[10] **1

# The effective model structure and $\infty$-groupoid objects

Nicola Gambino  Simon Henry  Christian Sattler  Karol Szumiło

(††thanks: This version of the paper reflects the one published in Forum of
Mathematics, Sigma (2022), Vol. 10:e34 1–59, submitted 9 March 2021, revised
19 January 2022, accepted 7 February 2022, available here. Two additional
minor typos have been fixed.)

###### Abstract

For a category $\mathcal{E}$ with finite limits and well-behaved countable
coproducts, we construct a model structure, called the effective model
structure, on the category of simplicial objects in $\mathcal{E}$,
generalising the Kan–Quillen model structure on simplicial sets. We then prove
that the effective model structure is left and right proper and satisfies
descent in the sense of Rezk. As a consequence, we obtain that the associated
$\infty$-category has finite limits, colimits satisfying descent, and is
locally Cartesian closed when $\mathcal{E}$ is, but is not a higher topos in
general. We also characterise the $\infty$-category presented by the effective
model structure, showing that it is the full sub-category of presheaves on
$\mathcal{E}$ spanned by Kan complexes in $\mathcal{E}$, a result that
suggests a close analogy with the theory of exact completions. 00footnotetext:
2020 Mathematics Subject Classification: 18N40 (primary), 18N60, 55U10.

## Introduction

Context and motivation. Over the past two decades, there has been an explosion
of interest in the connections between model categories and higher categories
[cisinski2020higher, Gepner-Kock, Joyal-Tierney, Lurie, Rezk-Segal, Szumilo].
This line of research led to the reformulation of significant parts of modern
homotopy theory in terms of higher category theory, the development of higher
topos theory [Toen-Vezzosi, Lurie] and is of great importance for Homotopy
Type Theory and the Univalent Foundations programme [Awodey-Warren,
berg2018univalent, Gepner-Kock, Kapulkin-Lumsdaine, Shulman]. Central to these
developments are model structures on categories of simplicial objects, i.e.,
functor categories of the form
$\mathsf{s}\mathcal{E}=[\Delta^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$,
where $\mathcal{E}$ is a category, as considered in [Quillen]*Section II.4,
[Goerss-Jardine]*Chapter II, [christensen2002quillen]*Theorem 6.3 and
[Hormann]. In particular, the category of simplicial sets equipped with the
Kan–Quillen model structure [Quillen] can be understood as a presentation of
the $\infty$-category of spaces, while categories of simplicial presheaves and
sheaves (i.e., simplicial objects in a Grothendieck topos) equipped with the
Rezk model structure [Rezk] and the Joyal–Jardine model structure [Brown,
Joyal, Jardine] can be seen as presentations of $\infty$-toposes and their
hypercompletions, respectively [DHI, Lurie].

The main contribution of this paper is to construct a new model structure,
which we call the effective model structure, on categories of simplicial
objects $\mathsf{s}\mathcal{E}$, assuming that $\mathcal{E}$ is merely a
countably lextensive category, i.e., a category with finite limits and
countable coproducts, where the latter are required to be van Kampen colimits
[CLW, Rezk]. The effective model structure is defined so that when
$\mathcal{E}=\mathsf{Set}$, we recover the Kan–Quillen model structure on
simplicial sets [Quillen]. We also prove several results on the effective
model structure and its associated $\infty$-category, which we discuss below.

The initial motivation for this work was the desire to establish whether our
earlier work on the constructive Kan–Quillen model structure [H, Gambino-
Sattler, GSS, Sattler] could be developed further so as to obtain a new model
structure on categories of simplicial sheaves. Indeed, in [H, GSS] we worked
with simplicial sets without using the law of excluded middle and the axiom of
choice, thus opening the possibility of replacing them with simplicial objects
in a Grothendieck topos. As we explored this idea, we realised that the
resulting argument admitted not only a clean presentation in terms of enriched
weak factorisation systems [RiehlE:catht]*Chapter 13, but also a vast
generalisation.

In fact, the existence of the effective model structure may be a surprise to
some readers, since assuming $\mathcal{E}$ to be countably lextensive is
significantly weaker than assuming it to be a Grothendieck topos and covers
many more examples (such as the category of countable sets and the category of
schemes). In particular, our arguments do not require the existence of all
small colimits, (local) Cartesian closure and local presentability, which are
ubiquitous in the known constructions of model structures.

One reason for the interest in the effective model structure is that, when
$\mathcal{E}$ is a Grothendieck topos, the effective model structure on
$\mathsf{s}\mathcal{E}$ differs from the known model structures on simplicial
sheaves and provides the first example of a peculiar combination of higher
categorical structure. Indeed, the associated $\infty$-category has finite
limits, colimits that satisfy descent and is locally Cartesian closed, but is
neither a higher Grothendieck topos [Lurie] nor a higher elementary topos in
the sense of [Shulman-ncatcafe, Rasekh], since its 0-truncation does not
always have a subobject classifier (see Example 11.8). In this case, the
effective model structure satisfies most of the axioms for a model topos
[Rezk], but is not combinatorial. One key point here is that the effective
model structure is not cofibrantly generated in the usual sense, but only in
an enriched sense. The relation between the effective model structure and
other model structures on categories of simplicial objects is discussed
further in Remark 9.10.

This situation can be understood by analogy with the theory of exact
completions in ordinary category theory [Carboni-Vitale]. There, it is known
that the exact completion of a (Grothendieck) topos need not be a
(Grothendieck) topos [Menni]. Indeed, we believe that the effective model
structure will provide a starting point for the development of a homotopical
counterpart of the theory of exact completions. As a first step in this
direction, we prove that the $\infty$-category associated to the effective
model structure on $\mathsf{s}\mathcal{E}$ is the full subcategory of the
$\infty$-category of presheaves on $\mathcal{E}$ spanned by Kan complexes in
$\mathcal{E}$, mirroring a corresponding description of the exact completion
of $\mathcal{E}$ in [Hu-Tholen]. We also make a conjecture (Conjecture 13.2)
on the relation between the effective model structure and $\infty$-categorical
exact completions, which we leave for future work. In the long term, we hope
that our work could be useful for the definition of a higher categorical
version of the effective topos [Hyland], which can be described as an exact
completion [Carboni].

Finally, our results may be of interest also in Homotopy Type Theory, since
they help to clarify how the simplicial model of Univalent Foundations
[Kapulkin-Lumsdaine], in which types are interpreted as Kan complexes, is
related to the setoid model of type theory [Hofmann], in which types are
interpreted as types equipped with an equivalence relation, by showing how not
only the latter [Emmenegger-Palmgren] but also the former is related to the
theory of exact completions. Furthermore, we expect that the effective model
structure may lead to new models of Homotopy Type Theory, another topic that
we leave for future research.

Main results. In order to outline our main results, let us briefly describe
the effective model structure, whose fibrant objects are to be thought of as
Kan complexes, or $\infty$-groupoids, in $\mathcal{E}$. In order to describe
the fibrations of the effective model structure, recall that, for
$E\in\mathcal{E}$, we have a functor

$\operatorname{Hom}_{\mathsf{sSet}}(E,-)\colon\mathsf{s}\mathcal{E}\to\mathsf{sSet}$
($\ast$)

sending $X\in\mathsf{s}\mathcal{E}$ to the simplicial set defined by
$\operatorname{Hom}_{\mathsf{sSet}}(E,X)_{n}=\operatorname{Hom}(E,X_{n})$, for
$[n]\in\Delta$. We can then define a map in $\mathsf{s}\mathcal{E}$ to be a
fibration in $\mathsf{s}\mathcal{E}$ if its image under the functor in
($\ast$) is a Kan fibration in $\mathsf{sSet}$ for every $E\in\mathcal{E}$.
Trivial fibrations are defined analogously. Our main results are the
following:

  * •

Theorem 9.9, asserting the existence of the effective model structure, whose
fibrations and trivial fibrations are defined as above;

  * •

Proposition 10.4 and Corollary 12.18, asserting that the effective model
structure is right and left proper, respectively, and Proposition 10.1,
showing that homotopy colimits in $\mathsf{s}\mathcal{E}$ satisfy descent;

  * •

Theorem 10.3 asserting that the $\infty$-category
$\operatorname{Ho}_{\infty}(\mathsf{s}\mathcal{E})$ associated to the
effective model structure has finite limits and $\alpha$-small colimits
satisfying descent when $\mathcal{E}$ is $\alpha$-lextensive, and Theorem
10.5, showing that $\operatorname{Ho}_{\infty}(\mathsf{s}\mathcal{E})$ is also
locally Cartesian closed when $\mathcal{E}$ is so.

  * •

Theorem 13.1, characterising the $\infty$-category associated to the effective
model structure.

