On a model invariance problem in Homotopy Type Theory

In this article the author endows the functor category [B(Z2),Gpd] with the structure of a type-theoretic fibration category with a univalent universe using the so-called injective model structure. It gives us a new model of Martin-L\"of type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in an other work) in the same underlying category together with the very same universe that turned out to be provably not univalent with respect to projective fibrations, provides an example of two Quillen equivalent model categories that host different models of type theory. Thus, we provide a counterexample to the model invariance problem formulated by Michael Shulman.


Introduction
This article is a contribution to the ongoing effort to find models of the Univalent Foundations [Bor17b] introduced by Vladimir Voevodsky. In particular, Shulman [Shu15b,Shu15a,Shu17] with his notion of type-theoretic fibration categories prompted the development of models of the Univalence Axiom in functor categories. In [Shu15b] Shulman endowed the functor category [D, C ], where D is an inverse category and C is a type-theoretic fibration category with a univalent universe, with the structure of a type-theoretic fibration category with a univalent universe by using the so-called Reedy model structure. In [Shu15a] Shulman endowed the category [D, sSet], where D is any elegant Reedy category and sSet is the category of simplicial sets, with the structure of a type-theoretic fibration category with a univalent universe, again by using the Reedy model structure. The reader should note that inverse categories are particular cases of elegant Reedy categories that are themselves particular cases of (strict) Reedy categories. Since Reedy categories do not allow non-trivial isomorphism, this kind of index categories has strong limitations. Moreover, it is useful to note that the class of elegant Reedy categories is precisely the class of index categories for which the Reedy model structure and the so-called injective model structure on a functor category are the same. Thus, despite technical challenges that might be difficult to overcome the injective model structure on a functor category seems a reasonable candidate to find models of the Univalence Axiom in functor categories. However, in [Shu17] Shulman used a different model structure to give models in EI-diagrams. An EI-category is a category where every endomorphism is an isomorphism, groups are particular interesting cases. According to Shulman "[He] constructed a model in a certain model category that presents the homotopy theory of presheaves on an EI-category, but it is not the injective model structure on a functor category. In the case of [the target This material is based upon work supported by grant GA CR P201/12/G028.

The injective type-theoretic fibration structure on Gpd Z2
We denote the functor category [B(Z 2 ), Gpd] simply by Gpd Z2 . The reader should note that an object in Gpd Z2 is nothing but a groupoid A equipped with an involution α : A → A, i.e. an automorphism satisfying the equation Definition 2.1. A type-theoretic fibration category is a category C with : (1) A terminal object 1.
(2) A subcategory of fibrations containing all the isomorphisms and all the morphisms with codomain 1. A morphism is called an acyclic cofibration if it has the left lifting property with respect to all fibrations. (3) All pullbacks of fibrations exist and are fibrations. (4) The pullback functor along any fibration has a right adjoint that preserves fibrations. (5) Every morphism factors as an acyclic cofibration followed by a fibration.
Remark 2.2. This category-theoretic structure corresponds to an interpretation into a category of a type theory with a unit type, dependent sums, dependent products, and intensional identity types. Definition 2.3. A type-theoretic model category is a model category M satisfying the following additional properties.
(i) The pullback functor along a fibration preserves acyclic cofibrations. (ii) The pullback functor g * along a fibration g has a right adjoint Π g .
Proposition 2.4. If M is a type-theoretic model category, then its full subcategory M f of fibrant objects is a type-theoretic fibration category.
Proof. Since 1 is a fibrant object, (1) is satisfied. The wide subcategory of M f with fibrations as morphisms contains all the isomorphisms between fibrant objects and by definition all the morphisms with codomain 1, so (2) holds. The fiber product associated with two fibrations between fibrant objects is fibrant, hence pullbacks of fibrations still exist and are fibrations. So, (3) is satisfied. Since the middleman in the factorization of any morphism between fibrant objects by an acyclic cofibration followed by a fibration is a fibrant object, (5) is still true. Last, by adjunction Π g preserves fibrations if and only if g * preserves acyclic cofibrations, so by (i) we have (4).
