On notions of compactness, object classifiers and weak Tarski universes

We prove a correspondence between $\kappa$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$-compact maps in their underlying quasi-categories for suitably large regular cardinals $\kappa$. We thus obtain a transition result between weakly universal small fibrations in the (type theoretic) injective Dugger-Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$-toposes in the sense of Lurie.


Introduction
A Grothendieck ∞-topos M is an accessible left exact localization of the presheaf (∞, 1)-category Ĉ over some small (∞, 1)-category C ([18, Section 6.1]).Hence, it is presented by a model topos M given by a left exact left Bousfield localization of the simplicial category sPsh(C) of simplicially enriched presheaves on an associated small simplicial category C, equipped with either the projective or the injective model structure ( [22,Section 6]).In the following we prove a correspondence between κ-small fibrations in such model toposes M, and relatively κ-compact maps in their associated Grothendieck ∞-toposes M = Ho ∞ (M) for regular cardinals κ large enough.This is motivated by the interpretation of univalent Tarski universes defined in Martin-Löf type theory ( [21]) as univalent fibrations universal for the class of κ-small fibrations for suitable cardinals κ ( [5], [26]), and their intended interpretation as object classifiers in higher topos theory for relatively κcompact maps as developed in [18,Section 6.1.6].Therefore, even though we prove an analogous (but slightly weaker result) result for the projective model structure and arbitrary localizations, the main result of this paper is the following.Theorem 3.21.Let C be a small simplicial category, T be a set of arrows in the underlying category of sPsh(C) and M be the left Bousfield localization L T sPsh(C) inj of sPsh(C) equipped with the injective model structure at the set T (in the classic sense of [13,Section 3.3]).Assume that the localization is left exact, and let κ be a sufficiently large regular cardinal.Then a morphism f ∈ Ho ∞ (M) is relatively κ-compact if and only if there is a κ-small fibration g ∈ sPsh(C) such that g f in Ho ∞ (M).
Corollary 4.1.Let M = L T sPsh(C) inj be a left exact left Bousfield localization as in Theorem 3.21, and let κ be a sufficiently large regular cardinal.Then a relatively κ-compact map p ∈ Ho ∞ (M) is a classifying map for all relatively κ-compact maps in Ho ∞ (M) if and only if there is a univalent κ-small fibration π ∈ M which is weakly universal for all κ-small fibrations in M such that p π in Ho ∞ (M).
As a prerequisite for the proof we are giving, in Section 2 we show that up to DK-equivalence every simplicial category can be replaced by the localization of a well founded poset as already observed by Shulman in [25].This allows us to replace simplicial presheaf categories over arbitrary small simplicial categories C by localizations of simplicial presheaf categories over well founded posets I.In Section 3, we will use that such model categories come equipped with a theory of minimal fibrations that will allow us to present relatively κ-compact maps in their underlying quasicategory by κ-small fibrations.Those can be pushed forward to κ-small projective fibrations in our original presheaf category over C making use of Dugger's ideas about universal homotopy theories 1 arXiv:1911.01895v3[math.CT] 24 Jan 2023 in [7].The move to the injective model structure then follows by Shulman's recent observation ([26, Section 8]) that the cobar construction on simplicial presheaf categories takes projective fibrations to injective ones.In Section 4 we explain the relevance of this result for the semantics of Homotopy Type Theory in higher topos theory, as Theorem 3.21 is necessary to translate Tarski universes in the syntax to object classifiers in an ∞-topos (when using the common semantics via type theoretic model categories given in [24,Section 4]).

Acknowledgments
The author would like to thank his advisor Nicola Gambino at the University of Leeds for the frequent discussions and references, as well as his feedback on all stages of the writing process.The author is also grateful to Karol Szumi lo and Mike Shulman for helpful discussions and, especially, to Mike Shulman for sharing his note on the presentation of small (∞, 1)-categories as localizations of inverse posets, which provided a crucial step to the proof of Theorem 3.21.Also thanks to Jarl Taxerås Flaten who realized that the assumption of a Mahlo cardinal in an earlier version of this paper is unnecessary (see Remark 4.2).
Most of the work for this paper was carried out as part of the author's PhD thesis, supported by a Faculty Award of the University of Leeds 110 Anniversary Research Scholarship.It was finished with support of the Grant Agency of the Czech Republic under the grant 22-02964S.

