# On the correspondence of object classifiers and Tarski universes

Raffael Stenzel

###### Abstract

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 relative $\kappa$-compact maps
in their underlying quasi-categories for suitably large inaccessible cardinals
$\kappa$. We thus obtain a transition result between weakly universal small
fibrations in the injective Dugger-Rezk-style standard presentations of model
toposes and object classifiers in Grothendieck $\infty$-toposes in the sense
of Lurie.

##  1 Introduction

A Grothendieck $\infty$-topos $\mathcal{M}$ is the left exact localization of
a presheaf $(\infty,1)$-category $\hat{\mathcal{C}}$. Hence, it is presented
by a model topos $\mathbb{M}$ of the form
$\mathcal{L}_{T}\mathrm{sPsh}(\mathbf{C})$ for some small simplicial category
$\mathbf{C}$, that is a left exact left Bousfield localization of the
simplicial category $\mathrm{sPsh}(\mathbf{C})$ equipped with either the
projective or the injective model structure. In the following we prove a
correspondence between relative $\kappa$-compact maps in Grothendieck
$\infty$-toposes $\mathcal{M}$ and $\kappa$-small fibrations in their
associated model topos $\mathbb{M}$. This is motivated by the interpretation
of univalent Tarski universes defined in type theory as univalent fibrations
universal for the class of $\kappa$-small fibrations, and their intended
interpretation as “object classifiers” in Higher Topos Theory, i.e.
classifying maps for relative $\kappa$-compact maps as developed in [14,
Chapter 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 3.16.

Let $\mathbf{C}$ be a small simplicial category, $T$ a set of arrows in
$\mathrm{sPsh}(\mathbf{C})$ and $\mathbb{M}$ be the left exact left Bousfield
localization $\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{inj}}$. Let
$\kappa$ be a sufficiently large inaccessible cardinal. Then a morphism
$f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact if and
only if there is a $\kappa$-small fibration $g\in\mathrm{sPsh}(\mathbf{C})$
such that $g\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

######  Corollary 4.1.

Let $\mathbb{M}\cong\mathcal{L}_{T}\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$
be a model topos, let $p$ be a fibration in $\mathbb{M}$. Let $\kappa$ be a
sufficiently large inaccessible cardinal. Then $p$ is univalent and weakly
universal for $\kappa$-small fibrations if and only if $p$ is a classifying
map for relatively $\kappa$-compact maps in
$\mathrm{Ho}_{\infty}(\mathbb{M})$.

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 [20]. This allows us
to replace simplicial presheaf categories over arbitrary small simplicial
categories $\mathbf{C}$ by 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 relative
$\kappa$-compact maps in their underlying quasi-category by $\kappa$-small
fibrations. Those can be pushed forward to $\kappa$-small fibrations in our
original presheaf category over $\mathbf{C}$ making use of Dugger’s ideas on
universal homotopy theories in [6]. The move to the injective model structure
then follows by Shulman’s recent observation ([21, 8]) that the cobar
construction on 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 3.16 is necessary to
translate Tarski universes in the syntax to object classifiers in a higher
topos (when using the common semantics via type theoretic model categories
given in [19, 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
Szumiło and Mike Shulman for helpful discussions and, especially, to Mike
Shulman for sharing his note on the presentation of small
$(\infty,1)$-categories as localizations of inverse posets, which provided a
crucial step to the proof of Theorem 3.16. 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.

##  2 Replacing simplicial categories with direct posets

In the following, simplicial categories - that is simplicially enriched
categories - will be denoted by bold faced letters $\mathbf{C}$ and ordinary
categories will be distinguished by blackboard letters $\mathbb{C}$.
$\mathbf{S}$ denotes the (simplicial) category of simplicial sets. By a
simplicial presheaf over $\mathbf{C}$ we mean a simplicially enriched presheaf
$X\colon\mathbf{C}^{op}\rightarrow\mathbf{S}$. Simplicial functors and
simplicial natural transformations form a simplicial category
$\mathrm{sPsh(\mathbf{C})}$ whose underlying ordinary category also will be
denoted by $\mathrm{sPsh(\mathbf{C})}$.

Mike Shulman noted in [20, Lemma 0.2] that every quasi-category can be
presented by the localization of a direct – in other words, well founded –
poset.111Shulman in fact argues for a presentation by inverse posets. But
since localization commutes with taking opposite categories, this amounts to
the same statement. Since the note is unpublished, in this section we present
a slight variation of his observation (with a different proof) and discuss the
resulting presentations of associated presheaf $\infty$-categories. Although
the following sections only will require the fact that every quasi-category
can be presented by the localization of an Eilenberg-Zilber Reedy category,
proving the stronger condition of posetality only requires about as much work
as the Eilenberg-Zilber Reedy condition itself.

Recall the following constructions and notation from [3]. A _relative
category_ is a pair $(\mathbb{C},V)$ such that $\mathbb{C}$ is a category and
$V$ is a subcategory of $\mathbb{C}$. A _relative functor_
$F\colon(\mathbb{C},V)\rightarrow(\mathbb{D},W)$ is a functor
$F\colon\mathbb{C}\rightarrow\mathbb{D}$ of categories such that
$F[V]\subseteq W$. The relative functor $F$ is a _relative inclusion_ if its
underlying functor of categories is an inclusion and $V=W\cap\mathbb{C}$. The
category of small relative categories and relative functors is denoted by
RelCat.

There are two canonical inclusions of the category $\mathbf{Cat}$ of small
categories into RelCat; for a category $\mathbb{C}$ and its discrete wide
subcategory $\mathbb{C}_{0}$, we obtain the associated minimal relative
category $\check{\mathbb{C}}:=(\mathbb{C},\mathbb{C}_{0})$ and the associated
maximal relative category $\hat{\mathbb{C}}:=(\mathbb{C},\mathbb{C})$.

In [3, Section 5.3], Barwick and Kan introduce a combinatorial sub-division
operation $\xi\colon\mathrm{RelCat}\rightarrow\mathrm{RelCat}$ and an
associated bisimplicial nerve construction
$N_{\xi}\colon\mathrm{RelCat}\rightarrow s\mathbf{S}$ giving rise to the
adjoint pair

$\textstyle{s\mathbf{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K_{\xi}}$$\textstyle{\mathrm{RelCat}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{\xi}}$

The left adjoint $K_{\xi}$ is given on representables by
$K_{\xi}(\Delta[m,n])=\xi(\check{[m]}\times\hat{[n]})$ and left Kan extension
along the Yoneda embedding. The authors of [3] have shown that the category
$\mathrm{RelCat}$ inherits a transferred model structure
$(\mathrm{RelCat},\mathrm{BK})$ from the Reedy model structure
$(s\mathbf{S},R_{v})$ which turns the pair $(K_{\xi},N_{\xi})$ into a Quillen
equivalence. By construction, the set $K_{\xi}[\mathcal{I}_{v}]$ forms a set
of generating cofibrations for the model structure in question, where

$\mathcal{I}_{v}:=\\{(\delta_{n}\colon\partial\Delta^{n}\hookrightarrow\Delta^{n})\Box^{\prime}(\delta_{m}\colon\partial\Delta^{m}\hookrightarrow\Delta^{m})\mid
n,m\in\mathbb{N}\\}$

is the generating set of monomorphisms in $s\mathbf{S}$ given by the pushout-
product $\Box^{\prime}$ associated to the box product
$\Box\colon\mathbf{S}\times\mathbf{S}\rightarrow s\mathbf{S}$ as presented in
[11, Section 2].

A central notion of [3] is that of “Dwyer maps” in RelCat. A relative functor
$F\colon(\mathbb{C},V)\rightarrow(\mathbb{D},W)$ is a _Dwyer inclusion_ if $F$
is a relative inclusion such that $\mathbb{C}$ is a sieve in $\mathbb{D}$ and
such that the cosieve $Z\mathbb{C}$ generated by $\mathbb{C}$ in $\mathbb{D}$
comes equipped with a strong deformation retraction
$Z\mathbb{C}\rightarrow\mathbb{C}$. 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

$K_{\xi}(\delta_{m}\Box^{\prime}\delta_{n})\colon
K_{\xi}((\Delta^{m}\Box\partial\Delta^{n})\cup_{\partial\Delta^{m}\Box\partial\Delta^{n}}(\partial\Delta^{m}\Box\Delta^{n}))\rightarrow
K_{\xi}(\Delta^{m}\Box\Delta^{n})$

of the model category $(\mathrm{RelCat},\mathrm{BK})$ are Dwyer maps of
relative posets ([3, Proposition 9.5]). It follows that every cofibration in
$(\mathrm{RelCat},\mathrm{BK})$ is a Dwyer map as shown in [3, Theorem 6.1].

######  Proposition 2.1.

The underlying category of a cofibrant object in
$(\mathrm{RelCat},\mathrm{BK})$ is a direct (i.e. well founded) poset.

  * Proof.