The goal of this section consists in proving that Gpd Z2 equipped with the socalled injective model structure is a type-theoretic model category, hence (Gpd Z2 ) f is a type-theoretic fibration category. We denote by 1 the terminal object in the category Gpd of groupoids, and by a slight abuse of notation 1 will also denote the terminal object of Gpd Z2 , namely the groupoid 1 together with the identity involution. The letter I will denote the groupoid with two distinct objects and one isomorphism φ : 0 → 1 between them. Recall that Gpd has a canonical model structure where the weak equivalences are the equivalences of groupoids. The fibrations are the functors with isomorphismlifting, namely the functors with the right lifting property with respect to the inclusion i : 1 ֒→ I. The cofibrations are the injective-on-objects functors. Given a combinatorial model category M and a small category I, there exists the injective model structure on [I, M ] (see [Lur09,A.3.3] for details). In this case a morphism f ∈ [I, M ] is a weak equivalence (resp. a cofibration) if f is an objectwise weak equivalence (resp. an objectwise cofibration). A morphism is a fibration if it has the right lifting property with respect to every acyclic cofibration (i.e. a morphism which is simultaneously a weak equivalence and a cofibration). Since Gpd together with its canonical model structure is combinatorial, there exists the injective model structure on Gpd Z2 . Given a morphism f in Gpd Z2 , f will denote its image under the forgetful functor that maps an equivariant functor to its underlying functor between groupoids.
Proposition 2.5. Let C be a category together with a distinguished class of morphisms called fibrations and I a small category. Moreover, assume that C is locally presentable and for every fibration g the pullback functor along g exists and has a right adjoint. Then for any objectwise fibration g in [I, C ] the pullback functor along g has a right adjoint.
. Knowing the nature of colimits in a slice category, it is enough to check that dom(g * (colim F )) is isomorphic to colim(dom • g * • F ). Since colimits in a functor category are pointwise, it is enough to check that for every x ∈ I. But pullbacks are pointwise in a functor category, so we have an isomorphism between dom(g * (colim F ))(x) and dom((g x ) * (colim F ) x ). Since g x is a fibration, by assumption (g x ) * has a right adjoint and so it preserves all small colimits. Moreover, one has an isomorphism of the form (colim As a consequence, one has the following sequences of isomorphisms , and finally Proposition 2.6. Let M be a type-theoretic model category whose underlying model category is combinatorial and I a small category. The category [I, M ] together with the injective model structure is a type-theoretic model category. Proof. Since a combinatorial model category is locally presentable as a category and fibrations with respect to the injective model structure are in particular objectwise fibrations, by 2.5 the pullback functor along a fibration has a right adjoint. Moreover, since pullbacks are pointwise in a functor category and acyclic cofibrations are objectwise with respect to the injective model structure, we conclude by (i) for M .
Since Gpd together with its canonical model structure is a type-theoretic model category [Shu15b, Examples 2.16] and is combinatorial as a model category, by 2.6 we conclude that Gpd Z2 together with the injective model structure is a typetheoretic model category. So, by 2.4 (Gpd Z2 ) f is a type-theoretic fibration category. If we wish to take this model of type theory further, we need a better control on the fibrations of the injective model structure. This is the topic of the next section.
3. The injective model structure on Gpd Z2 made explicit Notation 3.1. We denote byǏ the groupoid I together with the involution that maps φ to φ −1 . Also, we will denote by the groupoid that extendsǏ and its involution by a third point, fixed under the Z 2 -action, denoted 2 and a second nonidentity isomorphism ψ : 1 → 2 whose image by the involution is ψ • φ (there are only two non-identity isomorphisms in and their composition). The morphism i ′ :Ǐ ֒→ is the corresponding inclusion.
Remark 3.2. Note that i ′ is an objectwise acyclic cofibration, hence it is an acyclic cofibration with respect to the injective model structure. Moreover, note that the morphismǏ → 1 is an objectwise fibration. However, consider the following lifting problem,Ǐ _ Ǐ / / 1 . A diagonal filler cannot exist, since the fixed point 2 of should be mapped to a fixed point, but such a fixed point does not exist inǏ. So, there exist objectwise fibrations that are not fibrations with respect to the injective model structure.
Notation 3.3. Remember that {i}, where i is the inclusion 1 ֒→ I, is a set of generating acyclic cofibration with respect to the canonical model structure on Gpd. Moreover, the forgetful functor from Gpd Z2 to Gpd has a left adjoint S that maps a groupoid A to A A together with the involution that swaps the two copies of A. Proof. The implication (ii) ⇒ (i) is clear, since S(i) and i ′ are objectwise acyclic cofibrations, so they are acyclic cofibrations for the injective model structure. Moreover, the class of acyclic cofibrations is closed under pushouts and transfinite compositions.