Along the way, we prove several other results of independent interest. For
example, we characterise completely the cofibrations of the effective model
structure, which do not coincide with all monomorphisms (Theorem 4.6) and we
compare the effective model structure with model structures studied in
relation to Elmendorf’s Theorem (Theorem 11.7).

Novel aspects. This paper differs significantly from our work in [H, GSS,
Sattler] in both scope and technical aspects. Regarding scope, apart from
generalising the existence of the model structure from the case
$\mathcal{E}=\mathsf{Set}$ to that of a general countably lextensive category
$\mathcal{E}$, here we discuss a number of topics that are not even mentioned
for the case $\mathcal{E}=\mathsf{Set}$ in our earlier work, such as the
structure and characterisation of the $\infty$-category associated to the
effective model structure, the discussion of descent and the connections with
Elmendorf’s theorem.

Regarding the technical aspects, even if the general strategy for proving the
existence of the effective model structure is inspired from the case
$\mathcal{E}=\mathsf{Set}$ in [GSS], several new ideas are necessary to
implement it to the general case, as we explain below. This strategy involves
in three steps. First, we introduce the notions of a (trivial) fibration in
$\mathsf{s}\mathcal{E}$ as above and establish the existence of a fibration
category structure on the category of Kan complexes (assuming in fact only
that $\mathcal{E}$ has finite limits). Secondly, we construct the two weak
factorisation systems of the model structure, one given by cofibrations and
trivial fibrations and one given by trivial cofibrations and fibrations.
Thirdly, we show that weak equivalences (as determined by the two weak
factorisation systems) satisfy 2-out-of-3 by proving the so-called Equivalence
Extension Property (Proposition 8.3).

In order to realise this plan, we prove several results that are not necessary
for $\mathcal{E}=\mathsf{Set}$. We mention only the key ones. First, we
develop a new version of the enriched small object argument (Theorem 3.14),
which does not require existence of all colimits. In order to achieve it, we
analyse the colimits required for our applications and prove that they exist
in a countably lextensive category, exploiting crucially that some of the maps
involved are complemented monomorphisms. Secondly, we show that the fibration
category structure, where fibrations are defined as above, agrees with the
weak factorisation systems, defined in terms of enriched lifting properties
(Proposition 4.1). Thirdly, we obtain a characterisation of cofibrations in
categories of simplicial objects (Theorem 4.6), which requires a new, purely
categorical, argument that is entirely different to the one used in [H, GSS,
Sattler]. Finally, new ideas are required in the proof of the Equivalence
Extension Property (Proposition 8.3). For this, we need to construct
explicitly dependent products (i.e., pushforward) functors along cofibrations
(Theorem 6.5), which are not guaranteed to exist since $\mathcal{E}$ is not
assumed to be locally Cartesian closed. The existence of these pushforward
functors may be considered as a pleasant surprise since they are essential for
our argument and no exponentials are assumed to be present in $\mathcal{E}$.

The existence of the effective model structure is independent from that of the
constructive Kan–Quillen model structure on simplicial sets [GSS, H].
Actually, the use of enriched category theory here, especially for expressing
stronger versions of the lifting properties usually phrased in terms of mere
existence of diagonal fillers, makes explicit some of the informal conventions
adopted in [GSS, H] when treating the case $\mathcal{E}=\mathsf{Set}$. Also,
the proofs in [GSS, H] make use of structure on $\mathsf{Set}$ that is not
available in a countably lextensive category and therefore cannot be
interpreted as taking place in the so-called internal logic of $\mathcal{E}$
[Elephant]*Section D1.3. Even when $\mathcal{E}$ is a Grothendieck topos,
carrying out the proofs in the internal language of $\mathcal{E}$ [maclane-
moerdijk]*Chapter 6 would not make explicit the structure under consideration,
thus making it more difficult for the results to be accessible and applicable.

Outline of the paper. The paper is organised in four parts. The first,
including only Section 1, establishes the fibration category structure. The
second, including Sections 2, 3 and 4, introduces the two weak factorisation
systems, having first developed an appropriate version of the small object
argument. The third, including Sections 5, 6, 7, 8 and 9, establishes the
existence of the effective model structure, by constructing pushforward
functors and establishing the Frobenius and Equivalence Extension Property.
The fourth, including Sections 10, 11, 12 and 13, proves the key properties of
the effective model structure, namely descent and properness, their
$\infty$-categorical counterparts, and characterises its associated
$\infty$-category. Throughout the paper, we omit the proofs that can be
carried out with minor modifications from [H, GSS], but include the ones that
require new ideas.

Remark. The material in this paper is developed within ZFC set theory. Some of
the material, however, can be developed also in a constructive setting (see
footnotes and Appendix A for details).

Acknowledgements. We are grateful to André Joyal for questions and
discussions, which in particular led to Theorem 13.1, and to Dan Christensen
and Denis-Charles Cisinski for comments on an earlier version of the paper.
Nicola Gambino and Karol Szumiło gratefully acknowledge that this material is
based upon work supported by the US Air Force Office for Scientific Research
under awards number FA8655-13-1-3038 and FA9550-21-1-0007. Nicola Gambino was
also supported by EPSRC under grant EP/V002325/1. Simon Henry was partially
supported by an NSERC Discovery Grant. Christian Sattler was supported by
Swedish Research Council grant 2019-03765.

##  1 Kan fibrations

This section develops some simplicial homotopy theory in a category
$\mathcal{E}$ with finite limits. The category of simplicial objects in
$\mathcal{E}$ is defined by letting

$\mathsf{s}\mathcal{E}=_{\operatorname{def}}[\Delta^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]\,.$

In Definition 1.3 we introduce the notion of a fibration in
$\mathsf{s}\mathcal{E}$ with which we shall work throughout the paper. This
notion is defined using the enrichment of $\mathsf{s}\mathcal{E}$ in
$\mathsf{sSet}$ and generalises that of a Kan fibration in $\mathsf{sSet}$.
The main result of this section, Theorem 1.7, establishes a structure of a
fibration category on the category of fibrant objects in
$\mathsf{s}\mathcal{E}$. For applications throughout the paper, we also
establish a fiberwise version of this fibration category in Theorem 1.9. We
also introduce the notion of a _pointwise weak equivalence_ (Definition 1.6),
which provides the weak equivalences of these fibration categories. In the
subsequent sections we will extend these results to obtain the effective model
structure on $\mathsf{s}\mathcal{E}$, under the stronger assumption that
$\mathcal{E}$ is countably lextensive. The weak equivalences of the effective
model structure will not be the pointwise weak equivalences in general,
although the two notions will coincide for maps between fibrant objects.

Let us recall how the category $\mathsf{s}\mathcal{E}$ is enriched over
$\mathsf{sSet}$ with respect to the Cartesian monoidal structure. For a finite
simplicial set $K$ and $X\in\mathsf{s}\mathcal{E}$, we define
$K\mathop{\pitchfork}X\in\mathsf{s}\mathcal{E}$ via the end formula

$(K\mathop{\pitchfork}X)_{m}\mathbin{=_{\mathrm{def}}}\textstyle\int_{[n]\in\Delta}X_{n}^{(K\times\mathord{\Delta}[m])_{n}}\text{.}$
(1.1)

For $X,Y\in\mathsf{s}\mathcal{E}$, the simplicial hom-object is then defined
by letting111Here and in the following we use subscripts to indicate to which
category the hom-objects under consideration belong.

$\operatorname{Hom}_{\mathsf{sSet}}(X,Y)_{m}\mathbin{=_{\mathrm{def}}}\operatorname{Hom}_{\mathsf{Set}}(X,\mathord{\Delta}[m]\mathop{\pitchfork}Y)\text{.}$
(1.2)

This makes $\mathsf{s}\mathcal{E}$ into a $\mathsf{sSet}$-enriched category so
that the formula in (1.1) gives the cotensor (over finite simplicial sets)
with respect to the enrichment. Without further assumptions on $\mathcal{E}$,
$\mathsf{s}\mathcal{E}$ does not admit all cotensors or tensors over
simplicial sets. We often identify an object $E\in\mathcal{E}$ with the
constant simplicial object with value $E$. For example, for $E\in\mathcal{E}$
and $Y\in\mathsf{s}\mathcal{E}$ we write
$\operatorname{Hom}_{\mathsf{sSet}}(E,Y)$. Note that

$\displaystyle\operatorname{Hom}_{\mathsf{sSet}}(E,Y)_{m}$
$\displaystyle=\operatorname{Hom}_{\mathsf{Set}}(E,Y_{m})\text{,}$
$\displaystyle\operatorname{Hom}_{\mathsf{sSet}}(E,K\mathop{\pitchfork}Y)$
$\displaystyle=K\mathop{\pitchfork}\operatorname{Hom}_{\mathsf{sSet}}(E,Y)\text{.}$

