UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOPY TYPES. Denis-Charles Cisinski

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1 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES by Denis-Charles Cisinski Abstract. We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotoy tyes (obtained from Grothendieck s theory of test categories). In ractice, this means for instance that, aart from the homotoy theory of simlicial sets, intensional tye theory with the univalent axiom can be interreted in the homotoy theory of cubical sets (with connections or not), or of Joyal s cellular sets. We recall that any right roer cofibrantly generated model category structure on a (re)sheaf toos whose cofibrations are exactly the monomorhims is a tye theoretic model category in the sense of Shulman [Shu12, Definition 2.12]. This means that, u to coherence issues which are solved by Kaulkin, Lumsdaine, and Voevodsky [KLV12] and by Lumsdaine and Warren [LW], we can interret Martin- Löf intensional tye theory in such a model category. The urose of these notes is to rove the existence of univalent universes in suitable model categories for (local systems of) homotoy tyes, (such as simlicial sets or cubical sets): resheaves over a local test category in the sense of Grothendieck which is also elegant in the sense of Bergner and Rezk. We give two constructions. The first one uses Voevodsky s construction in the setting of simlicial sets as well as Shulman s extension to simlicial resheaves over elegant Reedy categories. This has the advantage of giving a rather short roof, but the disadvantage of giving a non-exlicit construction. The second one consists to develo the theory of minimal fibrations in the context of resheaves over an Eilenberg-Zilber Reedy category (which is a slightly more restrictive notion than the one of elegant Reedy category), following classical aroaches (as in [GZ67] for instance), and then to check how robust is Voevodsky s roof. The latter oint of view is much more general and also gives another roof of Shulman s construction of univalent universes, at least in the case of simlicial resheaves on Eilenberg-Zilber Reedy categories. The rearation of these notes started after discussions I had with Thierry Coquand about his joint work with Marc Bezem and Simon Huber on cubical sets [BCH14]. His kind invitation to give a talk at the Institut Henri Poincaré in Paris gave the decisive imulse to turn these into actual mathematics.

2 2 D.-C. CISINSKI Contents 1. First construction: reduction to the case of simlicial sets Minimal fibrations Second construction: extension of Voevodsky s roof References First construction: reduction to the case of simlicial sets We write for the category of simlices (the full subcategory of the category of small category whose objects are the non-emty finite totally ordered sets). For an integer n 0, we write n for the resheaf reresented by the totally ordered set {0,...,n}. If A is a small category, we write  for the category of resheaves of sets over A, and sâ =  for the category of simlicial resheaves over A. We will consider  as a full subcategory of sâ (by considering sets as constant resehaves on ). Given a cardinal κ, a morhism of resheaves X over a small category A will be said to have κ-small fibers if, for any object a of A and any section of over a, the fiber roduct a X, seen as a resheaf over A/a, is κ-accessible. Let A be an elegant Reedy category (see [BR13, Definition 3.2]). Let us consider the locally constant model structure on the category of simlicial resheaves over A, that is the left Bousfield localization of the injective model structure on the category of simlicial resheaves by mas of the form f 1 n : a n b n for any ma f : a b in A and any integer n 0. The fibrant objects of the locally constant model structure are thus the injectively fibrant simlicial resheaves X such that, for any ma a b in A, the induced morhism X b X a is a simlicial weak homotoy equivalence. Note that the locally constant model structure is right roer (this follows from [Cis06, Theorem and Corollary ], for instance), and is thus a tye theoretic model category. We let κ an inaccessible cardinal larger than the cardinal of the set of arrows of A. Proosition 1.1. There exists a univalent universe π : U U in sâ which classifies fibrations with κ-small fibers in the locally constant model structure. Proof. Let : V V a univalent universe in sâ with resect to the injective model structure, classifying fibrations with κ-small fibers (this exists by virtue of a result of Shulman; see [Shu13, Theorem 5.6]). Then, as the (, 1)-category associated to the locally constant model structure is an -toos (because it corresonds to the (,1)-category of functors from the -grouoid of A to the (,1)-category of - grouoids), one can aly a general result of Rezk [Lur09, Theorem ], and get the existence of a univalent universe u to homotoy with resect to the locally constant model structure; see [GK12, Proosition 6.10]. In other words, one can find a univalent fibration between fibrant objects q : W W of the locally constant model structure such that, for any fibration f : X of the locally constant model

3 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 3 structure, there exists an homotoy ullback square in sâ of the following form. Let us choose a (strict) ullback square f X W q W W w V q W w V and choose a factorization of the ma w into a trivial cofibration : W U followed by a fibration u : U V (with resect to the injective model structure). Then U is fibrant for the locally constant model structure. Moreover, if we ut U = U V V, then the ma W U is a trivial cofibration of the injective model structure (because the latter is right roer) and thus U is also fibrant with resect to the locally constant model structure. The rojection π : U U being a fibration of the injective model structure between fibrant objects of the locally constant model structure, it is a fibration for the locally constant model structure. By construction, the rojection π is a univalent universe u to homomotoy. It remains to rove that any fibration of the locally constant model structure can be obtained as a strict ullback of π. Let f : X be a fibration of the locally constant model structure (with κ-small fibers). Then the ma f can be obtained as a strict ullback of the universe. f X x V y The classifying ma y : V can be lifted to U u to simlicial homotoy. Indeed, there exists an homotoy ullback square in the locally constant model structure of the following form. f V X µ U λ This square is also an homotoy ullback square in the injective model structure: the comarison ma X U U is a weak equivalence between fibrant and cofibrant objects over with resect to the model structure on sâ/ induced by the locally constant model structure, and is thus a simlicial homotoy equivalence (in articular, a weak equivalence of the injective model structure), and the ma π being a fibration of the injective model structure (which is right roer), this roves our U π