Direct poset presentations of simplicial categories
In the following, simplicial categories -that is simplicially enriched categories -will be denoted by bold faced letters C and ordinary categories will be distinguished by blackboard letters C. S denotes the (simplicial) category of simplicial sets.By a simplicial presheaf over C we mean a simplicially enriched presheaf X : C op → S. Simplicial presheaves and simplicial natural transformations form part of a simplicial category sPsh(C) (via the usual end-construction, denoted [C op , S] in [16, Section 2.2]) whose underlying ordinary category will be denoted by sPsh(C).
Mike Shulman noted in [25, Lemma 0.2] that every quasi-category can be presented by the localization of a direct -in other words, well-founded -poset. 1 Since the note is unpublished, in this section we present a slightly stronger variation of his observation (with an accordingly slightly different proof) and discuss the resulting presentations of associated presheaf (∞, 1)-categories.Although the following sections only will require the fact that every (∞, 1)-category can be presented by the localization of an Eilenberg-Zilber category ( [4]), proving the stronger condition of posetality only requires about as much work as the Eilenberg-Zilber condition itself.
Recall the following constructions and notation from [3].A relative category is a pair (C, V ) such that C is a category and V is a subcategory of C. A relative functor The relative functor F is a relative inclusion if its underlying functor of categories is an inclusion and V = W ∩ C. The category of small relative categories and relative functors is denoted by RelCat.
There are two canonical inclusions of the category Cat of small categories into RelCat; for a category C and its discrete wide subcategory C 0 , we obtain the associated minimal relative category Č := (C, C 0 ) and the associated maximal relative category Ĉ := (C, C).
In [3, Section 4], Barwick and Kan introduce a combinatorial sub-division operation ξ : RelPos → RelPos on relative posets (considered as posetal relative categories) and an associated bisimplicial nerve construction N ξ : RelCat → sS giving rise to the adjoint pair The left adjoint K ξ is given by ) on representables and left Kan extension along the Yoneda embedding.The authors of [3] have shown that the category RelCat inherits a transferred model structure from the Reedy model structure (sS, R v ) which turns the pair (K ξ , N ξ ) into a Quillen-equivalence.By construction, the set K ξ [I v ] forms a set of generating cofibrations for the model structure in question, where denotes the generating set of monomorphisms in sS.Left Bousfield localization of both sides of (1) induces a model structure (RelCat, BK) such that (K ξ , N ξ ) is a Quillen-equivalence to Rezk's model structure (sS, CS) for complete Segal spaces.
It follows that the underlying quasi-category of the model category RelCat is equivalent to the quasi-category of small (∞, 1)-categories ( [17,Definition 1.3.4.15]).We will denote the thus associated small (∞, 1)-category to a relative category (C, V ) by Ho ∞ (C, V ) whenever it is not necessary to specify a specific model of (∞, 1)-category theory.
A central notion of [3] is that of a "Dwyer map" in RelCat.A relative functor F : (C, V ) → (D, W ) is a Dwyer inclusion if F is a relative inclusion such that C is a sieve in D and such that the cosieve ZC generated by C in D comes equipped with a strong deformation retraction ZC → C. The relative functor F is a Dwyer map if it factors as an isomorphism followed by a Dwyer inclusion, see [3,Section 3.5] for more details.
A major insight of the authors was that the generating cofibrations Proof.Since the empty relative category ∅ is a relative direct poset, it suffices to show that for every cofibration (P, V ) → (Q, W ) where (P, V ) is a relative direct poset also (Q, W ) is a relative direct poset.We show this by induction along the small object argument as follows.
The generating cofibrations (2) are maps between finite relative posets and such are clearly direct.Both Dwyer maps and relative posets are closed under coproducts and under pushouts between relative posets by [3,Proposition 9.2].Since Dwyer inclusions are inclusions of sieves, it is easy to see that both constructions preserve well-foundedness, too.Suppose we are given a transfinite composition of Dwyer maps A α → A β for α < β ≤ λ ordinals and A α relative inverse posets.Then, as stated in the proof of [3, Proposition 9.6], the colimit A λ is a relative poset.Suppose a = (a i | i < ω) is a descending sequence of arrows in A λ and let α < λ such that a 0 ∈ A α .Then the whole sequence a is contained in A α , because the inclusion A α → A λ is a Dwyer map by [3, Proposition 9.3] and so A α ⊆ A λ is a sieve.Therefore, the sequence a is finite.
In particular, every free cofibration ∅ → (P, V ) -that is every transfinite composition of pushouts of coproducts of generating cofibrations with domain ∅ -yields a relative direct poset (P, V ).But every cofibration ∅ → (Q, W ) is a retract of such, and hence every cofibrant object in RelCat is a relative direct poset.
Remark 2.2.The same proof shows that the cofibrant objects in the Thomason model structure on Cat are direct posets, using Thomason's original observation that the cofibrant objects in the Thomason model structure are posetal in the first place.
Let F ∆ : Cat → S-Cat be the Bar construction obtained in the standard way by the monad resolution associated to the free category functor F from the category RGraph of reflexive graphs to Cat ([9, Section 2.5]).Let U : Cat → RGraph be the corresponding right adjoint forgetful functor.Recall that F ∆ itself is not the left adjoint to the "underlying category" functor (•) 0 : S-Cat → Cat, but instead, as often remarked in the literature, a cofibrant replacement thereof.Furthermore, recall from [9, Section 4] the (standard) simplicial localization functor L ∆ : RelCat → S-Cat which takes a relative category (C, V ) to the simplicial category given in degree n ≥ 0 by The functor L ∆ : (RelCat, BK) → S-Cat is part of an equivalence of associated homotopy theories ( [2]), so that Ho ∞ (C, V ) L ∆ (C, V ).Indeed, it is pointwise equivalent to the hammock localization of a relative category ([8, Section 2]), and hence presents the (∞, 1)-categorical localization of C at V ([20, Remark 3.2, Theorem 3.8]).Yet it has the benefit of being a strict enriched construction: it is the localization of the simplicial category F ∆ (C) at the subcategory F ∆ (V ) 0 ⊆ F ∆ (C) 0 in the following sense.