The $\mathsf{sSet}$-enrichment allows us to define a notion of a homotopy
between morphisms of $\mathsf{s}\mathcal{E}$. Given maps $f_{0},f_{1}\colon
X\to Y$ in $\mathsf{s}\mathcal{E}$ (or one of its slice categories), a
_homotopy_ $H$ from $f_{0}$ to $f_{1}$, written $H\colon f_{0}\sim f_{1}$, is
a map

$H\colon X\to\mathord{\Delta}[1]\mathop{\pitchfork}Y$ (1.3)

that restricts to $f_{0}$ on $\\{0\\}\to\mathord{\Delta}[1]$ and to $f_{1}$ on
$\\{1\\}\to\mathord{\Delta}[1]$. It is _constant_ if it factors through the
canonical map
$\mathord{\Delta}[0]\mathop{\pitchfork}Y\to\mathord{\Delta}[1]\mathop{\pitchfork}Y$,
in which case $f_{0}=f_{1}$. Note that we can regard $H$ as a map
$\mathord{\Delta}[1]\to\operatorname{Hom}_{\mathsf{sSet}}(X,Y)$. This
generalises the usual notion of homotopy in simplicial sets. For each
$E\in\mathcal{E}$, the functor
$\operatorname{Hom}_{\mathsf{sSet}}(E,\mathord{\relbar})$ preserves homotopies
because it preserves the cotensor with $\mathord{\Delta}[1]$.

We need some definitions to introduce the notions of a Kan fibration and
trivial Kan fibration in $\mathsf{s}\mathcal{E}$. For a finite simplicial set
$K$, we define the _evaluation functor_
$\operatorname{ev}_{K}\colon\mathsf{s}\mathcal{E}\to\mathcal{E}$ via the end
formula

$\operatorname{ev}_{K}(X)=X(K)\mathbin{=_{\mathrm{def}}}\textstyle\int_{[n]\in\Delta}X_{n}^{K_{n}}\text{.}$
(1.4)

We will usually write $X(K)$ rather than $\operatorname{ev}_{K}(X)$ for
brevity. However, in some situations the notation $\operatorname{ev}_{K}(X)$
will be more convenient, see the definition of pullback evaluation below. The
end above exists since, by the finiteness of $K$, it can be constructed from
finite limits. For example, $X(\mathord{\Delta}[n])=X_{n}$ and
$X(\mathord{\Lambda}^{k}[2])=X_{1}\times_{X_{0}}X_{1}$. Also note that
$X(K)=(K\mathop{\pitchfork}X)_{0}$ and
$X(K\times\mathord{\Delta}[m])=(K\mathop{\pitchfork}X)_{m}$.

######  Remark 1.1.

There are two alternative ways of viewing the evaluation functor. First, since
$\mathcal{E}$ has finite limits, we can consider $X(K)$ as the value on $K$ of
the right Kan extension of
$X\colon\Delta^{\mathord{\mathrm{missing}}{op}}\to\mathcal{E}$ along the
inclusion of $\Delta$ into the category of finite simplicial sets. Secondly,
seeing $\mathcal{E}$ as a $\mathsf{Set}$-enriched category, we can view $X(K)$
as a weighted limit, namely the limit of $X$, viewed as a diagram in
$\mathcal{E}$, weighted by $K$, viewed as a diagram in $\mathsf{Set}$. Both of
these observations show that $X(K)$ is contravariantly functorial in $K$.

We write $\widehat{\operatorname{ev}}$ for the _pullback evaluation_ functor,
which is the result of applying the so-called Leibniz construction [Riehl-
Verity] to the two-variable functor $\operatorname{ev}$, i.e., the functor
sending a map $i\colon A\to B$ between finite simplicial sets and a morphism
$f\colon X\to Y$ of $\mathsf{s}\mathcal{E}$ to

$\displaystyle\widehat{\operatorname{ev}}_{i}(f)\colon\operatorname{ev}_{A}(X)\to\operatorname{ev}_{B}(X)\times_{\operatorname{ev}_{B}(Y)}\operatorname{ev}_{A}(Y)\text{
in }\mathcal{E}$ (1.5) $\displaystyle\text{also written as
}\widehat{\operatorname{ev}}_{i}(f)\colon X(A)\to
X(B)\times_{Y(B)}Y(A)\text{.}$

######  Remark 1.2.

We adopt the convention of prefixing with ‘pullback’ (or ‘pushout’) the name
of a two-variable functor to indicate the result of applying the Leibniz
construction to it. So for example, we shall say pushout product for what is
also referred to as Leibniz product or corner product.

We use standard notation for the sets of boundary inclusions and horn
inclusions,

$I_{\mathsf{sSet}}=\\{\mathord{\partial\mathord{\Delta}[n]}\to\mathord{\Delta}[n]\mid
n\geq 0\\}\text{ and
}J_{\mathsf{sSet}}=\\{\mathord{\Lambda}^{k}[n]\to\mathord{\Delta}[n]\mid n\geq
k\geq 0,n>0\\}\text{.}$ (1.6)

######  Definition 1.3.

We say that a morphism in $\mathsf{s}\mathcal{E}$ is

  * •

a _trivial Kan fibration_ if its pullback evaluations with all maps in
$I_{\mathsf{sSet}}$ are split epimorphisms;

  * •

a _Kan fibration_ if its pullback evaluations with all maps in
$J_{\mathsf{sSet}}$ are split epimorphisms.

Explicitly, a map $f\colon X\to Y$ in $\mathsf{s}\mathcal{E}$ is a Kan
fibration if the morphism

$X(\mathord{\Delta}[n])\to
X(\mathord{\Lambda}^{k}[n])\times_{Y(\mathord{\Lambda}^{k}[n])}Y(\mathord{\Delta}[n])$

in $\mathcal{E}$ has a section, for all $n\geq k\geq 0$ and $n>0$. For $Y=1$,
this means that the morphism

$X(\mathord{\Delta}[n])\to X(\mathord{\Lambda}^{k}[n])$

has a section, for all $n\geq k\geq 0$ and $n>0$, in which case we say that
$X$ is a _Kan complex_. Note that for $\mathcal{E}=\mathsf{Set}$, these
definitions reduce to the standard notions of a Kan fibration, trivial Kan
fibration and a Kan complex in simplicial sets. In the following, we shall
frequently write _fibration_ , _trivial fibration_ and _fibrant object_ , as
we do not consider other notions of fibrations.

Although we have not yet introduced cofibrations and trivial cofibrations in
$\mathsf{s}\mathcal{E}$, we can use the standard classes of cofibrations and
trivial cofibrations in $\mathsf{sSet}$, which are the saturations of the
generating sets $I_{\mathsf{sSet}}$ and $J_{\mathsf{sSet}}$, respectively.

The next proposition characterises fibrations and trivial fibrations by
reducing them to the corresponding notions in $\mathsf{sSet}$ in terms of the
$\mathsf{sSet}$-enrichment of $\mathsf{s}\mathcal{E}$, defined in (1.2).

######  Proposition 1.4.

Let $f\colon X\to Y$ be a map in $\mathsf{s}\mathcal{E}$. Then $f$ is a
(trivial) fibration if and only if, for all $E\in\mathcal{E}$, the map

$\operatorname{Hom}_{\mathsf{sSet}}(E,f)\colon\operatorname{Hom}_{\mathsf{sSet}}(E,X)\to\operatorname{Hom}_{\mathsf{sSet}}(E,Y)$

is a (trivial) fibration in $\mathsf{sSet}$.

###### Proof.

Note that the functors
$X(\mathord{\relbar})\colon\mathsf{sSet}^{\mathord{\mathrm{missing}}{op}}\to\mathcal{E}$
and
$\operatorname{Hom}_{\mathsf{sSet}}(\mathord{\relbar},X)\colon\mathcal{E}^{\mathord{\mathrm{missing}}{op}}\to\mathsf{sSet}$
are contravariantly adjoint. Thus for all maps $i\colon A\to B$ between finite
simplicial sets there is a bijective correspondence between the lifting
problems

${A}$${\operatorname{Hom}_{\mathsf{sSet}}(E,X)}$${X(B)}$${B}$${\operatorname{Hom}_{\mathsf{sSet}}(E,Y)}$${E}$${X(A)\times_{Y(A)}Y(B)}$$\widehat{\operatorname{ev}}_{i}(f)$

the latter of which is equivalent to the morphism on the right being a split
epimorphism (by setting $E=X(A)\times_{Y(A)}Y(B)$). ∎

If $i\colon A\to B$ is a map of finite simplicial sets and $p\colon X\to Y$ is
a morphism of $\mathsf{s}\mathcal{E}$, then we define the _pullback cotensor_
of $i$ and $p$ (cf. Remark 1.2) as the induced morphism

$i\mathbin{\widehat{\mathbin{\pitchfork}}}p\colon
B\mathbin{\pitchfork}X\to(A\mathbin{\pitchfork}X)\times_{A\mathbin{\pitchfork}Y}(B\mathbin{\pitchfork}X)\text{.}$

######  Lemma 1.5.

  1. (i)

The pullback cotensor in $\mathsf{s}\mathcal{E}$ of a cofibration between
finite simplicial sets and a fibration is a fibration. If the given
cofibration or fibration is trivial, then the result is a trivial fibration.