4 4 D.-C. CISINSKI assertion. Therefore, we obtain the homotoy ullback square below in the injective model structure. X V f uλ V The fibration being univalent, there exists a ma h : 1 V such that h {0} = y and h {1} = uλ. As the ma u is a fibration of the injective model structure, the commutative square {1} λ U 1 h V admits a lift l : 1 U. If we define y : U as y = l {0}, then we have uy = h {0} = y. In other words, we then obtain a strict ullback square f X x U y U which shows that π is a univalent universe for the locally constant model structure in the strict sense Assume now that A is a local test category in the sense of Grothendieck; see [Mal05, Definition and Theorem 1.5.6]. Then the category  is endowed with a roer combinatorial model structure with the monomorhisms as cofibrations, while the weak equivalences are the mas X which induce a simlicial weak homotoy equivalence N A/X N A/ (where A/X denotes the category of elements of the resheaf X, while N is the nerve functor); see [Cis06, Corollary and Theorem ]. We will refer to this model structure (which is in articular tye theoretic) as the Grothendieck model structure. Moreover, the inclusion functor  sâ is then a left Quillen equivalence with the locally constant model structure, with right adjoint the evaluation at zero functor. If, moreover, the category A is elegant, the universe π : U U obtained in Proosition 1.1 induces a fibration between fibrant objects π 0 : U 0 U 0 in Â. Theorem 1.3. The fibration π 0 : U 0 U 0 is a univalent universe in  which classifies fibrations with κ-small fibers. Proof. Remark that the category is an examle of a (local) test category and that the Grothendieck model structure on coincides with the usual model structure (whose fibrant objects are the Kan comlexes). Let D :  be a normalized cosimlicial resolution in the sense of [Cis06, Definition ] (this always exists: π u

5 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 5 see [Cis06, ] for a canonical examle). Note that, in articular, D 0 is the terminal resheaf. By virtue of [Cis06, Proosition , Corollaries and ], we then have a left Quillen equivalence from the locally constant model structure to the Grothendieck model structure with right adjoint given by the formula Real D : sâ  Sing D :  sâ Sing D (X) n = Hom(D n, X) (where Hom denotes the internal Hom of  with resect to the cartesian roduct). Let f : X be a fibration of  with κ-small fibers. Then, the ma Sing D (f ) being a fibration of the locally constant model structure with κ-small fibers, there exists a (strict) ullback square of the following form. Sing D (X) U Sing D (f ) Sing D () U π Evaluating at zero thus gives the (strict) ullback below. f X U 0 U 0 As evaluating at zero is a right Quillen equivalence, the ma π 0 is a univalent universe u to homotoy (because the fibration π has this roerty), and what recedes thus roves that it is a univalent universe in the strict sense. Corollary 1.4. For any elegant local test category A, the Grothendieck model structure on the category of resheaves of sets  suorts a model of intensional tye theory with deendent sums and roducts, identity tyes, and as many univalent universes as there are inaccessible cardinals greater than the set of arrows of A. The most well known examle of elegant local test categories is rovided by the category of simlices, the Grothendieck model structure on being then the standard model structure of simlicial sets. The receding corollary is in this case Voevodsky s theorem that univalence holds in simlicial sets; see [KLV12]. As the roof of this corollary relies on Voevodsky s results, the interesting examles for us are of course the other elegant (local) test categories. We give a few examles below. Examle 1.5. The category of cubical sets suorts a Grothendieck model structure. Indeed, cubical sets are resheaves on the category which is defined as follows. Let be the full subcategory of the category of sets whose objects are the π 0

6 6 D.-C. CISINSKI sets n = {0,1} n, n 0. This is a (strict) symmetric monoidal category (with cartesian roduct as tensor roduct). The category is defined as the smallest monoidal subcategory of generated by the mas δ 0 : 0 1 δ 1 : 0 1 σ : 1 0 where δ e is the constant ma with value e. For n 1, 1 i n and e = 0,1, one defines a ma δ i,e n = 1 i 1 δ e 1 n i : n 1 n, and for n 0 and 1 i n, a ma σ i n = 1 i 1 σ 1 n+1 i : n+1 n. The category is an elegant test category; see [Cis06, Cor , Pro and ]. Moreover, the Grothendieck model structure on admits the following exlicit descrition. We will consider the oneda embedding as an inclusion, and thus write n for the resheaf reresented by n for each n 0. The boundary of n is defined as the union of the images of the mas δ i,e n for 1 i n and e = 0,1, and will be denoted by n. We also define, for 1 i n and e = 0,1, the resheaf i,e n as the union in n of the images of the mas δ j,ε n for (j,ε) (i,e). Then the generating cofibrations (trivial cofibrations) of the Grothendieck model structure on are the inclusions n n, n 0 ( i,e n n, n 1, 1 i n, e = 0,1, resectively.) In other words, the fibrations are recisely the cubical Kan fibrations; see [Cis06, Theorem ]. Examle 1.6. Cubical sets with connections also give an examle of a category of resheaves over an elegant test category. The category c is defined as the smallest monoidal subcategory of generated by the mas δ 0, δ 1 and σ as above, as well as by the ma γ : 2 = given by γ(x,y) = su{x,y}. Cubical sets with connections are resheaves of sets on the category c. The category c is an elegant test category for the same reasons as for (alying [Cis06, Pro and ] for instance). But in fact, as was roved by Maltsiniotis [Mal09, Pro. 3.3], it has a better roerty: it is a strict test category, which means that the weak equivalences of the Grothendieck model category structure on the category of cubical sets with connections are closed under finite roducts (while this roerty is known to fail for cubical sets without connections). We define the inclusions n n and i,e n n in the same way as for cubical sets above. Note that the category of cubical sets with connections has a natural (non symmetric) closed monoidal structure induced