Since the empty relative category $\emptyset$ is a relative direct poset, it
suffices to show that for every cofibration
$(\mathbb{P},V)\hookrightarrow(\mathbb{Q},W)$ where $(\mathbb{P},V)$ is a
relative direct poset also $(\mathbb{Q},W)$ is a relative direct poset. We
show this by “induction along the small object argument” as follows.

The generating cofibrations $K_{\xi}(\delta_{m}\Box^{\prime}\delta_{n})$ are
maps between finite relative posets and such are clearly direct. Both Dwyer
maps and relative posets are closed under coproducts and under pushouts along
Dwyer maps between relative posets by [3, Proposition 9.2], and it is easy to
see that both constructions preserve well foundedness, too. Suppose we are
given a transfinite composition of Dwyer maps $A_{\alpha}\rightarrow
A_{\beta}$ for $\alpha<\beta\leq\lambda$ ordinals and $A_{\alpha}$ relative
inverse posets. Again by [3, Proposition 9.2], the colimit $A_{\lambda}$ is a
relative poset. Suppose $a=(a_{i}\mid i<\omega)$ is a descending sequence of
arrows in $A_{\lambda}$ and let $\alpha<\lambda$ such that $a_{0}\in
A_{\alpha}$. Then the whole sequence $a$ is contained in $A_{\alpha}$, because
the inclusion $A_{\alpha}\hookrightarrow A_{\lambda}$ is a Dwyer map by [3,
Proposition 9.3] and so $A_{\alpha}\subseteq A_{\lambda}$ is a sieve (see [3,
3.5]). Therefore, the sequence $a$ is finite.

In particular, every free cofibration $\emptyset\hookrightarrow(\mathbb{P},V)$
– that is every transfinite composition of pushouts of generating cofibrations
with domain $\emptyset$ – yields a relative direct poset $(\mathbb{P},V)$. But
every cofibration $\emptyset\hookrightarrow(\mathbb{Q},W)$ 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 $\mathrm{Cat}$ are direct posets, using Thomason’s original
observation that the cofibrant objects in the Thomason model structure are
posetal in the first place.

Let $F_{\Delta}\colon\mathrm{Cat}\rightarrow\mathbf{S}\text{-Cat}$ be the Bar
construction obtained in the standard way by monad resolution of the free
category functor $F$ from reflexive $\mathrm{Graphs}$ to $\mathrm{Cat}$.
Recall that $F_{\Delta}$ is not the left adjoint to the “underlying category”
functor, but, as often remarked in the literature, a cofibrant replacement of
this left adjoint. Furthermore, for example from [7], recall the (standard)
simplicial localization functor

$\mathcal{L}_{\Delta}\colon\mathrm{RelCat}\rightarrow\mathbf{S}\text{-Cat}$

which takes a relative category $(\mathbb{C},V)$ to the simplicial category
given in degree $n<\omega$ by

$\mathcal{L}_{\Delta}(\mathbb{C},V)_{n}=F_{\Delta}(\mathbb{C})_{n}[F_{\Delta}(V)_{n}^{-1}].$

The simplicial category $\mathcal{L}_{\Delta}(\mathbb{C},V)$ is in fact the
enriched localization of $F_{\Delta}(\mathbb{C})$ at $F_{\Delta}(V)$ in the
sense that, for every simplicial category $\mathbb{D}$ and
$\mathbf{S}\text{-Cat}(F_{\Delta}(\mathbb{C}),\mathbb{D})^{F_{\Delta}V\mapsto\mathrm{Iso}}$
the category of simplicial functors which map $F_{\Delta}V$ to the core of
$\mathbb{D}$, we obtain a natural isomorphism

$\mathbf{S}\text{-Cat}(F_{\Delta}(\mathbb{C}),\mathbb{D})^{F_{\Delta}W\mapsto\mathrm{Iso}}\cong\mathbf{S}\text{-Cat}(L_{\Delta}(\mathbb{C},V),\mathbb{D}).$

This universal property together with the corresponding observation that
presheaves
$X\colon\mathcal{L}_{\Delta}(\mathbb{C},V)^{op}\rightarrow\mathbf{S}$ are
exactly the presheaves $F_{\Delta}(\mathbb{C})^{op}\rightarrow\mathbf{S}$
which take maps in $V$ to isomorphisms in $\mathbf{S}$, enables us to prove
the following proposition in the same way as we would prove it for
localizations in ordinary category theory.

######  Proposition 2.3.

For $(\mathbb{C},V)\in\mathrm{RelCat}$ and $j\colon
F_{\Delta}(\mathbb{C})\rightarrow\mathcal{L}_{\Delta}(\mathbb{C},V)$ the
associated localization functor, the induced restriction

$j^{\ast}\colon\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))\rightarrow\mathrm{sPsh}(F_{\Delta}(\mathbb{C}))$

is fully faithful.

  * Proof.

See [23, Proposition 8.2.3]. ∎

Therefore, the map $j\colon
F_{\Delta}(\mathbb{C})\rightarrow\mathcal{L}_{\Delta}(\mathbb{C},V)$ induces
both a localization

$(j_{!},j^{\ast})\colon\mathrm{sPsh}(F_{\Delta}(\mathbb{C}))\rightarrow\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))$

and a colocalization

$(j^{\ast},j_{\ast})\colon\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))\rightarrow\mathrm{sPsh}(F_{\Delta}(\mathbb{C}))$

between simplicial presheaf categories. Equipping both sides with the
injective model structure, the pair $(j^{\ast},j_{\ast})$ becomes a Quillen
pair. Its derived adjoint pair on underlying quasi-categories is the fully
faithful left Kan extension
$j^{\ast}\colon\mathcal{P}(N(\mathbb{C})[V^{-1}])\rightarrow\mathcal{P}(N(\mathbb{C}))$
for $j\colon N(\mathbb{C})\rightarrow N(\mathbb{C})[V^{-1}]$ together with its
right adjoint $j_{\ast}$. It hence also gives rise to a colocalization of
underlying quasi-categories. Hence, equipping both sides with the injective
model structure, the pair $(j^{\ast},j_{\ast})$ becomes a homotopy
colocalization. Dually, equipping both sides with the projective model
structure, the pair $(j_{!},j^{\ast})$ becomes a homotopy localization.

######  Remark 2.4.

If one chooses to work with any other homotopical localization of
$(\mathbb{C},V)$ such as the hammock localization
$\mathcal{L}_{H}(\mathbb{C},V)$ an analogue of the functor $j^{\ast}$ still
exists and also induces a localization $(j_{!},j^{\ast})$ and colocalization
$(j^{\ast},j_{\ast})$ on underlying quasi-categories.

######  Lemma 2.5.

For $(\mathbb{C},V)\in\mathrm{RelCat}$, the functor
$j^{\ast}\colon\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))\rightarrow\mathrm{sPsh}(\mathbb{C})$
induces a Quillen equivalence

$(j_{!},j^{\ast})\colon\mathcal{L}_{y[V]}\mathrm{sPsh}(\mathbb{C})_{\mathrm{proj}}\rightarrow\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))_{\mathrm{proj}}.$

  * Proof.

The $(\infty,1)$-categorical content of this statement seems to be folklore
and was also used in [20, Lemma 0.1]. The Quillen pair

$(j_{!},j^{\ast})\colon\mathcal{L}_{y[V]}\mathrm{sPsh}(\mathbb{C})_{\mathrm{proj}}\rightarrow\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))_{\mathrm{proj}}$

is a homotopy localization as noted above, and it takes every map in $y[V]$ to
a weak equivalence. By [10, Proposition 3.3.18.(1)], we hence obtain a
homotopy localization

$(j_{!},j^{\ast})\colon\mathcal{L}_{y[V]}\mathrm{sPsh}(\mathbb{C})_{\mathrm{proj}}\rightarrow\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{C},V))_{\mathrm{proj}}.$

The fact that this Quillen pair is a Quillen equivalence can be seen on
underlying quasi-categories, where it follows that the induced reflective
localization is an equivalence by essentially the same computations we
performed in the proof of Proposition 2.3. ∎

Analogously, for the injective model structure one can show a dualized
statement. The localization functor
$\mathcal{L}_{\Delta}\colon\mathrm{RelCat}\rightarrow\mathbf{S}\text{-Cat}$
has a homotopy inverse, the “delocalization” or “flattening”

$\flat\colon\mathbf{S}\text{-Cat}\rightarrow\mathrm{RelCat},$

given by the Grothendieck construction of its input
$\mathbf{C}^{op}\colon\Delta\rightarrow\mathrm{Cat}$. This functor was
introduced in [8, Theorem 2.5] and is analysed in detail in [2].