Conversely, let f : A → B be an acyclic cofibration. Since f is an objectwise acyclic cofibration, f is (isomorphic to) the inclusion of a full subgroupoid of B which is equivalent to B. Let ((ObB \ ObA)/Z 2 , ) be the set of orbits of ObB \ ObA under the Z 2 -action together with a well-ordering, let λ be the order type of this well-ordered set, and let g : (ObB \ ObA)/Z 2 → λ be an order-preserving bijection. By transfinite recursion we define a λ-sequence X, where we add the elements of ObB \ ObA to A by following our well-ordering. Take X 0 := A. For γ such that γ + 1 < λ, let s be the element of (ObB \ ObA)/Z 2 that corresponds to γ + 1 under the bijection g. We built X γ+1 as follows. If s is a singleton with unique element x, i.e. x is fixed under the Z 2 -action, then f being essentially surjective there exists an isomorphism ϕ : y → x with y ∈ A. In this case X γ+1 is the following pushouť , where the upper horizontal morphism maps φ to β(ϕ) −1 • ϕ. Otherwise, the orbit s is {x, β(x)}, and we define X γ+1 as the following pushout , where the upper horizontal arrow maps 0 to y (and 1 to α(y)). Last, if γ is a limit ordinal, then X γ is colim δ<γ X δ . For every γ < λ, X γ is a full subgroupoid of B stable under the involution β on B, and f is the transfinite composition of the λ-sequence X. We recall the notion of a universe [Shu15b, Definition 6.12] in a type-theoretic fibration category.
Definition 4.1. A fibration p : U ։ U in a type-theoretic fibration category C is a universe if the following hold. Definition 4.2. Given a universe p : U → U in a type-theoretic fibration category, a small fibration, or a U -small fibration, is a pullback of p.
Remark 4.3. A universe in a type-theoretic fibration category interprets a universe type in type theory.
Also, we recall what it means for a universe in a type-theoretic fibration category to be univalent (see also [Shu15b, section 7]). Let Type be a universe in the type theory under consideration. Given two small types, i.e. two elements of Type, there is the type of weak equivalences between them. In a type-theoretic fibration category with a universe, this dependent type is represented by a fibration E ։ U × U . Moreover, there is a natural map U → E that sends a type to its identity equivalence. By (5) one can factor the diagonal map δ : U → U × U as an acyclic cofibration followed by a fibration in the following commutative diagram, . The universe p : U ։ U is univalent if the map U → E is a right homotopy equivalence, or equivalently (by the 2-out-of-3 property and the fact that U is fibrant like any object of a type-theoretic fibration category) if the dashed map is a right homotopy equivalence. Given κ an inaccessible cardinal, we recall [Bor17a] below the construction of a non-univalent universe in the type-theoretic fibration category Gpd Z2 together with the so-called projective model structure.
• The composition in U is given by . Note that U is a groupoid. Indeed, the inverse of the morphism (ρ, τ ) is given by . We equip U with the involutionυ as follows, . One denotes by U the "unpointed" version of U , i.e. objects are of the form (A, B, ϕ) and morphisms of the form (ρ, τ ), with its corresponding involution υ.
We define the morphism p in Gpd Z2 as the projection First, we will prove that the morphism p : U → U is a fibration between fibrant objects with respect to the injective model structure on Gpd Z2 .
Lemma 4.4. The morphism p : U → U is a fibration with respect to the injective model structure.
Lemma 4.5. The groupoids U and U together with their involutions υ and υ are fibrant objects of Gpd Z2 with respect to the injective model structure.
Proof. We start with U . It suffices by 3.5 to prove that the unique morphism from U to 1 has the right lifting property with respect to S(i) and i ′ . First, assume that we have the following lifting problemǏ . Let f (φ) be the pair (ρ, τ ) : (A, B, ϕ) → (B, A, ϕ −1 ). Since f is equivariant, one concludes (ρ −1 , τ −1 ) = (τ, ρ). Hence, one has τ = ρ −1 . We define a morphism j : → U by j(φ) = f (φ), j(2) = (A, A, τ • ϕ), and j(ψ) = (ϕ −1 , τ • ϕ). The reader can easily check that j is a diagonal filler. Second, U is a projective fibrant object and S(i) is a projective acylic cofibration, so U → 1 has the right lifting property with respect to S(i). Next, recall from 4.4 that p is a fibration and fibrations are closed under composition. Thus, we deduce the fibrancy of U from the fibrancy of U in the following commutative diagram Theorem 4.6. The morphism p : U → U is a universe in the type-theoretic fibration structure on (Gpd Z2 ) f given in section 2.