7 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 7 by the monoidal structure on c (using Day convolution). The following theorem is a rather direct consequence of its analog for cubical sets, but we state and rove it exlicitely for the convenience of the reader and for future reference. Theorem 1.7. The Grothendieck model category structure on the category of cubical sets with connections is a roer cofibrantly generated monoidal model category with generating cofibrations n n, n 0, and generating trivial cofibrations i,e n n, n 1, 1 i n, e = 0,1. Furthermore, the class of weak equivalences is closed under finite roducts. Proof. As the category c is elegant, it is clear that the boundary inclusions n n generate the class of monomorhisms. We have an obvious inclusion functor u : c which rovides an adjunction u! : c : u where u! is the left Kan extension of u. The functor u! is symmetric monoidal and sends n as well as i,ε n to their versions with connections (it is sufficient to rove this for n, which follows from [Cis06, Lemma ]). Therefore, the functor u! reserves monomorhisms, and, as any ma between reresentable cubical sets with connections is a weak equivalence, using the last assertion of [Cis06, ], we see immediately that the functor u! is a left Quillen functor. In articular, the inclusions i,ε n n are trivial cofibrations of cubical sets with connections. The theorem now follows straight away from [Cis06, Lemma ] which allows to aly [Cis06, Lemma and Theorem ] (the last assertion is due to the fact, that c is a strict test category). Examle 1.8. For 1 n ω, one can consider Joyal s category Θ n, which is to strict n-categories what is to categories (in articular, Θ 1 = ). The category Θ n can be thought of as the full subcategory of the category of strict n-categories whose objects are the strict n-categories freely generated on finite asting schemes of dimension n (Joyal also gave a descrition of Θn o as the category of n-disks). We know from [BR13, Cor. 4.5] that Θ n is elegant, and from [CM11, Examles 5.8 and 5.12] that Θ n is a strict test category. Therefore, the category of resheaves of sets on Θ n carries a Grothendieck model category structure in which the class of weak equivalences is closed under finite roducts. 2. Minimal fibrations 2.1. In this section, we fix once and for all an Eilenberg-Zilber category A. This means that A is a Reedy category with the following roerties. (EZ1) Any ma in A has a section in A. (EZ2) If two mas in A have the same set of sections, then they are equal.

8 8 D.-C. CISINSKI Such a Reedy category is elegant; see [BR13, Proosition 4.2]. We also consider given a model category structure on Â, whose cofibrations recisely are the monomorhisms. Given a reresentable resheaf a, we denote by a a the boundary inclusion (which means that a is the maximal roer subobject of a). These inclusions form a generating set for the class of cofibrations. We also choose an interval I such that the rojection I X X is a weak equivalence for any resheaf X on A (e.g. we can take for I the subobject classifier of the toos Â). We write I = {0} {1} I for the inclusion of the two end-oints of I Let h : I X be an homotoy. For e = 0,1, we write h e for the comosite X = {e} X I X h. Given a subobject S X, we say that h is constant on S if the restriction h I S factors through the second rojection I S S. Given a resheaf X, a section of X is a ma x : a X with a a reresentable resheaf. The boundary of such a section x is the ma x : a a x X. Definition 2.3. Let X be an object of Â. Two sections x,y : a X are -equivalent if the following conditions are satisfied. (i) These have the same boundaries: x = y. (ii) There exists an homotoy h : I a X which is constant on a, and such that h 0 = x and h 1 = y. We write x y whenever x and y are -equivalent. A minimal comlex is a fibrant object S such that, for any two sections x,y : a S, if x and y are -equivalent, then x = y. A minimal model of X is a trivial cofibration S X with S a minimal comlex. Proosition 2.4. Let X be a fibrant object. The -equivalence relation is an equivalence relation. Proof. This can be roved directly (exercise). Here is a fancy argument. The interval I defines an enrichment of the category  over the category of cubical sets: given two resheaves E and F over A, the cubical set Ma(E, F) is defined by Ma(E, F) n = HomÂ(E I n, F) for n 0, with I n the cartesian roduct of n coies of I. The functor Ma(E, ) is a right Quillen functor to the Grothendieck model category structure on the category of cubical sets (see Examle 1.6). Therefore, given a section x : a X, we can form the following ullback square. X(x) Ma(a, X) e x Ma(a, X)

9 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 9 (where e denotes the terminal cubical set). The section x can be seen as a global section of the cubical Kan comlex X(x). The relation of cubical homotoy is an equivalence relation on the set of oints of X(x). Finally, note that, if y : a X is another section such that x = y, then X(x) = X(y), from which we deduce right away this roosition. Proosition 2.5. Let ε {0,1}, and consider a fibrant object X, together with two mas h,k : I a X, with a reresentable, such that the restrictions of k and h coincide on I a {1 ε} a. Then the sections h ε and k ε are -equivalent. Proof. Put x = h ε, y = k ε, and z = h 1 ε = k 1 ε. We will use the cubical maing saces as in the roof of the receding roosition. We can think of h and k as two fillings in the commutative square {1 ε} 1 z h or k ξ Ma(a, X) Ma(a, X) in which ξ corresond to the restriction of h (or k) to I a. As mas in the homotoy category of ointed cubical sets over Ma(a, X), we must have h = k. Therefore, there exists a ma H : 2 Ma(a, X) with the following roerties. The restriction of H to {0} 1 = 1 (res. to {1} 1 = 1 ) is h (res. k), the restriction to 1 {1 ε} is constant with value z, and the following diagram commutes. 1 σ H 2 Ma(a, X) 1 ξ Ma(a, X) The restriction of H to {ε} defines a ma l : I a X which is constant on a and such that l 0 = x and l 1 = y. Lemma 2.6. Let X be a fibrant object, and x 0,x 1 : a X two degenerate sections. If x 0 and x 1 are -equivalent, then they are equal. Proof. For ε = 0,1, there is a unique coule ( ε,y ε ), where ε : a b ε is a slit eimorhism in A and y ε : b ε X is a non-degenerate section of X such that x ε = y ε ε. Let us choose a section s ε of ε. As x 0 and x 1 are degenerate and since x 0 = x 1, we have x 0 s 0 = x 1 s 0 and x 0 s 1 = x 1 s 1. On the other hand, we have y ε = x ε s ε. We thus have the equalities y 0 = y 1 1 s 0 and y 1 = y 0 0 s 1. These imly that the mas ε s 1 ε : b 1 ε b ε are in A + and that b 0 and b 1 have the same dimension. This means that ε s 1 ε is the identity for ε = 0,1. In other words, we have b 0 = b 1 and y 0 = y 1, and we also have roven that 0 and 1 have the same sections, whence are equal. Theorem 2.7. Any fibrant object has a minimal model.