Now, given a simplicial category $\mathbf{C}$, consider its delocalization
$\flat(\mathbf{C})\in\mathrm{RelCat}$. Cofibrantly replacing
$\flat(\mathbf{C})$ by some pair $(\mathbb{P},V)$ in RelCat yields a direct
relative poset $(\mathbb{P},V)$ weakly equivalent – i.e. Rezk-equivalent in
the language of [2] – to $\flat(\mathbf{C})$. Hence, by [2, Theorem 1.8], the
simplicial localization
$\mathcal{L}_{\Delta}(\mathbb{P},V)\in\mathrm{Cat}_{\Delta}$ is DK-equivalent
to the original simplicial category $\mathbf{C}$, i.e. there is a zig-zag of
DK-equivalences

$\mathbf{C}\xrightarrow{f_{1}}\dots\xleftarrow{f_{n}}\mathcal{L}_{\Delta}(\mathbb{P},V).$
($\ast$)

By [14, Proposition A.3.3.8] or [8, Theorem 2.1] and the sequence of maps in
$(\ast)$, we obtain a zig-zag of Quillen equivalences

$\textstyle{\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{P},V))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}^{\ast}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f_{1})_{!}}$$\scriptstyle{(f_{n})_{!}}$$\textstyle{\mathrm{sPsh}(\mathbf{C})_{\mathrm{proj}}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{\ast}}$

Further recall from [7, Proposition 2.6] that for every category $\mathbb{C}$
the canonical projection $\varphi\colon
F_{\Delta}\mathbb{C}\rightarrow\mathbb{C}$ is a DK-equivalence of simplicial
categories. So, to summarize, we have seen the following.

######  Proposition 2.6.

Let $\mathbf{C}$ be a simplicial category. Then there is a direct relative
poset $(\mathbb{P},V)$ together with a zig-zag of DK-equivalences

$\mathbf{C}\rightarrow\dots\leftarrow\mathcal{L}_{\Delta}(\mathbb{P},V)$

in $\mathrm{Cat}_{\Delta}$ which induces a zig-zag of Quillen pairs

$\textstyle{\mathrm{sPsh}(\mathbb{P})_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\ast}}$$\textstyle{(\mathrm{sPsh}(F_{\Delta}\mathbb{P}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{!}}$$\scriptstyle{j_{!}}$$\textstyle{\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{P},V))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\ast}}$$\scriptstyle{f_{n}^{\ast}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f_{1})_{!}}$$\scriptstyle{(f_{n})_{!}}$$\textstyle{(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{\ast}}$

such that $(j^{\ast},j_{\ast})$ is a homotopy localization and all other pairs
are Quillen equivalences.

∎

##  3 Comparing compactness in quasi-categories and model categories

We start by stating some facts about compactness in presheaf categories. Given
a small category $\mathbb{C}$, we denote the cardinality of $\mathbb{C}$ by

$|\mathbb{C}|:=\sum_{C,C^{\prime}\in\mathbf{C}}|\mathrm{Hom}_{\mathbb{C}}(C,C^{\prime})|.$

Given a (set-valued) presheaf $X\in\widehat{\mathbb{C}}$, its cardinality is
denoted by

$|X|:=\sum_{C\in\mathbb{C}}|X(C)|.$

Given a regular cardinal $\kappa>|\mathbb{C}|$, recall that a presheaf
$X\in\widehat{\mathbb{C}}$ is $\kappa$-small if $|X|<\kappa$, that is if all
its values $X(C)$ have cardinality smaller than $\kappa$. A map $f\colon
X\rightarrow Y$ in $\widehat{\mathbb{C}}$ is $\kappa$-small if all its
pullbacks along maps $Z\rightarrow Y$ with $\kappa$-small domain $Z$ are
$\kappa$-small presheaves.

Given a small simplicial category $\mathbf{C}$, we also denote the cardinality
of $\mathbf{C}$ by

$|\mathbf{C}|:=\sum_{C,C^{\prime}}|\mathrm{Hom}_{\mathbf{C}}(C,C^{\prime})|$

where the cardinality of the hom-spaces
$\mathrm{Hom}_{\mathbf{C}}(C,C^{\prime})\in\mathbf{S}$ is given by the
cardinality of presheaves defined above. Accordingly, given a regular cardinal
$\kappa>|\mathbf{C}|$, a simplicial presheaf $X\in\mathrm{sPsh}(\mathbf{C})$
is $\kappa$-small if all its values $X(C)$ are $\kappa$-small. A simplicial
natural transformation $f\colon X\rightarrow Y$ in $\mathrm{sPsh}(\mathbf{C})$
is $\kappa$-small if all its pullbacks along maps $Z\rightarrow Y$ with
$\kappa$-small domain $Z$ are $\kappa$-small simplicial presheaves.

The category $\mathrm{sPsh}(\mathbf{C})$ is locally presentable, generated by
the objects $yC\otimes\Delta^{n}$ for $C\in\mathbf{C}$ and $n\geq 0$ which we
refer to in the following as “the generators”. When an ordinary category
$\mathbb{C}$ is considered as discrete simplicial category, we have an obvious
isomorphism between $\mathrm{sPsh}(\mathbb{C})$ and the set-valued presheaf
category $\widehat{\mathbb{C}\times\Delta^{op}}$.

######  Notation.

For two cardinals $\mu<\kappa$ recall the sharply larger relation from [1,
Definition 2.12] and the special case “$\ll$” from [1, Example 2.13.(4)] and
[14, Definition 5.4.2.8] (the notation is chosen to be coherent with Lurie’s
definition in the quasi-categorical case used below). The relation
$\mu\ll\kappa$ from [14, Definition 5.4.2.8] which holds if and only if for
all cardinals $\kappa_{0}<\kappa$ and $\mu_{0}<\mu$ we have
$\kappa_{0}^{\mu_{0}}<\kappa$.

The order “$\ll$” is chosen in such a way that whenever $\mu\ll\kappa$ holds,
then $\mu<\kappa$ and $\mu$-accessibility of a quasi-category $\mathcal{C}$
implies $\kappa$-accessibility of $\mathcal{C}$ ([14, Proposition 5.4.2.11]).
As noted in [14], the order is unbounded in the class of cardinals as for any
cardinal $\mu$ we have $\mu\ll\mathrm{sup}(\tau^{\mu}\mid\tau<\mu)^{+}$. In
particular, we always find a regular cardinal sharply larger than a given
$\mu$. Furthermore, if $\mu$ is regular, then $\mu\ll\mu^{+}$, and whenever we
have $\lambda<\mu$ and $\mu\ll\kappa$, then also $\mu\ll\lambda$. Thus, for
any set $X$ of cardinals there is a regular cardinal $\mu$ such that
$\kappa\ll\mu$ for all $\kappa\in X$.

######  Lemma 3.1.

Let $\mathbf{C}$ be a small simplicial category and $\kappa\gg|\mathbf{C}|$ an
infinite regular cardinal. Then

  1. 1.

An object $X\in\mathrm{sPsh}(\mathbf{C})$ is $\kappa$-compact if and only if
it is $\kappa$-small.

  2. 2.

A map $f\in\mathrm{sPsh}(\mathbf{C})$ is relative $\kappa$-compact if and only
if it is $\kappa$-small.

  * Proof.

Let $\mathbf{C}$ be a small simplicial category. For part $(a)$, recall that a
presheaf $X$ is $\kappa$-compact if and only if it is the retract of a
$\kappa$-small $|\mathbf{C}|$-directed colimit of $|\mathbf{C}|$-compact
objects by [1, Remark 2.15]. But one can show that $\kappa\gg|\mathbf{C}|$
implies that all $|\mathbf{C}|$-compact objects are $\kappa$-small. Hence,
every $\kappa$-compact presheaf $X$ is a subobject of a $\kappa$-small colimit
of $\kappa$-small presheaves and hence $\kappa$-small. Vice versa, every
$\kappa$-small presheaf is a $\kappa$-small colimit of the generators
$yC\otimes\Delta^{n}$ for $C\in\mathbf{C}$ and $n\geq 0$, and hence
$\kappa$-compact. Part $(b)$ follows directly from part $(a)$. ∎

######  Lemma 3.2.

Let $\mathbf{C}$ and $\mathbf{D}$ be small simplicial categories and let

$\textstyle{F\colon\mathrm{sPsh}(\mathbf{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{sPsh}(\mathbf{D})\colon
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

be a simplicial adjoint pair. Let $\kappa\gg|\mathbf{C}|\cdot|\mathbf{D}|$ be
regular (and inaccessible) and suppose $F$ takes representables to
representables (preserves $\kappa$-small objects). Then $G$ preserves
$\kappa$-small maps.

  * Proof.