Lemma 2.3. For every relative category (C, V ) and every (potentially large) simplicial category D, the canonical simplicial functor
of hom-categories.
Here we consider S-Cat as a 2-category given by S-enriched categories, S-enriched functors and S-enriched natural transformations ([16, Section 1.2]).The right hand side of (3) denotes the full subcategory of S-Cat(F ∆ (C), D) spanned by those functors which take the arrows of F ∆ (V ) 0 to isomorphisms in D 0 .Thus, Lemma 2.3 states that F ∆ (V ) 0 ⊆ F ∆ (C) 0 is "S-well localizable" in the sense of [29].
Lemma 2.3 and Corollary 2.4 are not strictly necessary for the results of this paper, but may be beneficial to motivate the simplicial localization construction.
Proof of Lemma 2.3.First, each F ∆ (V ) n is generated as a category by the iterated degeneracies of F ∆ (V ) 0 .Thus, a simplicial functor F ∆ (C) → D takes all arrows in F ∆ (V ) n to isomorphisms in D n for all n ≥ 0 if and only if F 0 takes the arrows in F ∆ (V ) 0 to isomorphisms in D 0 .Using the universal property of the ordinary categorical localization Second, recall that S-Cat is cartesian closed ([16, 2.3]).Thus, Cordier's simplicial nerve construction N ∆ : S-Cat → S with left adjoint C ([18, Section 1.1.5])yields a simplicial enrichment of S-Cat itself.Thus, the fact that j * is fully faithful follows from the same objectwise argument but applied to the simplicial category where C(∆ 1 ) is the simplicial category with two objects 0, 1, and of simplicial presheaf categories is fully faithful.
Proof.Given simplicial presheaves X, Y ∈ sPsh(L ∆ (C, V )), we want to show that the induced map of simplicial sets is a bijection on all degrees n ≥ 0. But for each such integer n ≥ 0, the map j * (X, Y ) n is isomorphic to the restriction since simplicial presheaf categories are cotensored over S, and j * is cocontinuous.Thus, it is bijective by Lemma 2.3 applied to D = S.
Hence, the map j : and a colocalization (j * , j * ) : between simplicial presheaf categories.Equipping both sides with either the injective or the projective model structure, the restriction j * becomes a left, respectively right Quillen functor ([18, Proposition A.3.3.7]).By [10, Theorem 2.2] applied to the map of relative simplicial categories ([2, Section 2.2]), the restriction (4) remains fully faithful on associated homotopy theories.More precisely, equipping both sides with the projective model structure, the pair (j !, j * ) becomes a homotopy localization and induces a Quillen-equivalence where the right hand side denotes the according left Bousfield localization.Dually, equipping both sides in ( 4) with the injective model structure, the pair (j * , j * ) becomes a homotopy colocalization.The simplicial localization functor L ∆ : RelCat → S-Cat has a homotopy-inverse, the "delocalization" or "flattening" : S-Cat → RelCat, given by the Grothendieck construction of a given simplicial category C considered as a simplicial diagram C : ∆ op → Cat.This functor was introduced in [10, Theorem 2.5] and is analysed in detail in [2].Now, given a simplicial category C, consider its delocalization (C) ∈ RelCat.Cofibrantly replacing (C) with some pair (P, V ) in RelCat yields a direct relative poset (P, V ) weakly equivalent -i.e.Rezk-equivalent in the language of [2] -to (C).Hence, by [ Further recall from [9, Proposition 2.6] that for every category C the canonical projection ϕ : F ∆ C → C is a DK-equivalence of simplicial categories.So, to summarize, we have gathered the following chain of Quillen-equivalences.
Proposition 2.5.Let C be a small simplicial category.Then there is a direct relative poset (P, V ) together with a zig-zag of DK-equivalences in S-Cat which induces a zig-zag of simplicial Quillen-equivalences of the form