10 10 D.-C. CISINSKI Proof. Let X be a fibrant object. We choose a reresentant of each -equivalence class. A section of X which is a chosen reresentative of its -equivalence class will be called selected. By virtue of Lemma 2.6, we may assume that any degenerate section X is selected. Let E be the set of subobjects S of X such that any section of S is selected Then E is not emty: the image of any selected section of the 0-skeleton of X is an element of E. By Zorn s lemma, we can choose a maximal element S of E (with resect to inclusion). Remark that any selected section x : a X whose boundary x factors through S must belong to S. Indeed, if x is degenerate, then it factors through a hence through S. Otherwise, let us consider S = S Im(x). A non-degenerate section of S must either factor through S or be recisely equal to x. In any case, such a section must be selected, and the maximality of S imlies that S = S. We will rove that S is a retract of X and that the inclusion S X is an I-homotoy equivalence. This will rove that S is fibrant and thus a minimal model of X. Let us write i : S X for the inclusion ma. Consider triles (T,h,), where T is a subobject of X which contains S, : T S is a retraction (i.e. the restriction of to S is the identity), and h : I T X is a ma which is constant on S, and such that h 0 is the inclusion ma T X, while h 1 = i. Such triles are ordered in the obvious way: (T,h,) (T,h, ) if T T, with h I T = h and T =. By Zorn s lemma, we can choose a maximal trile (T,h,). To finish the roof, it is sufficient to rove that T = X. In other words, it is sufficient to rove that any non-degenerate section of X belongs to T. Let x : a X be a non-degenerate section which does not belong to T. Assume that the dimension of a is minimal for this roerty. Then x must factor through T, so that, if we define T to be the union of T and of the image of x in X, then we have a bicartesian square of the following form. a x T a x T We have a commutative square {0} a I a {0} a x X h(1 x) If we ut u = (h(1 x),x) : I a {0} a X, we can choose a ma H : I a X such that H 0 = x, while H I a = h(1 x). If we write y 0 = H 1, as h 1 factors through S, we see that the boundary y 0 must factor through S. Let y be the selected section -equivalent to y 0. We choose an homotoy K : I a X which is constant on a and such that K 0 = y 0 and K 1 = y. By an easy ath lifting argument (comosing the homotoies H and K), we see that we may choose H such that y = y 0. Note that, as y is selected with boundary in S, we must have y in S. We obtain the commutative

11 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 11 diagram I a 1I x I T H I a X so that, identifying I T with I a I a I T, we define h = (H,h) : I T X. Similarly, the commutative diagram a x T h a y S defines a ma = (y,) : T = a a T X. It is clear that the trile (T,h, ) extends (T,h,), which leads to a contradiction. Proosition 2.8. Let X be a fibrant object and i : S X a minimal resolution of X. Consider a ma r : X S such that ri = 1 S (such a ma always exists because i is a trivial cofibration with fibrant domain). Then the ma r is a trivial fibration. Proof. There exists a ma h : I X X which is constant on S and such that h 0 = ir and h 1 = 1 X : we can see i as a trivial cofibation between cofibrant and fibrant objects in the model category of objects under S, and r is then an inverse u to homotoy in this relative situation. Consider the commutative diagram below. a a u v X S r We want to rove the existence of a ma w : a X such that w a = u and rw = v. As X is fibrant, there exists a ma k : I a X whose retriction to I a is h(1 I u), while k 0 = iv. Let us ut w = k 1. Then w = w a = (h(1 I u)) 1 = h 1 u = u. It is thus sufficient to rove that v = rw. But k and h(1 I w) coincide on I a {1} a and thus, by virtue of Proosition 2.5, we must have k 0 (h(1 I w)) 0. In other words, we have iv irw. As ri = 1 S, this imlies that v rw, and, by minimality of S, that v = rw. Lemma 2.9. Let X be a minimal comlex and f : X X a ma which is I-homotoic to the identity. Then f is an isomorhism. Proof. Let us choose once and for all a ma h : I X X such that h 0 = 1 X and h 1 = f. We will rove that the ma f a : X a X a is bijective by induction on the dimension d of a. If a is of dimension < 0, there is nothing to rove because there is no such a. Assume that the ma f b : X b X b is bijective for any object b of dimension