Let
$\theta\colon\mathrm{sPsh}(\mathbf{C})(FX,Y)\rightarrow\mathrm{sPsh}(\mathbf{D})(X,GY)$
be the natural isomorphism associated to the adjunction
$F\rotatebox[origin={c}]{180.0}{$\vdash$}\hskip 1.20007ptG$. Let $f\colon
X\rightarrow Y$ be a $\kappa$-small map in $\mathrm{sPsh}(\mathbf{D})$ and
$g\colon yC\otimes\Delta^{n}\rightarrow GY$ be an element of $GY$. We have to
show that for every $C^{\prime}\in\mathbb{C}$ and $m\geq 0$ the hom-set
$\mathrm{sPsh}(\mathbf{C})(yC^{\prime}\otimes\Delta^{m},g^{\ast}GX)$ is
$\kappa$-small, but we have an isomorphism of hom-spaces between
$[yC^{\prime}\otimes\Delta^{m},g^{\ast}GX]_{\mathrm{sPsh}(\mathbf{C})}$ and
the pullack

$\displaystyle[yC^{\prime}\otimes\Delta^{m},yC\otimes\Delta^{n}]_{\mathrm{sPsh}(\mathbf{C})}\times_{[yC^{\prime}\otimes\Delta^{m},GY]_{\mathrm{sPsh}(\mathbf{C})}}[yC^{\prime}\otimes\Delta^{m},GX]_{\mathrm{sPsh}(\mathbf{C})}$
$\displaystyle\cong$
$\displaystyle([C^{\prime},C]_{\mathbf{C}}\times[yC^{\prime},1\otimes\Delta^{n}]_{\mathrm{sPsh}(\mathbf{C})})^{\Delta^{m}}\times_{[FyC^{\prime}\otimes\Delta^{m},Y]_{\mathrm{sPsh}(\mathbf{D})}}[FyC^{\prime}\otimes\Delta^{m},X]_{\mathrm{sPsh}(\mathbf{D})}.$

This yields an isomorphism on hom-sets between
$\mathrm{sPsh}(\mathbf{C})(yC^{\prime}\otimes\Delta^{m},g^{\ast}GX)$ and the
pullback

$\displaystyle(\mathbf{C}_{m}(C^{\prime},C)\times\mathrm{sPsh}(\mathbf{C})(yC^{\prime}\otimes\Delta^{m},1\otimes\Delta^{n}))\times_{\mathrm{sPsh}(\mathbf{D})(FyC^{\prime}\otimes\Delta^{m},Y)}\mathrm{sPsh}(\mathbf{D})(FyC^{\prime}\otimes\Delta^{m},X).$
(1)

The hom-set $\mathbf{C}_{m}(C^{\prime},C)$ is $\kappa$-small by assumption and
so is the entire left component of the pullback (1). If $F$ preserves
representables, the object $FyC^{\prime}$ is representable and hence
$\mathrm{sPsh}(\mathbf{D})(FyC^{\prime}\otimes\Delta^{m},X)$ is
$\kappa$-small, too. If $F$ preserves $\kappa$-small objects, the object
$FyC^{\prime}$ is $\kappa$-small and hence
$\mathrm{sPsh}(\mathbf{D})(FyC^{\prime}\otimes\Delta^{m},X)$ is $\kappa$-small
by inaccessibility of $\kappa$. Either way, it follows that the hom-set
$\mathrm{sPsh}(\mathbf{C})(yC^{\prime}\otimes\Delta^{m},g^{\ast}GX)$ is
$\kappa$-small by regularity of $\kappa$. ∎

The aim of this section is to compare this ordinary notion of compactness in a
simplicial presheaf category with the notion of compactness in its underlying
quasi-category as defined in [14, Definition 5.3.4.5] and [14, Definition
6.1.6.4]. The validity of this comparison was addressed in a question posted
in [15] by Shulman; for objects it is given in Proposition 3.5, for maps it is
given in Theorem 3.11 for the projective case and in Theorem 3.16 for the
injective case. A proof of the object-wise statement – that is Proposition 3.5
– was outlined by Lurie in the same post which in one direction coincides with
our proof given in Proposition 3.5. Before we state the theorems, we make the
following ad hoc construction and give one auxiliary folklore lemma.

Given a $\lambda$-accessible quasi-category $\mathcal{C}$ with generating set
$A$ and a regular cardinal $\mu\geq\lambda$, define the full subcategory
$J^{\mu}\subseteq\mathcal{C}$ recursively as follows. Let

$J^{\mu,0}_{0}:=A$

and $J^{\mu,0}$ be the full subcategory of $\mathcal{C}$ generated by
$J^{\mu,0}_{0}$. Whenever $\beta<\mu$ is a limit ordinal, let

$J^{\mu,\beta}_{0}=\bigcup_{\alpha<\beta}J^{\mu,\alpha}_{0}$

and $J^{\mu,\beta}$ the full subcategory generated by $J^{\mu,\beta}_{0}$. On
successors, given $J^{\mu,\alpha}$, let

$J^{\mu,\alpha+1}_{0}:=\\{\mathrm{colim}F\mid F\colon I\rightarrow
J^{\mu,\alpha},I\in\mathrm{QCat}\text{ is $\mu$-small and
$\lambda$-filtered}\\}$ (2)

(so we choose a set of representatives $V_{\mu}^{\Delta^{op}}$ for $\mu$-small
simplicial sets) and $J^{\mu,\alpha+1}$ be the corresponding full subcategory.
Eventually, we define the full subcategory $J^{\mu}$ of $\mathcal{C}$ to have
the set of objects

$J^{\mu}_{0}:=\bigcup_{\alpha<\mu}J^{\mu,\alpha}_{0}.$

The following lemma is noted in [14, Section 5.4.2] and a generalization of
the corresponding 1-categorical statement that can be found in [1, Remark
2.15] for accessible categories.

######  Lemma 3.3.

Let $\mathcal{C}$ and $\mathcal{D}$ be presentable quasi-categories.

  1. 1.

Suppose $\mathcal{C}$ is $\lambda$-presentable. Then, for every regular
$\mu\gg\lambda$, the $\mu$-compact objects in $\mathcal{C}$ are, up to
equivalence, exactly the retracts of objects in $J^{\mu}$.

  2. 2.

Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be an accessible functor. Then
there is a cardinal $\mu$ such that $F$ preserves $\kappa$-compact objects for
all regular $\kappa\gg\mu$.

  * Proof.

See [23, Lemma 8.3.4]. ∎

######  Notation 3.4.

The following group of statements will in each case claim that a certain
comparison holds for all $\kappa$ “sufficiently large” or “large enough”. That
means in each case there is a cardinal $\mu$ such that for all $\kappa\gg\mu$
the given statement holds true. Since we are not interested in a precise
formula for the lower bound $\mu$, we generally will not make the cardinal
$\mu$ explicit. Instead, we note that we will have to impose the condition on
$\kappa$ to be “large enough” only finitely often and eventually take the
corresponding supremum.

######  Proposition 3.5.

Let $\mathbb{M}$ be a combinatorial model category. Then for all sufficiently
large regular cardinals $\kappa$, an object
$C\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is $\kappa$-compact if and only if
there is a $\kappa$-compact $D\in\mathbb{M}$ such that $C\simeq D$ in
$\mathrm{Ho}_{\infty}(\mathbb{M})$.

  * Proof.

As observed in [15], one direction can be shown directly for every
combinatorial model category. Indeed, for $\kappa$ large enough,
$\kappa$-filtered colimits in $\mathbb{M}$ are homotopy colimits and the
$\kappa$-compact objects in $\mathbb{M}$ are exactly the $\kappa$-compact
objects in $N(\mathbb{M})$. So the localization
$N(\mathbb{M})\rightarrow\mathrm{Ho}_{\infty}(\mathbb{M})$ preserves
$\kappa$-filtered colimits and hence is $\kappa$-accessible. The statement now
follows from Lemma 3.3.

For the other direction, we note that by Dugger’s presentation theorem for
combinatorial model categories [5, Theorem 1.1] it suffices to consider left
Bousfield localizations of simplicial presheaf categories
$\mathrm{sPsh}(\mathbb{C})$. Indeed, given a combinatorial model category
$\mathbb{M}$ together with a category $\mathbb{C}$, a set
$T\subset\mathrm{sPsh}(\mathbb{C})$ of arrows and a Quillen equivalence

$\textstyle{\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L}$$\textstyle{\mathbb{M},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$

suppose we have shown the statement for
$\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}$ (or for
$\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{inj}}$ equivalently as
both model structures have the same underlying category and equivalent
underlying quasi-categories). Then, as both categories $\mathbb{M}$ and
$\mathrm{sPsh}(\mathbb{C})$ are presentable, we find $\kappa\gg|\mathbb{C}|$
large enough such that the right adjoint $R$ preserves $\kappa$-compact
objects. Certainly $\mathbb{L}L$ and $\mathbb{R}R$ preserve
$\kappa$-compactness in $\mathrm{Ho}_{\infty}(\mathbb{M})$, so given
$X\in\mathrm{Ho}_{\infty}(\mathbb{M})$ $\kappa$-compact, choose
$Y\in\mathrm{sPsh}(\mathbb{C})$ $\kappa$-small weakly equivalent to
$\mathbb{R}RX$. Without loss of generality $Y$ is cofibrant by [5, Proposition
2.3.(iii)] and so $L(Y)$ is $\kappa$-compact in $\mathbb{M}$ and presents $X$
in $\mathrm{Ho}_{\infty}(\mathbb{M})$. Therefore, assume
$\mathbb{M}=\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{inj}}$.
Furthermore, by Lemma 3.1 we can use the words “compact” and “small”
interchangeably.