Proof. It follows from 4.4 and 4.5 that p is a fibration in (Gpd Z2 ) f with respect to the injective model structure. Since small fibrations, i.e. pullbacks of p, and right adjointness, when they exist, are categorical notions they are the same in the projective and injective settings, hence conditions (i) and (ii) follow from their counterparts in [Bor17a,Thm.4.11]. Moreover, since projective acyclic cofibrations are in particular objectwise acyclic cofibrations, (iii) follows from its counterpart ibid. .
In the rest of this article we will prove that p is a univalent universe. The first step consists in constructing specific path objects in Gpd Z2 with respect to the injective model structure. Let f : A → C be a morphism in Gpd Z2 . By the universal property of the pullback, one gets the diagonal morphism δ as follows We define a groupoid P C A together with an involution π C A as follows. The objects of P C A are tuples (x, y, ϕ), where ϕ : x → y is an isomorphism in A such that f (ϕ) is the identity morphism. A morphism in P C A between (x, y, ϕ) and (x ′ , y ′ , ϕ ′ ) is a pair (ρ, τ ), where ρ : x → x ′ and τ : y → y ′ are isomorphisms in A such that f (ρ) = f (τ ) and ϕ ′ • ρ = τ • ϕ. The composition in P C A is given componentwise, i.e. (ρ ′ , τ ′ ) • (ρ, τ ) is (ρ ′ • ρ, τ ′ • τ ) whenever it makes sense. The inverse of (ρ, τ ) is (ρ −1 , τ −1 ). Define the involution π C A as follows , where α is the involution on A. We define δ 1 , δ 2 in Gpd Z2 as the following morphisms . The morphisms δ 1 and δ 2 are equivariant and δ = δ 2 • δ 1 .
Proposition 4.7. Given f : A → C in Gpd Z2 , P C A is a very good path object with respect to the injective fibration structure on Gpd Z2 .
Proposition 4.8. If f : A → C is a fibration and A is fibrant with respect to the injective model structure on Gpd Z2 , then P C A is a fibrant object.
Proof. First, since the class of fibrations is closed under pullbacks and f is a fibration, the first projection pr 1 : A × C A → A is a fibration. Then, using the facts that A is fibrant, the class of fibrations is closed under compositions, and the fact that δ 2 is a fibration as a result of 4.7, we conclude by considering the following commutative diagram Remark 4.9. The proposition 4.8 proves, under the assumption that f : A → C is a fibration in (Gpd Z2 ) f , that P C A is really a path object for our type-theoretic fibration structure on (Gpd Z2 ) f .
In the case where C := 1, A := U , and f : A → C is the unique morphism from U to 1, U being fibrant P 1 U lives in (Gpd Z2 ) f as noted in 4.9, and tracing through the interpretation of type theory we find that the space E of equivalences over U × U is isomorphic to P 1 U .
Corollary 4.10. In the type-theoretic fibration structure on (Gpd Z2 ) f given in section 2, the universe p : U → U satisfies the univalence property.
Proof. Recall (cf. the beginning of section 4) that we have to prove that the upper horizontal arrow in the following commutative diagram , which maps a small type to its identity equivalence, is a right homotopy equivalence. But this morphism is isomorphic to δ 1 , so by 4.7 and 4.5 it is an acyclic cofibration with a fibrant domain, hence a right homotopy equivalence.

Conclusion
Our new interpretation of the Univalent Foundations in the category [B(Z 2 ), Gpd] is an incremental progress in the direction of finding new type-theoretic fibration categories together with a universe satisfying the Univalence Axiom using the injective model structure on functor categories. This new model together with its previous twin model using the projective model structure on [B(Z 2 ), Gpd] provides a counterexample to Shulman's model invariance problem by showing that two Quillen equivalent model categories can host different interpretations of type theory. So, a Quillen equivalence between model categories is not trivial in the context of type theory and can make a difference with respect to the interpretation of the type theory under consideration. Last, a conjecture to the effect that "equivalent homotopy theories have equivalent internal type theories" should not mention model categories but type-theoretic fibration categories and an adequate notion of equivalence thereof.