  2. (ii)

Fibrations and trivial fibrations in $\mathsf{s}\mathcal{E}$ are closed under
composition, pullback, and retract.

  3. (iii)

Let $f\colon X\to Y$ and $g\colon Y\to Z$ be morphisms of
$\mathsf{s}\mathcal{E}$. If $f\colon X\to Y$ and $gf\colon X\to Z$ are trivial
fibrations, then so is $g\colon Y\to Z$.

###### Proof.

All the statements are proved in the same way: they hold for simplicial sets
(see, e.g., [Quillen]*Theorem II.3.3) and transfer to $\mathsf{s}\mathcal{E}$
using Proposition 1.4.222Constructively, part (i) is true in $\mathsf{sSet}$
by [GSS]*Corollary 1.3.4, part (ii) is evident and part (iii) is [GSS]*Lemma
1.3.6. Note that transferring (i) from $\mathsf{sSet}$ to
$\mathsf{s}\mathcal{E}$ relies on the fact that
$\operatorname{Hom}_{\mathsf{sSet}}(E,\mathord{\relbar})$ preserves pullbacks
and cotensors and hence pullback cotensors. ∎

######  Definition 1.6.

Let $f\colon X\to Y$ in $\mathsf{s}\mathcal{E}$. We say that $f$ is a
_pointwise weak equivalence_ if

$\displaystyle\operatorname{Hom}_{\mathsf{sSet}}(E,f)\colon\operatorname{Hom}_{\mathsf{sSet}}(E,X)\to\operatorname{Hom}_{\mathsf{sSet}}(E,Y)$

is a weak equivalence in $\mathsf{sSet}$ for all $E\in\mathcal{E}$.

For the next theorem, we use the definition of a fibration category as stated
in [GSS]*Section 1.6.

######  Theorem 1.7.

Let $\mathcal{E}$ be category with finite limits. Then pointwise weak
equivalences, Kan fibrations and trivial Kan fibrations equip the category of
Kan complexes in $\mathsf{s}\mathcal{E}$ with the structure of a fibration
category.

###### Proof.

Trivial fibrations are exactly the fibrations that are weak equivalences
because this holds in $\mathsf{sSet}$. We need to verify the following axioms.

  1. (F0)

$\mathsf{s}\mathcal{E}$ has a terminal object and all objects are fibrant,
which follows directly from the definitions.

  2. (F0)

Pullbacks along fibrations exist because $\mathcal{E}$ (and hence
$\mathsf{s}\mathcal{E}$) has all finite limits. Moreover, fibrations and
acyclic fibrations are closed under pullback by point (ii) of Lemma 1.5.

  3. (F0)

Every morphism factors as a weak equivalence followed by a fibration. By
[Brown]*p. 421, Factorization lemma it suffices to construct a path object,
i.e., a factorisation of the diagonal $X\to X\times X$. Such factorisation is
given by the cotensor $X\to\mathord{\Delta}[1]\mathop{\pitchfork}X\to X\times
X$. Applying $\operatorname{Hom}_{\mathsf{sSet}}(E,\mathord{\relbar})$ to this
factorisation gives

$\operatorname{Hom}_{\mathsf{sSet}}(E,X)\to\mathord{\Delta}[1]\mathop{\pitchfork}\operatorname{Hom}_{\mathsf{sSet}}(E,X)\to\operatorname{Hom}_{\mathsf{sSet}}(E,X)\times\operatorname{Hom}_{\mathsf{sSet}}(E,X)$

which is a well known factorisation of the diagonal of
$\operatorname{Hom}_{\mathsf{sSet}}(E,X)$ into a weak equivalence followed by
a fibration in $\mathsf{sSet}$ (since
$\operatorname{Hom}_{\mathsf{sSet}}(E,X)$ is a Kan complex by Proposition
1.4). See, e.g., [Goerss-Jardine]*p. 43. Hence
$X\to\mathord{\Delta}[1]\mathop{\pitchfork}X\to X\times X$ is also such
factorisation in $\mathsf{s}\mathcal{E}$.

  4. (F0)

Weak equivalences satisfy 2-out-of-6, which follows since this property holds
in $\mathsf{sSet}$. ∎

In view of our development in Section 8, we generalise Theorem 1.7 to the case
of a slice of $\mathsf{s}\mathcal{E}$ over a simplicial object $X$, which we
write $\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$. We then define
$\mathsf{s}\mathcal{E}\mathbin{\twoheaddownarrow}X$ to be the full subcategory
of $\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ spanned by the fibrations over
$X$.

First of all, let us recall that the enrichment of $\mathsf{s}\mathcal{E}$ in
simplicial sets, including the cotensor with finite simplicial sets, descends
to its slices. For $(A,f),(B,g)\in\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$,
the hom-object $\operatorname{Hom}_{\mathsf{sSet}}((A,f),(B,g))$ is the
pullback of $\operatorname{Hom}_{\mathsf{sSet}}(A,B)$ along the map $f\colon
1\to\operatorname{Hom}_{\mathsf{sSet}}(A,X)$. The cotensor of
$(A,f)\in\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ by a finite simplicial
set $K$ is the pullback of $K\mathop{\pitchfork}A$ along the map $X\to
K\mathop{\pitchfork}X$ (using the fact that the monoidal unit in
$\mathsf{sSet}$ is the terminal object). As before, for each $E$, the functor
$\operatorname{Hom}_{\mathsf{sSet}}(E,\mathord{\relbar})\colon\mathsf{s}\mathcal{E}\mathbin{\downarrow}X\to\mathsf{sSet}\mathbin{\downarrow}\operatorname{Hom}_{\mathsf{sSet}}(E,X)$
preserves these cotensors.

######  Lemma 1.8.

Let $X\in\mathsf{s}\mathcal{E}$. The pullback cotensor properties in part (i)
of Lemma 1.5 hold in $\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ as well.

###### Proof.

This follows from their validity in $\mathsf{s}\mathcal{E}$, i.e., part (i) of
Lemma 1.5 and the stability of fibrations and trivial fibration under
pullback, i.e., part (ii) of Lemma 1.5. ∎

######  Theorem 1.9.

Let $X\in\mathsf{s}\mathcal{E}$. Then pointwise weak equivalences, fibrations
and trivial fibrations equip the category
$\mathsf{s}\mathcal{E}\mathbin{\twoheaddownarrow}X$ with the structure of a
fibration category.

###### Proof.

All axioms are verified by the same argument as in the proof of Theorem 1.7.
For (F3), we use Lemma 1.8 which is a fiberwise version of part (i) of Lemma
1.5 used in the proof of Theorem 1.7. ∎

We conclude this section with a basic observation on homotopy equivalences.

######  Proposition 1.10.

Homotopy equivalences in $\mathsf{s}\mathcal{E}$ (and in particular, in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ for all
$X\in\mathsf{s}\mathcal{E}$) are pointwise weak equivalences.

###### Proof.

The functors $\operatorname{Hom}_{\mathsf{sSet}}(E,\mathord{\relbar})$
preserve homotopies and hence also homotopy equivalences. Thus the conclusion
follows from the fact that homotopy equivalences are weak equivalences in
$\mathsf{sSet}$. ∎

##  2 Lextensive categories and complemented inclusions

This section, Section 3 and Section 4 constitute the second part of the paper,
whose ultimate goal is to construct two weak factorisation systems on
$\mathsf{s}\mathcal{E}$, whose right classes of maps are the fibrations and
trivial fibrations of Section 1, assuming that $\mathsf{s}\mathcal{E}$ is a
countably lextensive category. This section recalls some basic facts about
lextensive categories. Throughout it, we consider a fixed category with finite
limits $\mathcal{E}$ and study diagrams in $\mathcal{E}$ indexed by a category
$D$. When convenient, we will regard cones under such diagrams as diagrams
over the category $D^{\rhd}$, obtained by adding a new terminal object $\star$
to $D$. We start by recalling the general notion of van Kampen colimit [Lurie,
Rezk] in our setting.