Now, if $X\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is $\kappa$-compact, we have
seen in Lemma 3.3.(1) that there is a $Y\in J^{\kappa}$ such that $X$ is a
homotopy retract of $Y$. But the representatives for the colimits in the
construction of $(J^{\kappa,\alpha}|\alpha<\kappa)$ can be chosen to be
homotopy colimits of strict diagrams $F\colon I\rightarrow\mathbf{S}$ for
$\kappa$-small categories $I$ in $\mathbb{M}$ by [14, Proposition 4.2.3.14]
and [13, Proposition 1.3.4.25]. Hence, they can be computed according to the
Bousfield-Kan formula

$\mathrm{hocolim}F=\mathrm{coeq}\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
47.12488pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\coprod_{i\rightarrow j}F(i)\otimes
N(j/I)^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.12488pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.12488pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.12488pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\coprod_{i}F(i)\otimes
N(i/I)^{\mathrm{op}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$

because $\mathbb{M}=\mathcal{L}_{T}\mathrm{sPsh}(\mathbb{C})$ is a simplicial
model category. But this choice of homotopy colimit is $\kappa$-small whenever
$F$ and $I$ are $\kappa$-small, and hence, by induction, every object
contained in $J^{\kappa}$ is in fact a $\kappa$-small object in
$\mathrm{sPsh}(\mathbb{C})$. Therefore, as $X$ is homotopy equivalent to a
subobject of $Y\in J^{\kappa}$, it is homotopy equivalent to a $\kappa$-small
presheaf. This proves the proposition. ∎

In the following we generalize Proposition 3.5 to relative $\kappa$-compact
maps. We begin with a special class of model categories.

######  Definition 3.6.

Let $\mathbb{M}$ be a model category such that all cofibrations are
monomorphisms. Say $\mathbb{M}$ has a _theory of minimal fibrations_ if there
is a pullback stable class $\mathcal{F}^{\mathrm{min}}_{\mathbb{M}}$ of
fibrations in $\mathbb{M}$ – the class of _minimal fibrations_ – such that the
following hold.

  1. 1.

Let $p\colon X\twoheadrightarrow Y$ and $q\colon X^{\prime}\twoheadrightarrow
Y$ be minimal fibrations. Then every homotopy equivalence between $X$ and
$X^{\prime}$ over $Y$ is an isomorphism.

  2. 2.

For every fibration $p\colon X\twoheadrightarrow Y$ in $\mathbb{M}$ there is
an acyclic cofibration $M\overset{\sim}{\hookrightarrow}X$ such that the
restriction $M\rightarrow Y$ is a minimal fibration.

######  Lemma 3.7.

Let $\mathbb{M}$ be a model category such that all cofibrations are
monomorphisms. Suppose $\mathbb{M}$ has a theory of minimal fibrations. Let
$T$ be a class of maps in $\mathbb{M}$ such that the left Bousfield
localization $\mathcal{L}_{T}(\mathbb{M})$ exists. Then the model category
$\mathcal{L}_{T}(\mathbb{M})$ has a theory of minimal fibrations.

  * Proof.

Given a model category $\mathbb{M}$ and a class $T$ of maps in $\mathbb{M}$ as
stated, simply define the class $\mathcal{F}^{\mathrm{min}}_{\mathbb{M}}$ of
minimal fibrations in $\mathcal{L}_{T}\mathbb{M}$ to be the class of
fibrations in $\mathcal{L}_{T}\mathbb{M}$ which are minimal fibrations in
$\mathbb{M}$. Pullback stability of $\mathcal{F}^{\mathrm{min}}_{\mathbb{M}}$
and Property (1) are immediate. For Property (2), let $p\colon
X\twoheadrightarrow Y$ be a fibration in $\mathcal{L}_{T}\mathbb{M}$. By the
assumption that $\mathbb{M}$ has a theory of minimal fibrations, there is an
acyclic cofibration $M\overset{\sim}{\hookrightarrow}X$ in $\mathbb{M}$ such
that the restriction $M\twoheadrightarrow Y$ is a minimal fibration in
$\mathbb{M}$. But $M\rightarrow X$ is a weak equivalence from the fibration
$M\twoheadrightarrow Y$ to the fibration $p\colon X\twoheadrightarrow Y$ over
$Y$. The latter is a fibration in $\mathcal{L}_{T}\mathbb{M}$ and it hence
follows by [10, Proposition 3.4.6] that $M\twoheadrightarrow Y$ is a fibration
in $\mathcal{L}_{T}\mathbb{M}$, too. ∎

######  Proposition 3.8.

Let $\mathbb{C}$ be a small category and $\mathbb{M}$ be a cofibrantly
generated model structure on the presheaf category $\widehat{\mathbb{C}}$ such
that all cofibrations are monomorphisms. Suppose $\mathbb{M}$ has a theory of
minimal fibrations. Then for all sufficiently large regular cardinals
$\kappa$, a morphism $f\colon C\rightarrow D$ in
$\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact if and only if
there is a $\kappa$-small fibration $p\in\mathbb{M}$ between fibrant objects
such that $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$

  * Proof.

For one direction, let $p\colon X\twoheadrightarrow Y$ be a $\kappa$-small
fibration between fibrant objects in $\mathbb{M}$. Given a map $g\colon
A\rightarrow Y$ with $\kappa$-compact domain $A$ in
$\mathrm{Ho}_{\infty}(\mathrm{sPsh}(\mathbb{C}))$, in order to show that the
(strict) pullback of $X$ along $g$ is $\kappa$-compact in
$\mathrm{Ho}_{\infty}(\mathbb{M})$, by part $(1)$ we can present $A$ by a
$\kappa$-small object $A^{\prime}$. Without loss of generality $A^{\prime}$ is
bifibrant by [5, Proposition 2.3.(iii)], so we obtain a map $g^{\prime}\colon
A^{\prime}\rightarrow Y$ presenting $g$. Also the pullback
$(g^{\prime})^{\ast}X$ is a homotopy pullback and it is $\kappa$-small by
assumption. Hence, it is $\kappa$-compact in
$\mathrm{Ho}_{\infty}(\mathbb{M})$ by part $(1)$. This shows that $p$ is
relative $\kappa$-compact in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

For the converse direction, assume that $f\colon C\rightarrow D$ is relative
$\kappa$-compact in $\mathrm{Ho}_{\infty}(\mathbb{M})$ and $p\colon
X\twoheadrightarrow Y$ is a fibration in $\mathbb{M}$ such that $Y$ fibrant in
$\mathbb{M}$ and $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$. By
Definition 3.6.(2) there is a subobject $M\subseteq X$ such that the
restriction $m\colon M\twoheadrightarrow Y$ of $p$ is a minimal fibration. As
$m$ and $p$ are homotopy equivalent over $Y$, the fibration $m$ is relative
$\kappa$-compact in $\mathrm{Ho}_{\infty}(\mathbb{M})$, too. We want to show
that $m$ is a $\kappa$-small fibration. Therefore, for $C\in\mathbb{C}$ let
$g\colon yC\rightarrow Y$ be an element of $Y$, so that we have to show that
the pullback $g^{\ast}M$ as depicted in the diagram

$\textstyle{g^{\ast}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\ast}m}$

$\cdot$$\lrcorner$

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{yC\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y.}$

is a $\kappa$-small object in $\widehat{\mathbb{C}}$. By [5, Proposition
2.3.(iii)] there is a $\kappa$-small fibrant replacement $RC$ of the
representable $yC$. Since the object $Y$ is fibrant, we obtain an extension
$g^{\prime}\colon RC\rightarrow Y$ of $g$ along the acylic cofibration
$yC\overset{\sim}{\hookrightarrow}RC$ and hence a factorization of the
following form.

$\textstyle{g^{\ast}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\ast}m}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{(g^{\prime})^{\ast}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{yC\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\sim}$$\textstyle{Y}$$\textstyle{RC\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$

All three faces of the diagram are pullback squares, hence, in order to show
that the object $g^{\ast}M$ is $\kappa$-small, it suffices to show that the
object $(g^{\prime})^{\ast}M$ is $\kappa$-small.

