ] Π I ((A 1 A 2 ) p) and c : A 1 (p e) + A 2 (p e) with (ϕ, e c. Note that for all u : [ϕ] and i : I P 1 u i!(a 1 : A 1 (p i)). g u i = inl a 1 P 2 u i!(a 2 : A 2 (p i)). g u i = inr a 2