Compactness in combinatorial model categories
We start with some facts about compactness in presheaf categories.Given a small category C, we denote the cardinality of C by Given a (set-valued) presheaf X ∈ C, its cardinality is denoted by Given a small simplicial category C, we also denote the cardinality of C by where the cardinality of the hom-objects C(C, C ) ∈ S is given by the cardinality of presheaves defined above.Accordingly, given a regular cardinal κ > |C|, a simplicial presheaf X ∈ sPsh(C) is κ-small if all its values X(C) are κ-small.A simplicial natural transformation f : X → Y in sPsh(C) is κ-small if all its pullbacks along maps Z → Y with κ-small domain Z are κ-small simplicial presheaves.κ if for all cardinals κ 0 < κ and µ 0 < µ, also κ µ0 0 < κ.The order " " is chosen in such a way that whenever µ κ holds, then µ < κ and µaccessibility of a quasi-category C implies κ-accessibility of C ([18, Proposition 5.4.2.11]).As noted in [18], the order is unbounded in the class of regular cardinals as for any regular cardinal µ we have µ sup{τ µ | τ < µ} + .In particular, we always find a regular cardinal sharply larger than a given regular µ.In fact, if µ is regular, then µ + is already sharply larger than µ ([1, Examples 2.13.(2)]).Whenever λ < µ is a regular cardinal and µ κ, then also λ κ.Thus, for any set X of cardinals there is a regular cardinal µ such that κ µ for all κ ∈ X. Recall that an object C in an accessible category C is κ-compact if its associated corepresentable preserves κ-directed colimits.A map f in C is relatively κ-compact if the pullback of f along any map with κ-compact domain in C is itself again κ-compact.

Lemma 3.3. Let C be a small simplicial category and κ
|C| an infinite regular cardinal.Then Proof.Let C be a small simplicial category.For Part 1, recall that a presheaf X is κ-compact if and only if it is a retract of a κ-small directed colimit of finite colimits of the generators yC ⊗ ∆ n via [1, Remark 2.15].But κ |C| being infinite and regular implies that all generators yC ⊗ ∆ n are κ-small, and hence so are all their finite colimits.Hence, every κ-compact presheaf X is a subobject of a κ-small colimit of κ-small presheaves and hence κ-small.Vice versa, every simplicial presheaf X is the colimit of its canonical diagram of generators yC ⊗ ∆ n ; whenever X is κ-small, so is the associated canonical diagram by the Yoneda Lemma.Thus, X is a κ-small colimit of the generators yC ⊗ ∆ n .Closing the generators under finite colimits gives a description of X as a κ-small directed colimit of κ-compact objects, so X is κ-compact again by [ The aim of this section is to compare this ordinary notion of compactness in a combinatorial model category M with the notion of compactness in its underlying (∞, 1)-category Ho ∞ (M, W) ([18, Definition 5.3.4.5, Definition 6.1.6.4]).The validity of this comparison was addressed in a question posted in [19] by Shulman; for objects it is given in Proposition 3.11 in a special case, and in Corollary 3.16 in general.For maps it is given in Theorem 3.15 in case M is (the left Bousfield localization of) a simplicial presheaf category equipped with the projective model structure, and in Theorem 3.21 in case it is such equipped with the injective model structure.An argument for the objectwise statement was outlined by Lurie in the same post, which in one direction coincides with our proof given in Proposition 3.11.Before we state the theorems, we make the following ad hoc construction, give one auxiliary folkore lemma, and define a simple axiomatic framework of minimal fibrations in a general model category that will provide a convenient set-up to state intermediate results in.
Given a λ-accessible quasi-category C with generating set A and a regular cardinal µ ≥ λ, define the full subcategory J µ ⊆ C recursively as follows.Let J µ,0 0 := A and J µ,0 be the full subcategory of C generated by J µ,0 0 .Whenever β < µ is a limit ordinal, let and J µ,β the full subcategory generated by J µ,β 0 .On successors, given J µ,α , let (so we choose a set of representatives V ∆ op µ for µ-small simplicial sets) and J µ,α+1 be the corresponding full subcategory.Eventually, we define the full subcategory J µ of C to have the set of objects The following lemma is noted in [18, Section 5.4.2], and is a generalization of the corresponding 1-categorical statement for accessible categories ([1, Remark 2.15]).Lemma 3.7.Let C and D be presentable quasi-categories.
1. Suppose C is λ-presentable.Then, for every regular µ λ, the µ-compact objects in C are, up to equivalence, exactly the retracts of objects in J µ .
2. Let F : C → D be an accessible functor.Then there is a cardinal µ such that F preserves κ-compact objects for all regular κ µ.
Notation 3.8.The following group of statements will in each case claim that a certain comparison holds for all κ "sufficiently large" or "large enough".That means in each case there is a cardinal µ such that for all κ µ the given statement holds true.As we are not interested in a precise formula for the lower bound µ, we generally will not make the cardinal µ explicit.we note that we will have to impose the condition on κ to be "large enough" only finitely often and eventually take the corresponding supremum.Definition 3.9.Let M be a model category.Say M has a theory of minimal fibrations if there is a pullback stable class F min M of fibrations in M -the class of minimal fibrations -such that the following hold.
1. Let p : X Y and q : X Y be minimal fibrations.Then every weak equivalence between X and X over Y is an isomorphism.