######  Definition 2.1.

Let $Y_{\bullet}\colon D\to\mathcal{E}$ be a diagram and assume
$Y_{\star}=\operatorname*{colim}_{d\in D}Y_{d}$ is its colimit in
$\mathcal{E}$. We say that $Y_{\star}$ is

  1. (i)

_universal_ , if it is preserved by pullbacks, i.e., if for every map
$X_{\star}\to Y_{\star}$, $X_{\star}$ is the colimit of the induced diagram
$X_{d}=X_{\star}\operatorname*{\times}_{Y_{\star}}Y_{d}$.

  2. (ii)

_effective_ , if given a Cartesian natural transformation $X\to Y$, the
diagram $X$ has a colimit $X_{\star}$, and all the squares

${X_{d}}$${X_{\star}}$${Y_{d}}$${Y_{\star}}$

are pullback squares, i.e., the extended natural transformation over
$D^{\rhd}$ is also Cartesian.

  3. (iii)

_van Kampen_ , if it is both universal and effective.

######  Lemma 2.2.

A colimit $Y_{\star}=\operatorname*{colim}_{d\in D}Y_{d}$ in $\mathcal{E}$ is
van Kampen if and only if it is preserved by the pseudo-functor
$\mathcal{E}^{\mathord{\mathrm{missing}}{op}}\to\mathsf{Cat}$ sending each
$X\in\mathcal{E}$ to the slice category $\mathcal{E}\mathbin{\downarrow}X$
(with morphisms acting by pullbacks). In other words, the slice category
$\mathcal{E}\mathbin{\downarrow}Y_{\star}$ is the pseudo-limit $\lim_{d\in
D}(\mathcal{E}\mathbin{\downarrow}Y_{d})$.

###### Proof.

Pullback along the structure morphisms of $Y_{\star}$ induces a functor
$P\colon\mathcal{E}\mathbin{\downarrow}Y_{\star}\to\lim_{d}(\mathcal{E}\mathbin{\downarrow}Y_{d})$.
We need to show that this functor is an equivalence if and only if the colimit
of $Y_{\bullet}$ is a van Kampen colimit.

An object of $\lim_{d}(\mathcal{E}\mathbin{\downarrow}Y_{d})$ can be
identified with a Cartesian transformation $X\to Y$. If colimits of diagrams
Cartesian over $Y_{\bullet}$ exist, then taking the colimit yields a left
adjoint to the functor above:

$\operatorname*{colim}\colon\lim_{d}(\mathcal{E}\mathbin{\downarrow}Y_{d})\leftrightarrows\mathcal{E}\mathbin{\downarrow}Y_{\star}\colon
P\text{.}$

Conversely, we claim that if $P$ has a left adjoint, then the left adjoint
computes the colimits of diagrams that are Cartesian over $Y_{\bullet}$.
Indeed, assume that the pullback functor
$P\colon\mathcal{E}\mathbin{\downarrow}Y_{\star}\to\lim_{d}(\mathcal{E}\mathbin{\downarrow}Y_{d})$
has a left adjoint $X_{\bullet}\mapsto X_{\star}$, and let $Z$ be an arbitrary
object of $\mathcal{E}$. A map $X_{\star}\to Z$ in $\mathcal{E}$ is the same
as a map $X_{\star}\to Z\times Y_{\star}$ in
$\mathcal{E}\mathbin{\downarrow}Y_{\star}$, which by the adjunction formula is
the same as a natural transformation $X_{d}\to Z\times Y_{d}$ over
$Y_{\bullet}$, but this is exactly the same as a natural transformation
$X_{d}\to Z$ in $\mathcal{E}$, and hence this shows that $X_{\star}$ is the
colimit of $X_{d}$.

Now, $Y_{\star}$ is universal if and only if the counit of this adjunction is
an isomorphism and it is effective if and only if the unit is an isomorphism.
Hence, the colimit $Y_{\star}$ of $Y_{\bullet}$ is van Kampen if and only if
the pullback functor described above has a left adjoint such that the unit and
counit of the adjunction are isomorphisms, i.e., if and only if it is an
equivalence. ∎

For example, an initial object $0$ is always vacuously effective and it is
universal if and only if it is _strict_ , i.e., if there is a morphism $X\to
0$, then $X$ is initial itself. Instead, a coproduct
$Y_{\star}=\coprod_{d}Y_{d}$ is van Kampen if and only if it is universal and
_disjoint_ , i.e., $Y_{d}\times_{Y_{\star}}Y_{d^{\prime}}$ is initial for
$d\neq d^{\prime}$. This can be seen inspecting the proof of [CLW]*Proposition
2.14.

######  Lemma 2.3.

Let $D$ be a small category. Let $Y_{\bullet}\colon C\to\mathcal{E}^{D}$ be a
diagram such that $Y_{\bullet}(d)$ admits a van Kampen colimit in
$\mathcal{E}$ for all $d\in D$. Then $Y_{\bullet}$ has a van Kampen colimit in
$\mathcal{E}^{D}$.

###### Proof.

If each $d\in D$, $\operatorname*{colim}_{c\in C}Y_{c}(d)$ exists in
$\mathcal{E}$, then it is functorial in $d$ and it is a colimit in
$\mathcal{E}^{D}$. In particular, an object over
$\operatorname*{colim}_{c}Y_{c}$ is a $D$-indexed diagram
$X(d)\to\operatorname*{colim}_{C}Y_{c}(d)$, which as these colimits are all
van Kampen is the same as a $(C\times D)$-indexed diagram $X_{c}(d)\to
Y_{c}(d)$ which is Cartesian in the $C$-direction, which in turn is the same
as a $C$-indexed diagram $X_{\bullet}\in\mathcal{E}^{D}$ which is Cartesian
over $Y_{\bullet}$, hence proving the lemma. ∎

We now recall the definition of various kinds of lextensive categories [CLW].

######  Definition 2.4.

Let $\mathcal{E}$ be a category with finite limits. For a regular cardinal
$\alpha$, we say that $\mathcal{E}$ is _$\alpha$ -lextensive_ if
$\alpha$-coproducts exist and are van Kampen colimits. Furthermore, we say
that $\mathcal{E}$ is

  1. (i)

_lextensive_ if it is $\omega$-lextensive, i.e., finite coproducts exist and
are van Kampen colimits,

  2. (ii)

_countably lextensive_ if it is $\omega_{1}$-lextensive, i.e., countable
coproducts exist and are van Kampen colimits,

  3. (iii)

_completely lextensive_ if it is $\alpha$-lextensive for all $\alpha$, i.e.,
all small coproducts exist and are van Kampen colimits.

######  Example 2.5.

There are numerous examples of lextensive categories.

  1. (i)

Any presheaf category is completely lextensive. In particular, for any group
$G$ the category of $G$-sets is countably lextensive.

  2. (ii)

More generally, any Grothendieck topos is completely lextensive. In fact,
Giraud’s theorem characterises Grothendieck toposes as the locally presentable
categories in which coproducts and (in an appropriate sense) quotients by
equivalence relations are van Kampen colimits.

  3. (iii)

The category of topological spaces is completely lextensive. The same is true
for many of its subcategories such as categories of Hausdorff spaces,
compactly generated spaces, weakly Hausdorff compactly generated spaces, etc.

  4. (iv)

The category of affine schemes is lextensive, the category of schemes is
completely lextensive.

  5. (v)

The category of countable sets is countably lextensive.

  6. (vi)

A category with finite limits $\mathcal{E}$ has the free coproduct completion
which can be constructed as the category
$\operatorname{\mathsf{Fam}}\mathcal{E}$ of families of objects in
$\mathcal{E}$. Explicitly, an object is pair $(S,(X_{s})_{s\in S})$ where $S$
is a set and $(X_{s})_{s\in S}$ is an $S$-indexed family of objects of
$\mathcal{E}$. A morphism
$(S,(X_{s}))\to(S^{\prime},(X^{\prime}_{s^{\prime}}))$ consists of a function
$f\colon S\to S^{\prime}$ and morphisms $X_{s}\to X^{\prime}_{f(s)}$ for all
$s\in S$. $\operatorname{\mathsf{Fam}}\mathcal{E}$ is completely lextensive.
The $\alpha$-coproduct completion,
$\operatorname{\mathsf{Fam}}_{\alpha}\mathcal{E}$, obtained by restricting to
$\alpha$-small families, is an $\alpha$-lextensive category.