12 12 D.-C. CISINSKI < d. Consider two sections x,y : a X such that f (x) = f (y). Then, as f is injective in dimension lesser than d, the equations f x = f (x) = f (y) = f y imly that x = y. On the other hand, we can aly Proosition 2.5 to the mas h(1 I x) and h(1 I y) for ε = 0, and we deduce that x y. As X is minimal, this roves that x = y. It remains to rove the surjectivity. Let y : a X be a section. For any ma σ : b a in A such that b is of degree lesser than d, there is a unique section x σ : b X such that f (x σ ) = σ (y) = yσ. This imlies that there is a unique ma z : a X such that f z = y. The ma I a 1 I z I a h X, together with the ma {1} a = a y X, define a ma ϕ = (h(1 I z),x), and we can choose a filling k in the diagram below. I a {1} a ϕ I a Let us ut x = k 0. Then x = z, and thus f (x) = y. Alying Proosition 2.5 to the mas k and h(1 I x) for ε = 1, we conclude that f (x) y. The object X being a minimal comlex, this roves that f (x) = y. Proosition Let X and be two minimal comlexes. Then any weak equivalence f : X is an isomorhism of resheaves. Proof. If f : X is a weak equivalence, as both x and are cofibrant and fibrant, there exists g : X such that f g and gf are homotoic to the identify of and of X, resectively. By virtue of the receding lemma, the mas gf and f g must be isomorhisms, which imly right away that f is an isomorhism. Theorem Let X be a fibrant object of Â. The following conditions are equivalent. (i) The object X is a minimal comlex. (ii) Any trivial fibration of the form X S is an isomorhim. (iii) Any trivial cofibration of the form S X, with S fibrant, is an isomorhism. (iv) Any weak equivalence X S, with S a minimal comlex, is an isomorhism. (v) Any weak equivalence S X, with S a minimal comlex, is an isomorhism. Proof. It follows immediately from Proosition 2.10 that condition (i) is equivalent to condition (iv) as well as to condition (v). Therefore, condition (v) imlies condition (iii): if i : S X is a trivial cofibration with S fibrant and X minimal, then S must be minimal as well, so that i has to be an isomorhism. Let us rove that condition (iii) imlies condition (ii): any trivial fibration : X S admits a section i : S X which has to be a trivial cofibration with fibrant domain, and thus an isomorhism. It is now sufficient to rove that condition (ii) imlies condition (i). By virtue of Theorem 2.7, there exists a minimal model of X, namely a trivial cofibration S X with S a minimal comlex. This cofibration has a retraction which, by virtue k X

13 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 13 of Proosition 2.8, is a trivial fibration. Condition (ii) imlies that S is isomorhic to X, and thus that X is minimal as well. Definition A fibration : X in  is minimal if it is a minimal comlex as an object of Â/ = Â/ for the induced model category structure (whose, weak equivalences, fibrations and cofibrations are the mas which have the corresonding roerty in Â, by forgetting the base). Proosition The class of minimal fibrations is stable by ullback. Proof. Consider a ullback square X u X v in which is a minimal fibration. Let x,y : a X two global sections which are -equivalent over (i.e. -equivalent in X, seen as a fibrant object of Â/). Then u(x) and u(y) are -equivalent in X over, and thus u(x) = u(y). As (x) = (y), this means that x = y. In other words, is a minimal fibration. Everything we roved so far about minimal comlexes has its counterart in the language of minimal fibrations. Let us mention the roerties that we will use later. Theorem For any fibration : X, there exists a trivial fibration r : X S and a minimal fibration q : S such that = qr. Proof. By virtue of Theorem 2.7 alied to, seen as a fibrant resheaf over A/, there exists a trivial cofibration i : S X such that q = S : S is a minimal fibration. As both X and S are fibrant (as resheaves over A/), the embedding i is a strong deformation retract, so that, by virtue of Proosition 2.8 (alied again in the context of resheaves over A/), there exists a trivial fibration r : X S such that ri = 1 S, and such that qr =. Remark In the factorisation = qr given by the receding theorem, q is necessarily a retract of. Therefore, if belongs to a class of mas which is stable under retracts, the minimal fibration must have the same roerty. Similarly, as r is a trivial fibration, if belongs to a class which is defined u to weak equivalences, then so does q. This means that this theorem can be used to study classes of fibrations which are more general than classes of fibrations of model category structures. Proosition For any minimal fibrations : X and : X, any weak equivalence f : X X such that f = is an isomorhism. Proof. This is a reformulation of Proosition 2.10 in the context of resheaves over A/.

14 14 D.-C. CISINSKI Lemma For any cofibration v : and any trivial fibration : X, there exists a trivial fibration : X and a ullback square of the following form. X u X v Proof. We use Joyal s trick. The ullback functor v : Â/ Â/ has a left adjoint v! and a right adjoint v. We see right away that v v! is isomorhic to the identity (i.e. that v! is fully faithful), so that, by transosition, v v is isomorhic to the identity as well. Moreover, the functor v reserves trivial fibrations because its left adjoint v reserves monomorhisms. We define the trivial fibration as v ( : X ). Proosition Consider a commutative diagram of the form X w i 0 1 X 1 X j 1 in which 0, 1 and 1 are fibrations, w is a weak equivalence, j is a cofibration, and the square is cartesian. Then there exists a cartesian square X 0 i 0 X 0 0 j 0 in which 0 is a fibration, as well as a weak equivalence w : X 0 X 1 such that 1w = 0 and i 1 w = w i 0. Proof. By virtue of Theorem 2.14, we can choose a trivial fibration r 1 : X 1 S and a minimal fibration q : S such that 1 = q r 1. Let us write S = S, and k : S S for the second rojection. The canonical ma r 1 : X 1 S is a trivial fibration (being the ullback of such a thing), and the rojection q : S is a minimal fibration by Proosition We have thus a factorisation 1 = qr 1. Moreover, the ma r 0 = r 1 w is a trivial fibration. To see this, let us choose a minimal model u : T X 0 of the fibration 0. Then the ma r 1 wu is a weak equivalence between minimal fibrations and is thus an isomorhism by Proosition This means that r 0 is isomorhic to a retraction of the ma u, and is therefore a trivial fibration by Proosition 2.8. The diagram we started from now has the following