Suppose M has a theory of minimal fibrations. Then the converse of Part 1 holds.
Proof.For Part 1 and κ large enough, κ-filtered colimits in M are homotopy colimits and the κ-compact objects in M are exactly the κ-compact objects in the quasi-category N (M).So the localization N (M) → Ho ∞ (M) preserves κ-filtered colimits and hence is κ-accessible.The statement now follows from Lemma 3.7.
For Part 2, we note that by our assumption and by Dugger's presentation theorem for combinatorial model categories [6, Theorem 1.1] it suffices to show the statement for objects C ∈ J κ on the one hand and left Bousfield localizations of simplicial presheaf categories sPsh(C) on the other.Indeed, given a combinatorial model category M together with a category C, a set T ⊂ sPsh(C) of arrows and a Quillen-equivalence of the form suppose we have shown the statement for all C ∈ J κ and all κ large enough in the case of L T (sPsh(C)) proj .Then, as both categories M and sPsh(C) are presentable, we find κ |C| large enough such that the right adjoint R preserves κ-compact objects.Certainly the derived functors LL and RR preserve κ-compactness in Ho ∞ (M), so whenever an object C ∈ Ho ∞ (M) is contained in J κ , we may choose a κ-compact presheaf D ∈ sPsh(C) weakly equivalent to RRX.Without loss of generality D is cofibrant by [6, Proposition 2.3.(iii)] and so L(D) is κ-compact in M and presents C in Ho ∞ (M).Now, every κ-compact object A ∈ Ho ∞ (M) is the retract of an object C ∈ J κ by Lemma 3.7.1.We thus may present A by a bifibrant object B ∈ M, and C by a κ-compact bifibrant object D ∈ M again via [6, Proposition 2.3.(iii)] and by the above.This yields a map j : B → D with homotopy retract r : D → B in M. Pick a minimal fibrant object ι : M → B. Since every acyclic cofibration between fibrant objects allows a retract ρ itself, we see that M is a homotopy retract of D. Hence, the composition (ρr)(jι) is homotopic to the identity 1 M and thus a homotopy equivalence ([14, Theorem 1.2.10.(iv)]).It follows that it is an isomorphism in virtue of minimality of M .Thus, M is a retract of D and as such κ-compact in M itself.
Therefore, assume M = L T (sPsh(C)) proj , and suppose C ∈ Ho ∞ (M) is contained in J κ .The representatives for the colimits in the construction of (J κ,α |α < κ) can be chosen to be homotopy colimits of strict diagrams F : I → M for κ-small categories I by [18, Proposition 4.2.3.14] and [17,Proposition 1.3.4.25].Hence, they can be computed according to the Bousfield-Kan formula 13,Example 18.3.6]).But this representative of the homotopy colimit is κ-compact whenever I is κ-small and furthermore each F (i) for i ∈ I in M is κ-compact.Hence, by induction, every object C ∈ J κ is presented by a κ-compact object D in sPsh(C).
In the following we generalize Proposition 3.11 to relatively κ-compact maps.
Proposition 3.12.Let M be a combinatorial model category.
1. Suppose the converse of Proposition 3.11.1 holds in M. Then for all sufficiently large regular cardinals κ, a morphism f : C → D in Ho ∞ (M) is relatively κ-compact if there is a relatively κ-compact fibration p ∈ M between fibrant objects such that p f in Ho ∞ (M).
2. Suppose M has a theory of minimal fibrations and κ-compact objects in M are closed under fiber products.Then the converse of Part 1. holds.
Proof.For Part 1, let p : X Y be a relatively κ-compact fibration between fibrant objects in M. Let g : A → Y be a map in Ho ∞ (M) with κ-compact domain.In order to show that the pullback of X along g is κ-compact in Ho ∞ (M), by assumption we can present A by a κ-compact object A in M. Without loss of generality A is bifibrant by [6, Proposition 2.3.(iii)],so we obtain a map g : A → Y presenting g.Also the pullback (g ) * X is a homotopy pullback and it is κ-compact in M by assumption.Hence, it is κ-compact in Ho ∞ (M) again by Proposition 3.11.This shows that p is relatively κ-compact in Ho ∞ (M).
For Part 2, assume that f : By assumption there is an acyclic cofibration M ∼ → X such that the restriction m : M Y of p is a minimal fibration.As m and p are homotopy equivalent over Y , the fibration m is relatively κ-compact in Ho ∞ (M), too.We want to show that m is a relatively κ-compact fibration in M. Therefore, let g : Z → Y be a map for some κ-compact object Z ∈ M; we have to show that the pullback is a κ-compact object in M as well.By [6, Proposition 2.3.(iii)]there is a κ-compact fibrant replacement RZ of Z. Since the object Y itself is fibrant, we obtain an extension g : RZ → Y of g along the acylic cofibration Z ∼ → RZ and hence a factorization of the following form.
All three faces of the diagram are pullback squares, and by assumption κ-compact objects in M are closed under fiber products.Hence, in order to show that the object g * M ∈ M is κ-compact, it suffices to show that the object (g As RZ is κ-compact in M, it also is κ-compact in the underlying quasi-category Ho ∞ (M) by Proposition 3.11, and hence so is the (homotopy-)pullback (g ) * M by our assumption on the morphism f that we started with.
By Proposition 3.11 and [6, Proposition 2.3.(iii)]we find a cofibrant κ-compact object X together with a weak equivalence e : X → (g ) * M .The composition (g ) * m • e : X → RZ is a map between κ-compact objects in M, and so we find a factorization X ∼ → RX RZ such that RX is κ-compact as well again by [6, Proposition 2.3.(iii)].We obtain a weak equivalence RX → (g ) * M between the respective fibrations over RZ as a lift to the resulting square.
Since M has a theory of minimal fibrations, there is an acyclic cofibration j : N ∼ → RX such that the restriction n : N RZ of RX RZ is a minimal fibration.Since j is an acyclic cofibration between fibrations, it has a retraction, and so N is still κ-compact in M.But the composition of weak equivalences N (g ) * M over RZ is a weak equivalence between minimal fibrations and hence is an isomorphism.Thus, (g ) * M is κ-compact in M. Corollary 3.13.Let P be an Eilenberg-Zilber category in the sense of [5,Section 2.1] and M = sPsh(P) inj be the category of simplicial presheaves on P equipped with the injective model structure.Then for all sufficiently large regular cardinals κ, a morphism f ∈ Ho ∞ (M) is relatively κ-compact if and only if there is a κ-small fibration p ∈ M between fibrant objects such that p f in Ho ∞ (M).
Proof.The model category M supports a theory of minimal fibrations as shown in [5, 2.13-2.16],and κ-small objects in (simplicial) presheaf categories are closed under fiber products for infinite κ.Thus the statement follows from Proposition 3.12 for M = sPsh(P) inj .