For $S\in\mathsf{Set}$ and $X\in\mathcal{E}$, we write $S\mathbin{\cdot}X$ for
the tensor of $X$ with $S$, when it exists. If $\mathcal{E}$ has countable
coproducts, then this tensor exists for countable $S$ and can be defined as

$S\mathbin{\cdot}X=\coprod_{s\in S}X\text{.}$ (2.1)

The global sections functor $\mathcal{E}(1,-)\colon\mathcal{E}\to\mathsf{Set}$
has a partial left adjoint, defined by mapping a countable set $S$ to

$\underline{S}\mathbin{=_{\mathrm{def}}}S\mathbin{\cdot}1=\coprod_{s\in
S}1\,.$ (2.2)

We extend this notation to diagram categories in a levelwise fashion: if
$\mathcal{E}$ has countable coproducts and $D$ a small category, then the
levelwise global sections functor $\mathcal{E}^{D}\to\mathsf{Set}^{D}$ has a
partial left adjoint, sending a levelwise countable diagram
$K\in\mathsf{Set}^{D}$ to $\underline{K}\in\mathcal{E}^{D}$, which is defined
by levelwise application of $S\mapsto\underline{S}$. These functors will be
used frequently in the paper. For example, we will use them in Section 4 to
transfer the sets of boundary inclusions and horn inclusions in (1.6) from
$\mathsf{sSet}$ to $\mathsf{s}\mathcal{E}$, so as to obtain generating sets
for weak factorisation systems in $\mathsf{s}\mathcal{E}$. We establish some
of their basic properties in the next lemmas.

######  Lemma 2.6.

If $\mathcal{E}$ is countably lextensive, then for every countable set $S$ and
$X\in\mathcal{E}$, we have $\underline{S}\times
X\mathrel{\cong}S\mathbin{\cdot}X$, naturally in $S$.

###### Proof.

Since $\mathcal{E}$ is countably lextensive, it is countably distributive.
Thus, product with $X$ preserves countable coproducts, in particular tensors
with countable sets. This reduces the claim to the natural isomorphism
$1\times X\mathrel{\cong}X$. ∎

The next lemma will be used, sometimes implicitly, in Section 4.

######  Lemma 2.7.

If $\mathcal{E}$ is countably lextensive, then the functor
$S\mapsto\underline{S}$ from countable sets to $\mathcal{E}$ preserves finite
limits.

###### Proof.

The functor $S\mapsto\underline{S}$ preserves terminal objects by definition.
It also preserves pullbacks. Indeed, every pullback diagram of (countable)
sets decomposes as a (countable) coproduct of product diagrams. These products
are preserved since products preserve countable coproducts in each variable by
lextensivity. ∎

The next lemma will be applied in Section 6.

######  Lemma 2.8.

Let $\mathcal{E}$ be an $\alpha$-lextensive category. If $D$ is a small
category and $S\colon D\to\mathsf{Set}$ is a functor which takes values in
$\alpha$-small sets, then there is an equivalence of categories

$\mathcal{E}^{D}\mathbin{\downarrow}\underline{S}\,\simeq\,\mathcal{E}^{D\mathbin{\downarrow}S}$

where $D\mathbin{\downarrow}S$ denotes the category of elements of $S$.

###### Proof.

The proof is similar to that of Lemma 2.2. There is a functor
$\mathcal{E}^{D\mathbin{\downarrow}S}\to\mathcal{E}^{D}\mathbin{\downarrow}\underline{S}$
which sends a functor $F\colon D\mathbin{\downarrow}S\to\mathcal{E}$ to the
functor $V\colon D\to\mathcal{E}$ defined by:

$V(d)=\coprod_{s\in S(d)}F(d,s)\text{.}$

It comes with an obvious map to $\underline{S}$, which was defined as
$\underline{S}(d)=\coprod_{s\in S(d)}1$. This functor has a right adjoint
$\mathcal{E}^{D}\mathbin{\downarrow}\underline{S}\to\mathcal{E}^{D\mathbin{\downarrow}S}$
sending a functor $V\colon D\to\mathcal{E}$ with a natural transformation
$V\to\underline{S}$ to the functor $F\colon
D\mathbin{\downarrow}S\to\mathcal{E}$ where $F(d,s)$ is defined as the
following pullback:

${F(d,s)}$${V(d)}$${1}$${\underline{S}(d)\text{.}}$

These two adjoints functor are equivalences. Indeed, the counit of this
adjunction is an isomorphism by universality of coproducts and the unit is an
isomorphism by effectivity of coproducts. ∎

We now turn our attention to the class of complemented inclusions. These will
be useful for construction of certain colimits whose existence is not
immediately obvious in lextensive categories and, especially, in their diagram
categories. First of all, recall that a morphism $i\colon A\to B$ in
$\mathcal{E}$ is a _complemented inclusion_ if it has a _complement_ , i.e., a
morphism $j\colon C\to B$ such that $i$ and $j$ exhibit $B$ as a coproduct of
$A$ and $C$ in $\mathcal{E}$. In other words, $i$ is isomorphic to the
coproduct inclusion $A\to A\sqcup C$. We will often say simply that $C$ is a
complement of $A$. The notation $A\rightarrowtail B$ will be sometimes used to
indicate complemented inclusions. Note that complemented inclusions are
sometimes (e.g., in our previous work [GSS, H]) called _decidable inclusions_
in reference to the notion of decidability in constructive logic.

######  Lemma 2.9.

  1. (i)

If $\mathcal{E}$ is lextensive, then the pushout of a complemented inclusion
along any morphism exists and is again a complemented inclusion. Moreover,
such pushouts are preserved by functors (and pseudo-functors) that preserve
finite coproducts and thus are van Kampen colimits.

  2. (ii)

If $\mathcal{E}$ is countably lextensive, then the colimit of a sequence of
complemented inclusions exists and is again a complemented inclusion.
Moreover, such colimits are preserved by functors (and pseudo-functors) that
preserve countable coproducts and thus are van Kampen colimits.

###### Proof.

If $i\colon A\to B$ is a complemented inclusion with complement $C$, then the
pushout of $i$ along $A\to D$ is $C\sqcup D$. Similarly, if $i_{k}\colon
A_{k}\to A_{k+1}$ are complemented inclusions with complements $C_{k+1}$, then
$\operatorname*{colim}_{k}A_{k}$ is $\coprod_{k}C_{k}$ (where $C_{0}=A_{0}$).
The claims on preservation by functors then follow immediately.

These presentations of colimits as coproducts remain when we consider
$\mathcal{E}$ as a bicategory. Recall from Lemma 2.2 that a colimit is van
Kampen exactly if it is preserved by a certain pseudo-functor. Since (finite
or countable) coproducts are assumed van Kampen, so are the presented
colimits. ∎

######  Lemma 2.10.

Assume $\mathcal{E}$ is lextensive.

  1. (i)

complemented subobjects in $\mathcal{E}$ are closed under finite unions.

  2. (ii)

complemented inclusions in $\mathcal{E}$ are closed under finite limits, i.e.,
if $X\to Y$ is a natural transformation between finite diagrams in
$\mathcal{E}$ that is a levelwise complemented inclusion, then so is the
induced morphism $\lim X\to\lim Y$.

###### Proof.

The proof of [GSS]*Lemma 1.1.4 applies verbatim. ∎

######  Lemma 2.11.

Assume that $\mathcal{E}$ is countably lextensive. Then the full subcategory
of $[\omega,\mathcal{E}]$ consisting of sequences of complemented inclusions
has finite limits which are preserved by the colimit functor (sending each
sequence to its colimit in $\mathcal{E}$).

###### Proof.

First note that the category of sequences of complemented inclusions has
finite limits by part (ii) of Lemma 2.10. Moreover, part (ii) of Lemma 2.9
implies that colimits of such sequences exist. It suffices to show that this
colimit functor preserves terminal objects and pullbacks. Terminal objects are
preserved since $\omega$ is a connected category (it has an initial object).
For the case of pullbacks, we consider a span $A\to C\leftarrow B$ of
sequences of complemented inclusions. We need to show that the map

$\operatorname*{colim}_{k\in\omega}A_{k}\times_{C_{k}}B_{k}\to\operatorname*{colim}A\times_{\operatorname*{colim}C}\operatorname*{colim}B$

is invertible. We decompose this map into three factors:

${\operatorname*{colim}_{k\in\omega}A_{k}\times_{C_{k}}B_{k}}$${\operatorname*{colim}A\times_{\operatorname*{colim}C}\operatorname*{colim}B\hbox
to0.0pt{\text{.}\hss}}$${\operatorname*{colim}_{k\in\omega}A_{k}\times_{\operatorname*{colim}C}B_{k}}$${\operatorname*{colim}_{i,j\in\omega}A_{i}\times_{\operatorname*{colim}C}B_{j}}$

The left map is invertible even before taking colimits because
$C_{k}\to\operatorname*{colim}C$ is a monomorphism. The bottom map is
invertible because the diagonal functor $\omega\to\omega\times\omega$ is final
(it has a left adjoint). The right map is invertible by universality of the
van Kampen colimits $\operatorname*{colim}A$ and $\operatorname*{colim}B$
(part (ii) of Lemma 2.9). ∎

Let $D$ be a small category. We say that a morphism $\varphi\colon
F\rightarrow G$ in $\mathcal{E}^{D}$, is a levelwise complemented inclusion if
its components $\varphi_{d}\colon F_{d}\to G_{d}$, for $d\in D$, are
complemented inclusions in $\mathcal{E}$. Note that this is considerably less
restrictive than asking for $\varphi$ to be a complemented inclusion in
$\mathcal{E}^{D}$.