We know that $RC$ is also $\kappa$-compact in the underlying quasi-category
$\mathrm{Ho}_{\infty}(\mathbb{M})$ by part $(1)$ and so is the map
$(g^{\prime})^{\ast}m$ by our assumption on the morphism $f$. The underlying
quasi-category $\mathrm{Ho}_{\infty}(\mathbb{M}/RC)$ is equivalent to the
overcategory $\mathrm{Ho}_{\infty}(\mathbb{M})_{/RC}$ by [23, Lemma 7.2.1],
and is generated under (homotopy) colimits by the $\kappa$-small collection of
maps with codomain $RC$ and representable domain by [6, Proposition 2.9].
Recall that the domain functor

$\mathrm{dom}\colon\mathrm{Ho}_{\infty}(\mathbb{M})_{/RC}\rightarrow\mathrm{Ho}_{\infty}(\mathbb{M})$

both preserves and reflects colimits by [14, Proposition 1.2.13.8]. Therefore,
by Lemma 3.3.(2) the functor dom preserves $\kappa$-small objects. Dually, it
is not hard to see that the functor dom also reflects $\kappa$-compact
objects. We obtain that the map
$r^{\ast}m\in\mathrm{Ho}_{\infty}(\mathbb{M})_{/RC}$ is a $\kappa$-compact
object. By part $(1)$ applied to the combinatorial model category
$\mathbb{M}/RC$ (observing that the chosen $\kappa$ for $\mathbb{M}$ also
works for $\mathbb{M}/RC$ since $RC$ is $\kappa$-small), we obtain a
$\kappa$-small fibration $q\colon Z\twoheadrightarrow RC$ together with a
homotopy equivalence $Z\simeq(g^{\prime})^{\ast}M$ over $RC$. Again by [4,
Theorem 2.14] there is a subobject $N\subseteq Z$ such that the restriction
$n\colon N\twoheadrightarrow RC$ of $q\colon Z\rightarrow RC$ is a minimal
fibration. Clearly $n$ is still $\kappa$-small. But the induced homotopy
equivalence $N\simeq(g^{\prime})^{\ast}M$ over $RC$ is a homotopy equivalence
between minimal fibrations and hence turns out to be an isomorphism by [4,
Proposition 2.16]. Therefore, $(g^{\prime})^{\ast}M$ is $\kappa$-small. ∎

######  Corollary 3.9.

Let $\mathbb{P}$ be an Eilenberg-Zilber Reedy category in the sense of [4,
Section 2.1] and $\mathbb{M}=\mathrm{sPsh}(\mathbb{P})_{\mathrm{inj}}$ the
category of simplicial presheaves on $\mathbb{P}$ equipped with the injective
model structure. Then for all sufficiently large regular cardinals $\kappa$, a
morphism $f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact
if and only if there is a $\kappa$-small fibration $p\in\mathbb{M}$ between
fibrant objects such that $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

  * Proof.

The model category $\mathbb{M}$ supports a theory of minimal fibrations as
shown in [4, 2.13-2.16], thus one direction follows immediately from
Proposition 3.8. For the other direction we can proceed exactly as we did in
the proof of Proposition 3.8.

∎

###  3.1 The projective case

We now make use of the observations in Section 2 to generalize Corollary 3.9
to the category of simplicial presheaves over arbitrary small simplicial
categories.

Therefore, we make use of the following adaption of [6, Proposition 5.10,
Corollary 6.5], which in virtue of the simplicial enrichment of all involved
categories and the available literature on the general theory of such in fact
is easier to show than the original.

######  Lemma 3.10.

Let $\mathbf{C}$ be a small simplicial category, and $\mathbb{M}$,
$\mathbb{N}$ be simplicial model categories together with a simplicial Quillen
equivalence $(L,R)\colon\mathbb{N}\xrightarrow{\sim}\mathbb{M}$. Let $T$ be a
class of arrows in $\mathrm{sPsh}(\mathbf{C})$ and
$(F,G)\colon\mathcal{L}_{T}(\mathrm{sPsh(\mathbf{C})})_{\mathrm{proj}}\rightarrow\mathbb{M}$
a simplicial Quillen pair. Then there is a Quillen pair
$(F^{\prime},G^{\prime})\colon\mathcal{L}_{T}(\mathrm{sPsh(\mathbf{C})})_{\mathrm{proj}}\rightarrow\mathbb{N}$
such that the functors $L\circ F^{\prime}$ and $F$ are Quillen-homotopic in
the sense of [6, Definition 5.9]. In other words, simplicial Quillen pairs
with domain $\mathcal{L}_{T}(\mathrm{sPsh(\mathbf{C})})_{\mathrm{proj}}$ can
be lifted along Quillen equivalences up to homotopy.

  * Proof.

Let $\lambda$ and $\rho$ denote cofibrant and fibrant replacements
respectively and $\mathbb{L}=\lambda^{\ast}$, $\mathbb{R}=\rho^{\ast}$ denote
their associated left and right derivations of functors. Let the composition

$\lambda\mathbb{R}(R)Fy\colon\mathbf{C}\rightarrow\mathbb{N}$

be denoted by $p$. Note that the left and right derivation $\mathbb{L}$ and
$\mathbb{R}$ of simplicial functors may be chosen to be simplicial again by
[18, Corollary 13.2.4], thus $p$ is a simplicial functor and we can consider
the simplicially enriched left Kan extension

$\textstyle{\mathbf{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y}$$\scriptstyle{p}$$\textstyle{\mathbb{N}.}$$\textstyle{\mathrm{sPsh}(\mathbf{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Lan}_{y}p}$

We claim that $F^{\prime}:=\mathrm{Lan}_{y}p$ is the left Quillen functor we
are looking for. First, let us construct the Quillen homotopies connecting
$L\circ F^{\prime}$ and $F$.

Recall that, as explained for instance in [12, 4.31], for every presheaf
$X\in\mathrm{sPsh}(\mathbf{C})$ the object $\mathrm{Lan}_{y}p(X)$ is the
colimit of $p$ weighted by $X$, i.e.

$F^{\prime}=\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip 2.0pt}\star p.$

The left Quillen functor $L\colon\mathbb{N}\rightarrow\mathbb{M}$ is a left
adjoint and hence preserves weighted colimits, thus we have that $L\circ
F^{\prime}\cong\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip 2.0pt}\star Lp$.
Further, from [16, Theorem 3.3] remember that the weighted colimit functor

$\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip 2.0pt}\star\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip
2.0pt}\colon\mathrm{sPsh}(\mathbf{C})_{\mathrm{proj}}\times([\mathbf{C},\mathbb{M}]_{\mathbf{S}})_{\mathrm{inj}}\rightarrow\mathbb{M}$

is a left Quillen bifunctor. In particular, for cofibrant presheaves
$X\in\mathrm{sPsh}(\mathbf{C})_{\mathrm{proj}}$ the $X$-weighted colimit

$X\star\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip
2.0pt}\colon([\mathbf{C},\mathbb{M}]_{\mathbf{S}})_{\mathrm{inj}}\rightarrow\mathbb{M}$

is a left Quillen functor. But both $Lp\cong\mathbb{L}(L)\mathbb{R}(R)Fy$ and
$Fy$ are cofibrant objects in
$([\mathbf{C},\mathbb{M}]_{\mathbf{S}})_{\mathrm{inj}}$, the former because
representables are projectively cofibrant and $F$ preserves cofibrant objects,
and the latter because $L$ preserves cofibrant objects. Thus, if
$\rho_{Fy}\colon Fy\rightarrow r(Fy)$ denotes an injective fibrant replacement
of $Fy$, the counit $\varepsilon_{r(Fy)}\colon Lp\Rightarrow r(Fy)$ of the
Quillen-equivalence $(L,R)$ induces a span of natural weak equivalences
between the cofibrant objects $Lp$, $r(Fy)$ and $Fy$. Thus, for cofibrant
presheaves $X\in(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}$, we obtain a zig
zag of natural weak equivalences between $X\star Lp$ and $X\star Fy$. But
$\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip 2.0pt}\star Fy$ is just $F$ (by
[12, 4.51]), thus we have constructed a span of Quillen homotopies between
$L\circ F^{\prime}$ and $F$.

Second, the fact that
$F^{\prime}\colon(\mathrm{sPsh}(\mathbf{C})_{\mathrm{proj}}\rightarrow\mathbb{N}$
is a left Quillen functor with right adjoint $G^{\prime}(N)=[p\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip 2.0pt},N]_{\mathbb{N}}$ was basically already
shown above (following for instance, as it were, from [16, Theorem 3.3.]).

We are left to show that, third, the Quillen pair

$(F^{\prime},G^{\prime})\colon\mathrm{sPsh}(\mathbf{C})_{\mathrm{proj}}\rightarrow\mathbb{N}$

descends to the localization at $T$ whenever $F$ does so. That is, we have to
show that in this case every arrow $f\in T$ is send to a weak equivalence in
$\mathbb{N}$. Without loss of generality all arrows $f\in T$ have cofibrant
domain and codomain. Then, given $f\in T$, the arrow $F(f)$ is a weak
equivalence in $\mathbb{M}$, and so is $LF^{\prime}(f)\in\mathbb{M}$ since $F$
and $LF^{\prime}$ are Quillen-homotopic. Thus, $\mathbb{R}(R)(LF^{\prime}(f))$
is a weak equivalence in $\mathbb{N}$, but this arrow is weakly equivalent to
$F^{\prime}(f)$ since $(L,R)$ is a Quillen equivalence. This concludes the
proof. ∎

######  Theorem 3.11.

Let $\mathbf{C}$ be a small simplicial category,
$T\subset\mathrm{sPsh}(\mathbf{C})$ be a set of maps and
$\mathbb{M}=\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}$. Then
for all sufficiently large inaccessible cardinals $\kappa$, a morphism
$f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact if and
only if there is a $\kappa$-small fibration $p\in\mathbb{M}$ between fibrant
objects such that $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

  * Proof.