15 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 15 form. X w i 0 1 X 1 X 1 r 0 S r 1 k S r 1 Moreover, both squares are cartesian. This means that we can relace j by k. In other words, without loss of generality, it is sufficient to rove the roosition in the case where 0, 1 and 1 are trivial fibrations. Under these additional assumtions, we obtain a cartesian square q X 0 i 0 j X 0 q in which 0 is a trivial fibration by Lemma The lifting roblem 0 j 0 X 0 X 0 i 0 w X 1 i 1 X 1 w 0 1 has a solution because i 0 is a cofibration and 1 a trivial fibration. Moreover, any lift w must be a weak equivalence because both 0 and 1 are trivial fibrations. Remark One can rove the receding roosition in a much greater generality, without using the theory of minimal fibrations: the roof of [KLV12, Theorem 3.4.1] can be carried out in any toos endowed with a model category structure whose cofibrations recisely are the monomorhisms. Lemma Assume that the model category structure on  is right roer. For any trivial cofibration v : and any minimal fibration : X, there exists a minimal fibration : X and a ullback square of the following form. X u X v Proof. Let us factor the ma v as a trivial cofibration u : X X followed by a fibration : X. By virtue of Theorem 2.14, we can factor into a trivial

16 16 D.-C. CISINSKI fibration q : X X followed by a minimal fibration : X. We thus get a commutative square X u X v with u = qu. The rojection X is a minimal fibration (Proosition 2.13). On the other hand, the model category structure being right roer, the comarison ma X X is a weak equivalence over. Therefore, Proosition 2.16 imlies that this comarison ma is an isomorhism, and thus that this commutative square is cartesian. Proosition Assume that the model category structure on  is right roer. For any trivial cofibration v : and any fibration : X, there exists a fibration : X and a ullback square of the following form. X u X v Proof. By virtue of Theorem 2.14, there exists a factorisation of as = qr with r a trivial fibration and q a minimal fibration. We can extend q and then r, using Lemmata 2.20 and 2.17 successively. 3. Second construction: extension of Voevodsky s roof 3.1. Let A be a small category. A class of resheaves C on A is saturated by monomorhisms is it satisfies the following roerties. (a) The emty resheaf is in C. (b) For any ushout square X u i X v in which X, X and are in C and i is a monomorhism, then is in C. (c) For any well ordered set α and any functor X : α  such that the natural ma X i X j is a monomorhism for any i < j in α, if X i is in C for any i α, then lim i α X i is in C. (d) Any retract of an object in C is in C. Proosition 3.2. If A is an elegant Reedy category, any class C of resheaves on A which is saturated by monomorhisms and contains the reresentable resheaves contains all the resheaves on A. i

17 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 17 Proof. As the boundary inclusions a a form a generating family for the class of monomorhisms, it is sufficient to rove that the boundaries a belong to C for any reresentable resheaf a. We roceed by induction on the dimension d of a. If d 0, then the only roer subobject of a is the emty resheaf, which belongs to C by definition. If d > 0, consider the set E of roer subobjects K of a which are in C. It is clear that E is non-emty because the emty subobject of a is an element of E. Note that a subobject K of a is roer if and only if the identity of a is not contained in the set of non-degenerate sections of K. Therefore, roer subobjects are stable by arbitrary unions in a. Since any totally ordered set has a cofinal well ordered subset, by Zorn s lemma, there exists a maximal element K in E. Let us rove that K = a. If not, then let us choose a section u : b a of a which does not belong to K. We may choose u such that the dimension of b is minimal with resect to this roerty. Thus u must be non-degenerate, and the boundary u : b a must factor through K. We then have a ushout square of the following form, where L a denotes the union of K and of the image of u in a. b u K b u L By induction, b is in C, and as b is reresentable, it belongs to C. As K is in C as well, L must be in C, which gives a contradiction. Therefore, the boundary a = K is in C. Proosition 3.3. If A is an elegant Reedy category, then any A-localizer is regular. In other words, for any model category structure on  whose cofibrations recisely are the monomorhisms, and for any resheaf X on A, the family of sections of X exhibit X as an homotoy colimit of reresentable resheaves (see [Cis06, Définition ]). Proof. This follows right away from [Cis06, Exemle , Proosition ] and from the receding roosition. A consequence of the receding roosition is that, in the category of resheaves on an elegant Reedy category, the notion of weak equivalence is local in the following sense. Corollary 3.4. Let A be an elegant Reedy category, and assume that  is endowed with a model category structure whose cofibrations recisely are the monomorhisms. Consider a commutative triangle of the form X f q S

18 18 D.-C. CISINSKI in which both and q are fibrations. Then f is a weak equivalence if and only if, for any reresentable resheaf a and any section s : a S, the induced morhism a S X a S is a weak equivalence. Proof. This obviously is a necessary condition (because, the ullback functor along a S is a right Quillen functor from Â/S to Â/a, and thus, by Ken Brown s lemma, reserves weak equivalences between fibrant objects). The converse follows from the receding roosition and from [Cis06, Corollaire ]. One can also give a more elementary roof using directly Proosition 3.2 as follows. Relacing A by A/S, we may assume without loss of generality that S is the terminal object. The class of resheaves Z such that the induced ma Z X Z is a weak equivalence is saturated and contains the reresentable resheaves, so that it contains the terminal object by Proosition Let A be an elegant Reedy category, and assume that  is endowed with a roer model category structure whose cofibrations recisely are the monomorhisms. Given two fibrations : X S and q : S, we will write (3.5.1) Hom S (X, ) S for the ma corresonding to the internal Hom of Â/S through the equivalence Â/S Â/S. In other words, given a ma T S, morhisms from T to Hom S (X, ) over S corresond bijectively to morhisms of the form T S X T S over T. Given any ma T S, we have the canonical ullback square below. (3.5.2) Hom T (T S X, T S ) T Hom S (X, ) Remark that, if and q are fibrations, then the ma (3.5.1) is a fibration as well: as our model category structure is right roer, the ullback functor along is a left Quillen functor from Â/S to Â/S, so that its right adjoint is a right Quillen functor, hence reserves fibrant objects. In this case, we define a subresheaf Eq S (X, ) Hom S (X, ) by requiring that, for any ma T S, a section of Hom S (X, ) over T factors through Eq S (X, ) if and only if the corresonding ma T S X T S is a weak equivalence. This actually defines a subresheaf of Hom S (X, ) over S recisely because of Corollary 3.4. Proosition 3.6. Under the assumtions of aragrah 3.5, for any fibrations X S and S, the structural ma Eq S (X, ) S is a fibration. Proof. Consider a commutative square of the following form K k Eq S (X, ) S j L S