######  Corollary 2.12.

Let $D$ be a small category.

  1. (i)

If $\mathcal{E}$ is lextensive, then pushouts along levelwise complemented
inclusions exist, are computed levelwise and are van Kampen colimits in
$\mathcal{E}^{D}$.

  2. (ii)

If $\mathcal{E}$ is countably lextensive, then colimits of sequences of
levelwise complemented inclusions exist, are computed levelwise and are van
Kampen colimits in $\mathcal{E}^{D}$.

###### Proof.

This follows immediately from Lemmas 2.3 and 2.9. ∎

######  Lemma 2.13.

Let $D$ be a small category. If $\mathcal{E}$ is lextensive, then the pushout
products of levelwise complemented inclusions in $\mathcal{E}^{D}$ with
arbitrary morphisms exist. Moreover, the pushouts involved are van Kampen.

###### Proof.

By universality of coproducts, levelwise complemented inclusions are closed
under pullbacks. Thus a pushout computing a pushout product with a levelwise
complemented inclusion is a pushout along a levelwise complemented inclusion.
They are van Kampen by Corollary 2.12. ∎

The following statement will be needed in Section 4 to prove Lemma 4.5.

######  Lemma 2.14.

Let $\mathcal{C}$ be a category, $P$ a poset with binary meets,
$X\in\mathcal{C}$ an object and

$A=(A_{p}\hookrightarrow X\ |\ p\in P)$

a diagram of subobjects of $X$ closed under intersection, i.e., such that
$A_{p}\cap A_{q}=A_{p\cap q}$. Then if $A$ has a van Kampen colimit, the
colimit is also a subobject of $X$.

###### Proof.

We assume that $\operatorname*{colim}_{p\in P}A_{p}$ exists and is a van
Kampen colimit, and we show that the diagonal map $\operatorname*{colim}_{p\in
P}A_{p}\to F=\left(\operatorname*{colim}_{p\in
P}A_{p}\right)\times_{X}\left(\operatorname*{colim}_{p\in P}A_{p}\right)$ is
an isomorphism. First, we form pullbacks:

${A_{p}\cap
A_{q}}$${F_{q}}$${A_{q}}$${F_{p}}$${F}$${\operatorname*{colim}_{q}A_{q}}$${A_{p}}$${\operatorname*{colim}_{p}A_{p}}$${X}$

Using that the colimits are van Kampen, we have that
$F=\operatorname*{colim}_{p}F_{p}$ and
$F_{p}=\operatorname*{colim}_{q}A_{q}\cap A_{p}$ and hence
$F=\operatorname*{colim}_{p,q}A_{p}\cap A_{q}$ with the two maps
$F\to\operatorname*{colim}_{p}A_{p}$ being induced by the maps $A_{p}\cap
A_{q}\to A_{p}$ and $A_{p}\cap A_{q}\to A_{q}$. We conclude by observing that
$\operatorname*{colim}_{p}\left(A_{p}\cap A_{q}\right)=A_{q}$. Indeed the map
$P\to(\downarrow q)$ that send $p\in P$ to $p\cap q$ is right adjoint to the
inclusion of $(\downarrow q)$ to $P$, so it is a final functor. It hence
follows that

$\operatorname*{colim}_{p\in P}A_{p\cap q}=\operatorname*{colim}_{p\leqslant
q}A_{p}=A_{q}$

So this implies that $F=\operatorname*{colim}_{q}A_{q}$, with the projection
map $F\to\operatorname*{colim}_{q}A_{q}$ being the identity, hence proving
that $\operatorname*{colim}_{q}A_{q}\to X$ is a monomorphism. ∎

We prove a statement relating van Kampen colimits and the pullback evaluation
$\widehat{\operatorname{ev}}$ functor, defined in (1.5). This statement will
be needed in Section 8.

######  Lemma 2.15.

Let $D$ be a small category. Let $Y\colon
C\to[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$ be a diagram with
levelwise van Kampen colimit $\operatorname*{colim}Y$. Let $p\colon X\to Y$ be
a Cartesian transformation, which we regard as a $C$-indexed diagram of arrows
in $[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$.

Let $q\colon A\to B$ be a map in
$[D^{\mathord{\mathrm{missing}}{op}},\mathsf{Set}]$ with $B$ representable
such that $[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$ supports
evaluation at $A$. Then $\widehat{\operatorname{ev}}_{q}$ (valued in arrows of
$\mathcal{E}$) preserves the colimit of $p$, the resulting colimit is computed
separately on source and target, and all maps of the colimit cocone are
pullback squares.

###### Proof.

First note that by levelwise effectivity of $\operatorname*{colim}Y$, we
obtain $\operatorname*{colim}X$ (and hence $\operatorname*{colim}p$). The
square $p_{c}\to\operatorname*{colim}p$ is a pullback for all $c\in C$.

Consider the functor $F$ sending an arrow $M\to N$ in
$[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$ to the sequence of arrows

${M(B)}$${M(A)\times_{N(A)}N(B)}$${N(B)\hbox to0.0pt{\text{.}\hss}}$

The first arrow is the pullback evaluation at $q$ of $M\to N$. Evaluation
preserves limits, in particular pullbacks. By pullback pasting, the action of
$F$ on a map of arrows that is a pullback is a pasting of pullback squares.

Let us inspect the action of $F$ on the colimit cocone of $p$. It will suffice
to show that it results in objectwise colimit cocones. Since the maps of the
colimit cocone of $p$ are pullback squares, we obtain pastings of pullback
squares upon applying $F$. Recall that $\operatorname{ev}_{B}$ is computed by
evaluation at the object representing $B$. So by assumption,
$(\operatorname*{colim}Y)(B)=\operatorname{ev}_{B}(\operatorname*{colim}Y)$ is
colimit of $\operatorname{ev}_{B}\circ Y$ and van Kampen. The claim follows by
universality of this van Kampen colimit. ∎

##  3 An enriched small object argument

The goal of this section is to develop a version of the small object argument
that allows us to construct weak factorisation systems on the category of
simplicial objects $\mathsf{s}\mathcal{E}$, where $\mathcal{E}$ is a countably
lextensive category. In view of our application to both simplicial objects in
Section 4 and semisimplicial objects in Section 12, we develop our small
object argument for diagram categories $\mathcal{E}^{D}$ in general.
Importantly, our weak factorisation systems are enriched, in the sense of
[RiehlE:catht]. We will be constructing
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched weak factorisation systems
on $\mathcal{E}^{D}$, where where $\operatorname{\mathsf{Psh}}\mathcal{E}$
denotes the category of presheaves over $\mathcal{E}$. This is because the
category of diagrams $\mathcal{E}^{D}$ is not necessarily
$\mathcal{E}$-enriched, but it is
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched, as we now recall.

For $E\in\mathcal{E}$ and $X\in\mathcal{E}^{D}$, we define $E\times
X\in\mathcal{E}^{D}$ by letting

$(E\times X)_{d}\mathbin{=_{\mathrm{def}}}E\times X_{d}\,.$ (3.1)

Given $X,Y\in\mathcal{E}^{D}$, we then define the hom-object
$\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(X,Y)\in\operatorname{\mathsf{Psh}}\mathcal{E}$
by letting:

$\begin{array}[]{rccl}\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(X,Y)\colon&\mathcal{E}^{{\mathord{\mathrm{missing}}{op}}}&\to&\mathsf{Set}\\\
&E&\mapsto&\operatorname{Hom}_{\mathsf{Set}}(E\times X,Y)\end{array}$

This makes $\mathcal{E}^{D}$ into a
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched category, so that the
formula in (3.1) provides the tensor of
$E\in\operatorname{\mathsf{Psh}}{\mathcal{E}}$ and $X\in\mathcal{E}^{D}$ with
respect to this enrichment. When the presheaf is representable, the
representing object is denoted by $\operatorname{Hom}_{\mathcal{E}}(X,Y)$.