Let $\mathbf{C}$ be a small simplicial category and
$T\subset\mathrm{sPsh}(\mathbf{C})$ a set of maps. Combining Lemma 2.5 and
Proposition 2.6, we obtain a zig-zag of Quillen equivalences

$\textstyle{\mathcal{L}_{\varphi_{!}[y[V]]}\mathrm{sPsh}(\mathbb{P})_{\mathrm{inj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\textstyle{\mathcal{L}_{\varphi_{!}[y[V]]}\mathrm{sPsh}(\mathbb{P})_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\ast}}$$\scriptstyle{\mathrm{id}}$$\textstyle{\mathcal{L}_{y[V]}(\mathrm{sPsh}(F_{\Delta}\mathbb{P}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{!}}$
$\textstyle{\mathcal{L}_{y[V]}(\mathrm{sPsh}(F_{\Delta}\mathbb{P}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{!}}$$\textstyle{\mathrm{sPsh}(\mathcal{L}_{\Delta}(\mathbb{P},V))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\ast}}$$\scriptstyle{f_{n}^{\ast}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f_{1})_{!}}$$\scriptstyle{(f_{n})_{!}}$$\textstyle{(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{\ast}}$

such that $\mathbb{P}$ is an Eilenberg-Zilber Reedy category. This yields a
zig-zag of Quillen equivalences

$\textstyle{\mathcal{L}_{(\varphi_{!}[y[V]]\cup\bar{T})}\mathrm{sPsh}(\mathbb{P})_{\mathrm{inj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\bar{T}\subset\mathrm{sPsh}(\mathbb{P})$ is obtained from
$T\subset\mathrm{sPsh}(\mathbf{C})$ by transferring $T$ along the finitely
many Quillen equivalences successively. We denote the union
$\varphi_{!}[y[V]]\cup\bar{T}\subset\mathrm{sPsh}(\mathbb{P})$ short-handedly
by $U$.

By Lemma 3.10 this chain of Quillen equivalences induces a single Quillen
equivalence

$\textstyle{\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\mathcal{L}_{U}(\mathrm{sPsh}(\mathbb{P}))_{\mathrm{inj}}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$
(3)

The Bousfield localization
$\mathcal{L}_{U}(\mathrm{sPsh}(\mathbb{P}))_{\mathrm{inj}}$ has a theory of
minimal fibrations by Lemma 3.7. Now, let $\kappa\gg|\mathbf{C}|,|\mathbb{P}|$
be inaccessible large enough such that Corollary 3.9 applies to $\mathbb{P}$
and large enough such that Proposition 3.8 applies to
$\mathcal{L}_{U}(\mathrm{sPsh}(\mathbb{P}))_{\mathrm{inj}}$.

For one direction, let $f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ be relative
$\kappa$-compact. Since the pair (3) is a Quillen equivalence, the quasi-
category $\mathrm{Ho}_{\infty}(\mathbb{M})$ is equivalent to the underlying
quasi-category of $\mathcal{L}_{U}(\mathrm{sPsh}(\mathbb{P})_{\mathrm{inj}}$.
Then, by Proposition 3.8, there is a $\kappa$-small fibration $p\colon
X\twoheadrightarrow Y$ between fibrant objects in
$\mathcal{L}_{U}(\mathrm{sPsh}(\mathbb{P}))_{\mathrm{inj}}$ presenting $f$ in
$\mathrm{Ho}_{\infty}(\mathbb{M})$. By Lemma 3.3 (or its ordinary categorical
analogon), we know that for $\kappa\gg|\mathbf{C}|,|\mathbb{P}|$ the left
adjoint $F$ preserves $\kappa$-compact objects. Hence, since
$\kappa\gg|\mathbf{C}|,|\mathbb{P}|$ is inaccessible, by Lemma 3.2 the right
Quillen functor $G$ preserves $\kappa$-small maps. Thus, $Gp\colon
GX\twoheadrightarrow GY$ is a $\kappa$-small fibration between fibrant objects
presenting $f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

The proof of other direction is exactly as in Proposition 3.8. ∎

######  Corollary 3.12.

Let $\mathbb{M}$ be a combinatorial model category. Let
$\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}$ be the
presentation of $\mathbb{M}$ from Dugger’s representation theorem for
combinatorial model categories in [5]. Then for all sufficiently large
inaccessible cardinals $\kappa$, a morphism
$f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact if and
only if there is a $\kappa$-small fibration
$p\in\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}$ between
fibrant objects such that $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

∎

######  Remark 3.13.

The reason why in Corollary 3.12 we don’t obtain the comparison result for
$\mathbb{M}$ itself is that there is no obvious reason why the Quillen
equivalence

$\textstyle{\mathcal{L}_{T}(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L}$$\textstyle{\mathbb{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$

given by Dugger’s presentation theorem should preserve $\kappa$-small maps.
While the right adjoint certainly does preserve such maps, the left adjoint
does not seem to exhibit any properties with that respect.

###  3.2 The injective case

In this section we prove an analogous result for the injective model structure
and get rid of the condition on fibrancy of the bases whenever the
localization is left exact. We will make use of Shulman’s results [21] in two
ways. Therefore, applied to the special case relevant for this paper, recall
the forgetful functor

$U\colon\mathrm{sPsh}(\mathbf{C})\rightarrow\mathbf{S}^{\mathrm{Ob}(\mathbf{C})}$

with right adjoint

$G\colon\mathbf{S}^{\mathrm{Ob}(\mathbf{C})}\rightarrow\mathrm{sPsh}(\mathbf{C}).$

The functor $G$ takes objects $W\in\mathbf{S}^{\mathrm{Ob}(\mathbf{C})}$ to
the presheaf evaluating

$G(W)(C):=\prod_{C^{\prime}\in\mathbf{C}}W(C^{\prime})^{[C^{\prime},C]_{\mathbf{C}}}\in\mathbf{S}.$

The adjoint pair $(U,G)$ gives rise to a comonad on
$\mathrm{sPsh}(\mathbf{C})$ with standard resolution

$\mathsf{C}_{\bullet}(G,UG,U\mbox{\hskip 3.0pt\text@underline{\ \ }\hskip
2.0pt})\colon\mathrm{sPsh}(\mathbf{C})\rightarrow\mathrm{sPsh}(\mathbf{C})^{\Delta}.$

The associated cobar construction $\mathsf{C}(G,UG,U\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip
2.0pt})\colon\mathrm{sPsh}(\mathbf{C})\rightarrow\mathrm{sPsh}(\mathbf{C})$ is
then defined as the pointwise totalization

$\mathrm{Tot}(\mathsf{C}_{\bullet}(G,UG,U\mbox{\hskip 3.0pt\text@underline{\ \
}\hskip 2.0pt}))=\int_{[n]\in\Delta}(\mathsf{C}_{n}(G,UG,U\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip 2.0pt}))^{\Delta^{n}}.$

A crucial observation of Shulman is that the cobar construction takes
(acyclic) projective fibrations to pointwise weakly equivalent (acyclic)
injective fibrations. More precisely, the natural coaugmentation
$\eta\colon\mathrm{id}\Rightarrow\mathsf{C}(G,UG,U\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip 2.0pt})$ is a pointwise weak equivalence, and
the arrow $\mathsf{C}(G,UG,Up)$ is an (acyclic) injective fibration whenever
$p$ is a (acyclic) projective fibration. All this is covered in [21, Section
8] in much greater generality.

In fact, it is not hard to see that the cobar construction preserves
$\kappa$-smallness (for $\kappa$ large enough).

######  Lemma 3.14.

Let $\mathbf{C}$ be a small simplicial category and $f\colon X\rightarrow Y$
be a $\kappa$-small map in $\mathrm{sPsh}(\mathbf{C})$. Then
$\mathsf{C}(G,UG,Uf)$ is $\kappa$-small, too.

  * Proof.

Clearly, the forgetful functor preserves $\kappa$-smallness. If the right
adjoint $G$ preserves $\kappa$-smallness, too, then for every $\kappa$-small
map $f\colon X\rightarrow Y$ in $\mathrm{sPsh(\mathbf{C}})$, the map
$\mathsf{C}_{\bullet}(G,UG,Uf)$ of cosimplicial objects is levelwise
$\kappa$-small. Thus we are only left to show that totalization preserves
$\kappa$-smallness. All in all, this leaves us to show that the two functors
$G\colon\mathbf{S}^{\mathrm{Ob}(\mathbf{C})}\rightarrow\mathrm{sPsh}(\mathbf{C})$
and $\mathrm{Tot}(\mathsf{C}_{\bullet}(G,UG,U\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip
2.0pt}))=\int_{[n]\in\Delta}(\mathsf{C}_{n}(G,UG,U\mbox{\hskip
3.0pt\text@underline{\ \ }\hskip 2.0pt}))^{\Delta^{n}}$ preserve
$\kappa$-smallness of maps. Both proofs consist of rather elementary checks
with no suprises. ∎

Therefore, we directly obtain an analogue of Theorem 3.11 for the injective
model structure as follows.