19 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 19 in which the ma j is a trivial cofibration. As the structural ma Hom S (X, ) S is a fibration, one can find a lift l in the solid commutative square below. K k Hom S (X, ) j L The equation lj = k means that we have a ullback square of the form l S K S X k K S j S 1 X L S X l j S 1 X L S in which the two vertical mas are weak equivalence (by right roerness) as well as k. Therefore, the ma l : L Hom S (X, ) factors through Eq S (X, ), which roduces a lift in the commutative square we started from. Definition 3.7. Let A be a small category. A strongly roer model category structure on  is a roer model category structure whose cofibrations recisely are the monomorhisms, and such that the notion of fibration is local over A in the following sense: any morhism of resheaves : X, the following conditions are equivalent. (i) The ma : X is a fibration. (ii) For any reresentable resheaf a and any section a, the first rojection a X a is a fibration Let A be a small category, and assume that  is endowed with a strongly roer model category structure. Consider an infinite regular cardinal κ which is greater than the cardinal of the set of arrows of A. We will use the construction of Hofmann and Streicher [Str14] of a universe of fibrations with κ-small fibers. We denote by Set κ some full subcategory of the category of sets of cardinal lesser than κ, such that, for any cardinal α < κ, there exists a set of cardinal α in Set κ. Let W be the resheaf whose set of sections over an object a of A is the set of functors (A/a) o Set κ. Given a ma f : a b in A, the recomosition with the induced functor A/a A/b defines the corresonding ma f : W b W a. Similarly, let W be the resheaf whose set of sections consists of coules (X,s), where X is a resheaf on A/a with values in Set κ (i.e. an element of W a ), and s is a global section of X. Forgetting the sections defines a mohism of resheaves ρ : W W. Note that, since we have canonical equivalences Â/a Â/a, any element X of W a determines canonically a morhism X : X a; in fact, one can identify the elements of W a as the data of a resheaf X on A with values in Set κ, together with a ma X a, as well

20 20 D.-C. CISINSKI as with secified cartesian squares f (X) X b for any morhism f : b a in A (with f (X) a resheaf with values in Set κ ). We define the resheaf U as the subresheaf of W whose sections over a reresentable resheaf a are the elements X such that the corresonding morhism X : X a is a fibration. We define U by the following ullback square. The following lemma is straightforward. π U U Lemma 3.9. Assume that there are two cartesian squares of the following form. X u X f W a ρ W X v If has κ-small fibers and v is a monomorhism, then there exists a ma y : W such that y v = y. In articular, the case where is emty tells us that a morhism of resheaves over A has κ-small fibers if and only if it can be obtained as a ullback of the morhism ρ : W W. Proosition Under the assumtions of aragrah 3.8, let : X be ma with κ-small fibers, and choose a classifying cartesian square. X Then is a fibration if and only if the classifying ma y factors through U W. y Proof. This is a reformulation of the last art of the definition of strong roerness. Corollary Assume that there are two cartesian squares of the following form. X u X W ρ W x y W ρ W X x U v y π U

21 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 21 If is a fibration with κ-small fibers and if v is a monomorhism, then there exists a ma y : U such that y v = y. Theorem Under the assumtions of aragrah 3.8, if A is an Eilenberg-Zilber category and if κ is an inaccessible cardinal, the ma π : U U is a univalent fibration between fibrant objects which classifies fibrations with κ-small fibers. Proof. The fact that U is fibrant is a reformulation of roosition 2.21 and of Corollary The fact that π is a fibration follows straight away from Proosition Let π 0 : U 0 = U U U U and π 1 : U 1 = U U U U be the ullbacks of the fibration π along the first and second rojection of U U to U, resectively. By virtue of Proosition 3.6, we have a canonical fibration (s,t) : Eq U U (U 0, U 1 ) U U. As the ullback of the fibration π i along the diagonal U U U is canonically isomorhic to π for i = 0, 1, the fibration (s, t) has a canonical section over the diagonal U U U, which rovides a morhism id : U Eq U U (U 0, U 1 ) such that (s,t)id is the diagonal (or equivalently, such that sid = tid = 1 U ). The roerty that π is univalent means that this ma id is a weak equivalence. It is thus sufficient to rove that the fibration t : Eq U U (U 0, U 1 ) U is a trivial fibration. Consider a cofibration j :. Then a commutative square ξ Eq U U (U 0, U 1 ) j ξ U consists essentially of a commutative diagram of the form X w i 0 1 X 1 0 in which 0, 1 and 1 are fibrations (with κ-small fibers), w is a weak equivalence, and the square is cartesian (where the trile ( 0,w, 1 ) correonds to ξ, the fibration 1 corresonds to ξ, and the cartesian square to the equation ξ j = tξ). Therefore, Proosition 2.18 together with Corollary 3.11 give a ma ζ : Eq U U (U 0, U 1 ) such that tζ = ξ and ζj = ξ. Proosition Let A and B be a small categories, with B having the structure of an elegant Reedy category. Assume that  is endowed with a strongly roer model category structure, and consider the associated injective model category structure on the category 1 j t X 1 1