The projective case
We now make use of the observations in Section 2 to generalize Corollary 3.13 to the category of simplicial presheaves over arbitrary small simplicial categories.
Therefore, we want to make use of [7, Proposition 5.10, Corollary 6.5] which shows that any zig-zag M 1 ← • • • → M n of Quillen-equivalences between combinatorial model categories can be reduced to a single Quillen-equivalence whenever either M 1 or M n is a "standard presentation" of the form L T sPsh(C) proj for some small category C and some set of maps T ⊂ sPsh(C).In our case however, we we wish to start with model categories of the form L T sPsh(C) proj for general small simplicial categories C instead.Therefore, we show a simplicially enriched version of [7,Proposition 5.10] first.We claim that F := Lan y p is the left Quillen functor we are looking for.First, let us construct the Quillen homotopies connecting L • F and F .
Recall that, as explained for instance in [16, 4.31], for every presheaf X ∈ sPsh(C) the object Lan y p(X) is the colimit of p weighted by X, i.e. the former because representables are projectively cofibrant and F preserves cofibrant objects, and the latter because L preserves cofibrant objects.Thus, if ρ F y : F y → r(F y) denotes an injective fibrant replacement of F y, the counit ε r(F y) : Lp ⇒ r(F y) of the Quillen-equivalence (L, R) induces a span of natural weak equivalences between the cofibrant objects Lp, r(F y) and F y. Thus, for cofibrant presheaves X ∈ (sPsh(C)) proj , we obtain a zig-zag of natural weak equivalences between X Lp and X F y.But F y is just F (by [16, 4.51]), so we have constructed a span of Quillen homotopies between L • F and F .
Second, the fact that F : sPsh(C) proj → N is a left Quillen functor with right adjoint G (N ) = N(p , N ) was basically already shown above (following for instance, as it were, from [11,Theorem 3.3.]).
We are left to show that, third, the Quillen pair (F , G ) : sPsh(C) proj → N descends to the localization at T whenever F does so.That is, we have to show that every arrow f ∈ T is mapped to a weak equivalence by F in N assuming every such arrow is mapped to a weak equivalence by F in M. Without loss of generality all arrows f ∈ T have cofibrant domain and codomain.Then, given f ∈ T , the arrow F (f ) is a weak equivalence in M, and so is LF (f ) ∈ M since F and LF are Quillen-homotopic.Thus, R(R)(LF (f )) is a weak equivalence in N, but this arrow is weakly equivalent to F (f ) since (L, R) is a Quillen equivalence.It follows that F (f ) is a weak equivalence in N itself.
Theorem 3.15.Let C be a small simplicial category, T ⊂ sPsh(C) be a set of maps and M = L T (sPsh(C)) proj .Then for all sufficiently large regular cardinals κ, a morphism f ∈ Ho ∞ (M) is relatively κ-compact if and only if there is a κ-small fibration p ∈ M between fibrant objects such that p f in Ho ∞ (M).
Proof.Let C be a small simplicial category and T ⊂ sPsh(C) be a set of maps.By Proposition 2.5 we obtain a relative poset (P, V ) and a zig-zag of Quillen-equivalences of the form This yields a zig-zag of Quillen-equivalences where T ⊂ sPsh(P) is obtained from T ⊂ sPsh(C) by transferring T along the finitely many Quillen-equivalences successively.We denote the union y[V ] ∪ T ⊂ sPsh(P) short-handedly by U .