######  Proposition 3.15.

Let $\mathbf{C}$ be a small simplicial category,
$T\subset\mathrm{sPsh}(\mathbf{C})$ be a set of maps and
$\mathbb{M}=\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{inj}}$. Then
for all sufficiently large inaccessible cardinals $\kappa$, a morphism
$f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact if and
only if there is a $\kappa$-small fibration $p\in\mathbb{M}$ between fibrant
objects such that $p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

  * Proof.

Let $f$ be relative $\kappa$-compact in $\mathrm{Ho}_{\infty}(\mathbb{M})$. By
Theorem 3.11 there is a fibration $p\colon X\twoheadrightarrow Y$ between
fibrant objects in
$\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{proj}}$ such that
$p\simeq f$ in the underlying quasi-category. Hence, by Lemma 3.14 and [21,
Section 8] the map

$\mathsf{C}(G,UG,Up)\colon\mathsf{C}(G,UG,UX)\twoheadrightarrow\mathsf{C}(G,UG,UY)$

is a $\kappa$-small injective fibration between injectively fibrant objects.
But since the coaugmentations $\eta_{X}$ and $\eta_{Y}$ are pointwise weak
equivalences, the objects $\mathsf{C}(G,UG,UX)$ and $\mathsf{C}(G,UG,UY)$ are
$T$-local and thus fibrant in $\mathbb{M}$. Thus, the map
$\mathsf{C}(G,UG,Up)$ is a fibration in $\mathbb{M}$.

The other direction follows immediately from Theorem 3.11 since every
injective fibration is a projective fibration. ∎

Note that if $\mathbb{M}$ satisfies the fibration extension property for
$\kappa$-small maps ([23, Definition 2.2.1]), we can get rid of the fibrancy
condition on the bases of small maps. That is, because in that case every
$\kappa$-small fibration is weakly equivalent to a $\kappa$-small fibration
with fibrant base. Since every left exact left Bousfield localization of
$\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$ is a type theoretic model topos by
[21, Corollary 8.31, Theorem 10.5] and hence has univalent universes for
$\kappa$-small fibrations, the class $S_{\kappa}$ satisfies the fibration
extension property in every left exact left Bousfield localization of
$\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$.

######  Theorem 3.16.

Let $\mathbf{C}$ be a small simplicial category,
$T\subset\mathrm{sPsh}(\mathbf{C})$ be a set of maps such that the
localization
$\mathbb{M}=\mathcal{L}_{T}(\mathrm{sPsh}(\mathbf{C}))_{\mathrm{inj}}$ is left
exact. Then for all sufficiently large inaccessible cardinals $\kappa$, a
morphism $f\in\mathrm{Ho}_{\infty}(\mathbb{M})$ is relative $\kappa$-compact
if and only if there is a $\kappa$-small fibration $p\in\mathbb{M}$ such that
$p\simeq f$ in $\mathrm{Ho}_{\infty}(\mathbb{M})$.

##  4 The correspondence of weak Tarski universes and object classifiers

We conclude by commenting briefly on the relevance of these results for
Homotopy Type Theory. Let $\mathcal{C}$ be a presentable quasi-category and
let $\mathbf{C}$ be a small simplicial category with a set $T$ of arrows in
$\mathrm{sPsh}(\mathbf{C})$ such that the localization
$\mathbb{M}:=\mathcal{L}_{T}\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$ presents
$\mathcal{M}$. Then $\mathbb{M}$ is a type theoretic model category as shown
in [9, Section 7]. Shulman recently has shown in [21] (among other results)
that this presentation $\mathbb{M}$ in fact is a type theoretic model topos,
and hence exhibits an infinite sequence of univalent strict Tarski universes
whenever $\mathcal{M}$ is an $\infty$-topos. In fact it is reasonable to
assume that in this case the object classifiers in $\mathcal{M}$ at least
yield univalent weak Tarski universes as claimed in the Introduction of [9].
Here, by “weak Tarski universe” we understand an inaccessible cardinal
$\kappa$ together with a fibration that is weakly universal for the class of
$\kappa$-small fibrations. Weak universality of a fibration $p\colon
E\twoheadrightarrow B$ for a class $S$ of fibrations in turn means that $p$ is
univalent and that for all fibrations $q\colon X\twoheadrightarrow Y$ in $S$
there is a map $w\colon X\rightarrow B$ such that $q$ is the homotopy pullback
of $p$ along $w$. Clearly, every univalent strictly universal fibration is a
weakly universal fibration for the same class of maps whenenever the model
category is right proper.

Then it is easy to see that a univalent weakly universal fibration for a
pullback stable class $S$ of fibrations in $\mathbb{M}$ yields a classifying
object for the class $\mathrm{Ho}_{\infty}[S]$ of morphisms in $\mathcal{M}$
and that, vice versa, every classifying object for a pullback stable class $T$
of morphisms in $\mathcal{M}$ yields a univalent weakly universal fibration
for the class

$\bar{T}:=\\{f\in\mathcal{F}_{\mathbb{M}}\mid
f\in\mathrm{Ho}_{\infty}(\mathbb{M})\text{ is in $T$}\\}$

of maps in $\mathbb{M}$. There is one class of maps in each case which is
relevant for the construction of strict Tarski universes in $\mathbb{M}$ on
the one hand, and the construction of object classifiers in $\mathcal{M}$ on
the other. That is, given a sufficiently large inaccessible cardinal $\kappa$,
the class $S_{\kappa}$ of $\kappa$-small fibrations in
$\mathrm{sPsh}(\mathbf{C})$ and the class $T_{\kappa}$ of relative
$\kappa$-compact maps in $\mathcal{M}$. In the former case, the common
constructions of univalent universal fibrations
$\pi_{\kappa}\colon\tilde{U}_{\kappa}\twoheadrightarrow U_{\kappa}$ use
various functorial closure properties of $S_{\kappa}$ and the fact that an
infinite sequence of inaccessible cardinals yields a cumulative hierarchy of
universal fibrations in this way. In the latter case, [14, Theorem 6.1.6.8]
characterizes $\infty$-toposes in terms of classifying objects $V_{\kappa}$
for $T_{\kappa}$ for all sufficiently large cardinals $\kappa$.

While the associated classifying map
$\pi_{\kappa}\colon\tilde{V}_{\kappa}\rightarrow V_{\kappa}$ lifts to a
fibration in $\mathbb{M}$ which is weakly universal for $\bar{T}_{\kappa}$,
and $U_{\kappa}$ descends to a classifying object for the class
$\mathrm{Ho}_{\infty}[S_{\kappa}]$, it is a priori unclear whether
$S_{\kappa}=\bar{T}_{\kappa}$ or $T_{\kappa}=\mathrm{Ho}_{\infty}[S_{\kappa}]$
hold. In other words, without a comparison of smallness notions as considered
in Section 3, it is not clear whether the categorical construction of (either
weak or strict) universal $\kappa$-small fibrations in $\mathbb{M}$ – which
models Tarski universes in the associated type theory – also models universes
in the underlying quasi-category.

But Theorem 3.16 shows that both $S_{\kappa}=\bar{T}_{\kappa}$ and
$T_{\kappa}=\mathrm{Ho}_{\infty}[S_{\kappa}]$ hold true. In other words, we
have shown the following.

######  Corollary 4.1.

Let $\mathbb{M}\cong\mathcal{L}_{T}\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$
be a model topos, let $p$ be a fibration in $\mathbb{M}$. Let $\kappa$ be a
sufficiently large inaccessible cardinal. Then $p$ is univalent and weakly
universal for $\kappa$-small fibrations if and only if $p$ is a classifying
map for relatively $\kappa$-compact maps in
$\mathrm{Ho}_{\infty}(\mathbb{M})$.

∎

######  Remark.

Let us finish with a note on the existence of “sufficiently large” cardinals
that has been a standing assumption along the way and that only has been
addressed very briefly when introduced in Notation 3.4. It is interesting to
remember that to be sufficiently large means to be contained in the club of
cardinals sharply larger than a specified cardinal associated to the small
simplicial category $\mathbf{C}$ – or the $\infty$-Grothendieck topos
$\mathcal{M}$ that is.222See [22] for an illuminating discussion on directly
related issues. Thus, if we start out with an $\infty$-topos $\mathcal{M}$ and
wish to show that its type theoretic presentation
$\mathbb{M}:=\mathcal{L}_{T}\mathrm{sPsh}(\mathbf{C})_{\mathrm{inj}}$ exhibits
a universal fibration, we need an object classifier $U_{\kappa}$ in
$\mathcal{M}$ classifying relative $\kappa$-compact maps for $\kappa$ _in_ the
sharply large club associated to $\mathcal{M}$. The same holds if we want to
show that a given cumulative hierarchy of universal fibrations in $\mathbb{M}$
yields a corresponding hierarchy of object classifiers in $\mathcal{M}$. Thus,
it seems that 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 a Mahlo cardinal, rather than
only an inaccessible (or an infinite sequence of such).

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Department of Mathematics and Statistics, Masaryk University, Kotlářská 2,
Brno 61137, Czech Republic

E-mail address<EMAIL_ADDRESS>