22 22 D.-C. CISINSKI of resheaves on B with values in  (for which cofibrations and weak equivalences are defined termwise with resect to evaluation at objects of B). Then this defines a strongly roer model category structure on  B. Proof. Given a resheaf X on C = A B and a resheaf F on B, we obtain a resheaf Hom B (F, X) on A whose sections over an object a are given by Hom B (F, X) a = Hom B (F, X a) where X a is the resheaf on B obtained by evaluating X at a. Given an object b of B, the oneda lemma for resheaves over B gives the identification X b = Hom B (b, X), and we set M b X = Hom B (b, X). The injective model category structure on the category of resheaves on B with values in  coincides with the Reedy model structure. This means that a morhism of resheaves : X on C is a fibration if and only if, for any object b of B, the induced ma q b : X b b Mb M b X is a fibration of Â. Consider a ma : X such that, for any reresentable resheaf c on C and any section c, the canonical ma c X c is a fibration. Let b be an object of B. We want to rove that q b is a fibration of Â. But the model category structure on  being strongly roer, it sufficient to rove the existence of lifts in commutative diagrams of the from j K L a X b q b b Mb M b X in which j is a trivial cofibration and a is a reresentable resheaf on A. Such a lifting roblem is equivalent to a lifting roblem of the form L b K b L b a b in which, for any resheaves E and F on A and B resectively, we write E F for the cartesian roduct of the ullbacks of E and F along the rojections A B A and A B B resectively. But c = a b is then a reresentable resheaf on C, and we are reduced to a lifting roblem of the following form. L b K b c X X L b c

23 UNIVALENT UNIVERSES FOR ELEGANT MODELS OF HOMOTOP TPES 23 As the rojection c X c is a fibration, this achieves the roof. Examles of Eilenberg-Zilber test categories are the simlicial category, the cubical category (1.5), the cubical category with connections c (1.6), and Joyal s categories Θ n for 1 n ω (1.8). Once we are here, we can get a much more exlicit roof of Corollary 1.4, at least in the case of Eilenberg-Zilber local test categories: we aly Theorem 3.12 to the Grothendieck model structure on the category of resheaves on an Eilenberg-Zilber local test category (see 1.2), which is meaningful for we have the following result. Theorem Let A be an elegant local test category. The Grothendieck model category structure on  is strongly roer. Proof. We already know that the Grothendieck model category structure is roer (this does not use the roerty that A is an elegant Reedy category and is true for any local test category). Let : X be morhism such that, for any section a, the induced ma a X a is a fibration. We want to rove that is a fibration. Note that A/ is a again an elegant local test category and that, under the identification Â/ Â/, the Grothendieck model structure on Â/ coincides with the model category structure on Â/ induced by the Grothendieck model structure on Â. Therefore, relacing A by A/, we may assume that is the terminal object. We thus have a resheaf X on A such that a X a is a fibration for any reresentable resheaf a, and we want to rove that X is fibrant. We will consider the minimal model structure on  (corresonding to the minimal A-localizer; see [Cis06, Théorème 1.4.3]), and will rove first that X is fibrant for the minimal model structure. Let us choose an interval I such that the rojection Z I Z belongs to the minimal A-localizer (e.g. I might be the subobject classifier; see [Cis06, 1.3.9]). By virtue of [Cis06, Remarque , Proosition ], we have to check that the ma from X to the terminal resheaf has the right lifting roerty with resect to the inclusions of the form I a {ε} a I a for any reresentable resheaf a and ε = 0,1. But lifting roblems of shae I a {ε} a I a are in bijection with lifting roblems of the form v u X I a {ε} a (,u) I a (r 2,v) r 2 a X where : I a {ε} a a is the restriction of the second rojection I a a. Hence X is fibrant for the minimal model structure (because the rojection being a

24 24 D.-C. CISINSKI a fibration for the Grothendieck model structure, it is also a fibration for the minimal model structure). By virtue of Proosition 3.3 and [Cis06, Proosition ], the Grothendieck model category structure is the left Bousfield localization of the minimal model category structure on  by the set of mas between reresentable resheaves. It is thus sufficient to rove that, for any ma between reresentable resheaves u : a b, the ma Ma(b, X) Ma(a, X) is a weak equivalence (where the maing saces are constructed from the minimal model structure). The latter is equivalent to the ma Ma b (b,b X) Ma b (a,b X) where Ma b denotes the maing sace with resect to the model category structure on Â/b induced by the minimal model structure on Â. The rojection from b X to b being a fibration of the Grothendieck model category structure, we deduce that X is local with resect to the left Bousfield localization by the mas between reresentable resheaves, and thus that X is fibrant in the Grothendieck model category structure. Remark The receding theorem, together with Proosition 3.13, gives a new roof, in the case of Eilenberg-Zilber categories, of Shulman s result that the injective model structure for simlicial resheaves suorts a model of intensional tye theory with univalent universes [Shu13, Theorem 5.6]. Remark The roof of Theorem 3.14 would have been much easier if we would have exhibited a generating set of trivial cofibrations of the Grothendieck model category structure of the form K a with a reresentable. This haens in ractice (e.g. horn inclusions for simlicial sets, oen boxes for cubical sets), but I don t know if this is true for a general elegant local test category. In fact, there is a candidate for a counter-examle. Let Ω be the category of finite rooted trees considered by Weiss and Moerdijk for their notion of dendroidal sets. In a short note in rearation (in collaboration with D. Ara and I. Moerdijk), it will be shown that Ω is a test category. Although Ω is not an elegant Reedy category for the simle reason that it is not a Reedy category, for any normal dendroidal set X (i.e. such that, for any tree T, the automorhisms of T act freely on the set of sections of X over T), the category Ω/ X is an Eilenberg-Zilber Reedy category. As a consequence, given any reasonable model of the oerad E (that is any weakly contractible normal dendroidal set), the category Ω/E is an Eilenberg-Zilber test category. On the other hand the dendroidal horns are well understood: the right lifting roerty with resect to dendroidal inner horns define the -oerads, which are models for toological (coloured) symmetric oerads. But if we consider the right lifting roerty with resect to all dendroidal horns, Bašić and Nikolaus [BN12] have shown that we obtain models of infinite loo saces. This means that we have an Eilenberg-Zilber test category Ω/E for which there really is no natural candidate for a generating family of trivial cofibrations with reresentable codomains.