By Proposition 3.14 this chain of Quillen-equivalences induces a single Quillen-equivalence The left Bousfield localization L U sPsh(P) inj has a theory of minimal fibrations by Lemma 3.10.Thus, let κ |C|, |P| be regular and, first, large enough such that Corollary 3.13 applies to P, second, large enough such that Proposition 3.12 applies to L U sPsh(P) inj , and third, large enough such that both F and G preserves κ-compact objects (via Lemma 3.7 or its ordinary categorical analogon as right adjoints between locally presentable categories are accessible again).Now, let f ∈ Ho ∞ (M) be relatively κ-compact.Since the pair ( 7) is a Quillen-equivalence, the quasi-category Ho ∞ (M) is equivalent to the underlying quasi-category of L U sPsh(P) inj .Then, by Proposition 3.12, there is a κ-small fibration p : X Y between fibrant objects in L U sPsh(P) inj presenting f in Ho ∞ (M).By assumption both adjoints preserve κ-compact objects, so by Corollary 3.6 the right Quillen functor G preserves κ-small maps.Thus, Gp : GX GY is a κ-small fibration between fibrant objects presenting f in Ho ∞ (M).
In particular, the converse of Proposition 3.11.1 holds in M for every such κ, and so the other direction follows directly from Proposition 3.12.1.2. Let L T (sPsh(C)) proj be the presentation of M from Dugger's representation theorem for combinatorial model categories in [6].Then for all sufficiently large regular cardinals κ, a morphism f ∈ Ho ∞ (M) is relatively κ-compact if and only if there is a κ-small fibration p ∈ L T (sPsh(C)) proj between fibrant objects such that p f in Ho ∞ (M).Remark 3.17.The reason why in Corollary 3.16.2we don't obtain the comparison result for M itself is that there is no obvious reason why the Quillen-equivalence given by Dugger's presentation theorem should preserve relatively κ-compact maps.While the right adjoint certainly does preserve such maps, the left adjoint does not seem to exhibit any properties with that respect.
weak or strict) universal κ-small fibrations in M -which models Tarski universes in the associated type theory -also models universes in the underlying quasi-category.Theorem 3.21 however does show T κ = Ho ∞ [S κ ].In other words, we obtain the following corollary.
Corollary 4.1.Let M = L T sPsh(C) inj be a model topos, and let κ be a sufficiently large regular cardinal.Then a relatively κ-compact map p ∈ Ho ∞ (M) is a classifying map for all relatively κ-compact maps in Ho ∞ (M) if and only if there is a univalent κ-small fibration π ∈ M which is weakly universal for all κ-small fibrations in M such that p π in Ho ∞ (M).
Remark 4.2.Let us finish with a note on the closure under standard type formers of a given Tarski universe, with regards to the existence of "sufficiently large" regular cardinals that has been a standing assumption along the way (Notation 3.8).Given a model topos of the form M = L T sPsh(C) inj , for a regular cardinal κ to be sufficiently large means to be contained in the class Shl(λ) of regular cardinals sharply larger than a specified cardinal λ associated to the small simplicial category C -or to the ∞-topos M that is. 2 For a universal fibration π κ ∈ M as in Corollary 4.1 to be closed under the standard type formers in an appropriate sense ([26, Section 6]), the cardinal κ has to be assumed to be strongly inaccessible.Thus, if we start with an ∞-topos M and wish to show that its type theoretic presentation M := L T sPsh(C) inj exhibits a universal fibration for κ-small fibrations that is closed under all standard type formers, we need an object classifier V κ in M classifying relatively κ-compact maps for a strongly inaccessible κ in Shl(λ).The same holds if we want to show that a given cumulative hierarchy of universal fibrations associated to strong inaccessibles κ i in M yields a corresponding hierarchy of object classifiers in M. Thus, the translation of the categorical structure together with a universe (or even a cumulative infinite hierarchy of such) between Homotopy Type Theory and higher topos theory requires the existence of inaccessibles within any such given class of sharply larger cardinals.Fortunately, for strongly inaccessible cardinals κ we have λ κ if and only if λ < κ.We hence do not need to make any large cardinal assumptions beyond the existence of the inaccessibles themselves.

Remark 3 . 1 .Notation 3 . 2 .
Again, a map f : X → Y in sPsh(C) is κ-small if and only if for all C ∈ C, the map f (C) :X(C) → Y (C) is a κ-small map of simplicial sets.The category sPsh(C) is locally finitely presentable ([15, Examples 3.4, Proposition 7.5]), generated by (finite colimits of) the objects yC ⊗ ∆ n for C ∈ C and n ≥ 0 which we refer to in the following as the generators.When an ordinary category C is considered as a discrete simplicial category, we have an obvious isomorphism between sPsh(C) and the set-valued presheaf category C × ∆ op .To develop a general theory of accessible categories, Adámek and Rosický introduce for pairs of regular cardinals µ < κ the sharply larger relation ([1, Definition 2.12]) and a special case " " in [1, Example 2.13.(4)] which is used as well in[18, Definition 5.4.2.8]  to develop a theory of accessible (∞, 1)-categories.Here, µ

1 ,Lemma 3 . 4 .Remark 3 . 5 .Corollary 3 . 6 .
Remark 2.15].Part 2 follows directly from Part 1 by definition.Let C and D be locally presentable categories and let F : C / / D : G o o be an adjoint pair.Let κ be a regular cardinal such that 1. both F and G preserve κ-compact objects, 2. κ-compact objects in C are closed under fiber products.Then G preserves relatively κ-compact maps.Proof.Straight-forward.Although the class of µ-compact objects in a locally κ-presentable category is not necessarily closed under fiber products for all µ κ, the class of such µ is unbounded in the class of regular cardinals sharply larger than κ ([26, Proposition 4.5]).It nevertheless is a trivial property in the case of (simplicial) presheaf categories whenever µ is infinite by Lemma 3.3, since µ-smallness of sets is closed under finite limits.Let C and D be small simplicial categories and let F : sPsh(C) / / sPsh(D) : G o o be an adjoint pair.Let κ max(|C|, |D|) be regular and suppose both F and G preserve κ-small objects.Then G preserves κ-small maps.Proof.Follows directly from Lemma 3.3.2,Lemma 3.4 and Remark 3.5.

2 .Lemma 3 . 10 .
For every fibration p : X Y in M there is an acyclic cofibration M ∼ → X such that the restriction M → Y is a minimal fibration.Let M be a model category with a theory of minimal fibrations.Let T be a class of maps in M such that the left Bousfield localization L T M exists.Then the model category L T M has a theory of minimal fibrations.Proof.Given a model category M and a class T of maps in M as stated, simply define the class F min T of minimal fibrations in L T M to be the class of fibrations in L T M which are minimal fibrations in M. Pullback stability of F min T is immediate.Property 1 follows readily, as T -local weak equivalences between T -local fibrations are weak equivalences in M itself.For Property 2, let p : X Y be a fibration in L T M. By the assumption that M has a theory of minimal fibrations, there is an acyclic cofibration M ∼ → X in M such that the restriction M Y is a minimal fibration in M.But M → X is a weak equivalence from the fibration M Y to the fibration p : X Y over Y .The latter is a fibration in L T M and it hence follows by [13, Proposition 3.4.6]that M Y is a fibration in L T M, too.Proposition 3.11.Let M be a combinatorial model category.

Proposition 3 . 14 .
Let C be a small simplicial category, and M, N be simplicial model categories together with a simplicial Quillen-equivalence (L, R) : N ∼ − → M. Let T be a class of arrows in sPsh(C) such that its left Bousfield localization exists, and let (F, G) : L T sPsh(C) proj → M be a simplicial Quillen pair.Then there is a simplicial Quillen pair (F , G ) : L T sPsh(C) proj → N such that the functors L • F and F are Quillen-homotopic in the sense of [7, Definition 5.9].In other words, simplicial Quillen pairs with domain L T sPsh(C) proj can be lifted up to homotopy along Quillen-equivalences. Proof.Let λ and ρ denote cofibrant and fibrant replacements respectively and L = λ * , R = ρ * denote their associated left and right derivations of functors.Let the composition λR(R)F y : C → N be denoted by p.Note that the left and right derivation L and R of simplicial functors may be chosen to be simplicial again by [23, Corollary 13.2.4],thus p is a simplicial functor and we can consider the simplicially enriched left Kan extension C The left Quillen functor L : N → M is a left adjoint and hence preserves weighted colimits, thus we have that L • F ∼ = Lp.Furthermore, by[11, Theorem 3.3]  the weighted colimit functor: sPsh(C) proj × [C, M] inj → Mis a left Quillen bifunctor.In particular, for cofibrant presheavesX ∈ sPsh(C) proj the X-weighted colimit X : [C, M] inj → Mis a left Quillen functor.But both F y and Lp ∼ = L(L)R(R)F y are cofibrant objects in [C, M] inj :

Corollary 3 . 16 .
Let M be a combinatorial model category.1.For all sufficiently large regular cardinals κ, an object C in Ho ∞ (M) is κ-compact if and onlyif there is a κ-compact object D ∈ M such that C D in Ho ∞ (M).

Proof. For Part 1 ,
one direction is exactly Proposition 3.11.1.For the other direction, let L T sPsh(C) proj L / / M R o o be the Quillen-equivalence from Dugger's representation theorem.Then for κ large enough and every κ-compact object C ∈ Ho ∞ (M), we obtain a κ-compact object D ∈ L T sPsh(C) proj from Theorem 3.15 which presents C. As the left adjoint L preserves κ-compact objects, we find a κ-compact fibrant replacement of L(D) in M which presents C. Part 2 is just a special case of Theorem 3.15.