Using the enrichment, we can define an internal version of the familiar
lifting problems involved in the definition of a weak factorisation systems.
For morphisms $i\colon A\to B$ and $p\colon X\to Y$ in $\mathcal{E}^{D}$, we
define the _presheaf of lifting problems_ of $i$ against $p$ by letting

$\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)\mathbin{=_{\mathrm{def}}}\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(A,X)\times_{\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(A,Y)}\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(B,Y)\,.$

When the relevant hom-objects are representable, then so is
$\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)$. In this
case, we write $\operatorname{Prob}_{\mathcal{E}}(i,p)$ for its representing
object and call it the _object of lifting problems_ of $i$ against $p$. Note
that the induced pullback hom of $i$ and $p$ (cf. Remark 1.2) has the form

$\widehat{\operatorname{Hom}}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)\colon\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(B,X)\to\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)$
(3.2)

Again, if the objects are representables, we have also an induced pullback hom
in $\mathcal{E}$, which has the form

$\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,p)\colon\operatorname{Hom}_{\mathcal{E}}(B,X)\to\operatorname{Prob}_{\mathcal{E}}(i,p)\,.$
(3.3)

We are ready to define the $\operatorname{\mathsf{Psh}}{\mathcal{E}}$-enriched
counterparts of the standard lifting properties.

######  Definition 3.1.

Let $i\colon A\to B$ and $p\colon X\to Y$ be morphisms of $\mathcal{E}^{D}$.

  * •

We say that $i$ has the _$\operatorname{\mathsf{Psh}}\mathcal{E}$ -enriched
left lifting property_ with respect to $p$ and that $p$ has the
_$\operatorname{\mathsf{Psh}}\mathcal{E}$ -enriched right lifting property_
with respect to $i$ if the induced pullback hom in (3.2) is a split
epimorphism in $\operatorname{\mathsf{Psh}}\mathcal{E}$.

  * •

We say that $i$ has the _$\mathcal{E}$ -enriched left lifting property_ with
respect to $p$ and that $p$ has the _$\mathcal{E}$ -enriched right lifting
property_ with respect to $i$ if the induced pullback hom in (3.3) exists and
is a split epimorphism in $\mathcal{E}$.

Since the Yoneda embedding is fully faithful and preserves pullbacks, as soon
as all relevant $\mathcal{E}$-valued hom-objects exist, the
_$\operatorname{\mathsf{Psh}}\mathcal{E}$ -enriched left lifting property_ and
_$\mathcal{E}$ -enriched left lifting property_ are equivalent and
$\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)$ is
represented by $\operatorname{Prob}_{\mathcal{E}}(i,p)$.

In both $\operatorname{\mathsf{Psh}}\mathcal{E}$ and $\mathcal{E}$, the class
of split epimorphisms is the right class of a weak factorization system, with
left class given by complemented inclusions. As such, it enjoys a number of
standard closure properties. Our notions of enriched lifting property are
defined from this class via the pullback hom. Because of this, the classes of
maps defined below by an $\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched
lifting property will inherit corresponding closure properties. For example,
split epimorphisms are closed under retracts. Thus, classes of maps defined by
an $\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched lifting property are
closed under retracts.

As is usual, we extend the terminology of enriched lifting properties from
maps to classes of maps on either side by universal quantification.

######  Definition 3.2.

Let $I=\\{i\colon A_{i}\to B_{i}\\}$ be a set of morphisms of
$\mathcal{E}^{D}$.

  * •

An _(enriched) $I$-fibration_ is a morphism with the enriched right lifting
property with respect to $I$.

  * •

An _(enriched) $I$-cofibration_ is a morphism with the enriched left lifting
property with respect to $I$-fibrations.

When the left map of a $\operatorname{\mathsf{Psh}}{\mathcal{E}}$-enriched
lifting problem comes from $\mathsf{Set}^{D}$ via levelwise application of the
operation in (2.2), we may simplify the lifting problem (assuming some
technical conditions hold). Indeed, the pullback hom (3.3) reduces to a
pullback evaluation. We record this in the next couple of statements, which
are phrased using $D^{{\mathord{\mathrm{missing}}{op}}}$ instead of $D$ in
order to exploit the language of representable functors. We make use of the
evaluation functor
$\operatorname{ev}_{K}\colon[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]\to\mathcal{E}$
defined for finite colimits $K$ of representables by letting:

$\operatorname{ev}_{K}(X)=\textstyle\int_{d\in
D^{{\mathord{\mathrm{missing}}{op}}}}X_{d}^{K_{d}}.$

This generalises the evaluation functor defined in Eq. 1.4, which is the case
$D=\Delta$. As in Remark 1.1, we may equivalently view
$\operatorname{ev}_{K}(X)$ as the $K$-weighted limit of $X$, which implies
that $\operatorname{ev}$ is a (partial) two-variable functor.

######  Lemma 3.3.

Let $K\in[D^{\mathord{\mathrm{missing}}{op}},\mathsf{Set}]$ be levelwise
countable.

  1. (i)

There is an isomorphism
$(E\times\underline{K})_{d}\mathrel{\cong}K_{d}\mathbin{\cdot}E$ natural in
$K$, $E\in\mathcal{E}$, and $d\in D$.

  2. (ii)

Assume that $K$ is a finite colimit of representables. Then the hom-presheaf
$\operatorname{Hom}_{\operatorname{\mathsf{Psh}}{\mathcal{E}}}(\underline{K},X)$
is representable for $X\in[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$
and we have an isomorphism
$\operatorname{Hom}_{\mathcal{E}}(\underline{K},X)\mathrel{\cong}\operatorname{ev}_{K}(X)$,
natural in $K$ and $X\in[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$.

###### Proof.

Part (i) follows from Lemma 2.6. For part (ii), part (i) implies that
$\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(\underline{K},X)$
is naturally isomorphic to the $\mathcal{E}$-presheaf
$E\mapsto\operatorname{Hom}_{\mathsf{Set}}(d\mapsto K_{d}\mathbin{\cdot}E,X)$.
A representing object for it is by definition the $K$-weighted limit of $X$,
i.e., $\operatorname{ev}_{K}(X)$. This exists in our setting for $K$ a finite
colimit of representables. ∎

######  Proposition 3.4.

Let $i\colon A\to B$ be a map in
$[D^{\mathord{\mathrm{missing}}{op}},\mathsf{Set}]$ between objects that are
levelwise countable and finite colimits of representables and let $p\colon
X\to Y$ be a map in $[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$. Then
the following are equivalent:

  1. (i)

$\underline{i}\colon\underline{A}\to\underline{B}$ has the
$\mathcal{E}$-enriched left lifting property with respect to $p$,

  2. (ii)

the pullback evaluation $\widehat{\operatorname{ev}}_{i}(p)$ is a split
epimorphism in $\mathcal{E}$.

###### Proof.

This is an immediate consequence of part (ii) of Lemma 3.3. ∎

Proposition 3.4 will be used in Section 4 to relate (trivial) Kan fibrations
in $\mathsf{s}\mathcal{E}$ in the sense of Definition 1.3 with fibrations in
the sense of Definition 3.2 with respect to the images in
$\mathsf{s}\mathcal{E}$ of horn inclusions (boundary inclusions, respectively)
under the operation
$\underline{(\mathord{\relbar})}\colon\mathsf{Set}\to\mathcal{E}$.

We now turn our attention to $\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched
weak factorisation systems.

######  Definition 3.5.

A _$\operatorname{\mathsf{Psh}}\mathcal{E}$ -enriched weak factorisation
system_ on $\mathcal{E}^{D}$ is a pair $(\mathscr{L},\mathscr{R})$ of classes
of morphisms of $\mathcal{E}^{D}$ such that:

  * •

a morphism belongs to $\mathscr{L}$ if and only if it has the
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched left lifting property with
respect to $\mathscr{R}$;

  * •

a morphism belongs to $\mathscr{R}$ if and only if it has the
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched right lifting property with
respect to $\mathscr{L}$;

  * •

every morphism of $\mathcal{E}^{D}$ factors as an $\mathscr{L}$-morphism
followed by an $\mathscr{R}$-morphism.

The classes $\mathscr{L}$ and $\mathscr{R}$ in the above definition are closed
under retract as they are characterized by
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched lifting properties.

We will abbreviate “$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched lifting
property” to “enriched lifting property”, but we will be explicit about cases
where it coincides with the $\mathcal{E}$-enriched lifting property.

######  Lemma 3.6.

Let $(\mathscr{L},\mathscr{R})$ be an enriched weak factorisation system.

  1. (i)

A morphism is in $\mathscr{L}$ if and only if it has the ordinary left lifting
property with respect to $\mathscr{R}$.

  2. (ii)

A morphism is in $\mathscr{R}$ if and only if it has the ordinary right
lifting property with respect to $\mathscr{L}$.

In particular, $(\mathscr{L},\mathscr{R})$ is also an ordinary weak
factorisation system.

###### Proof.

For (i), a morphism of $\mathscr{L}$ has the ordinary left lifting property
with respect to $\mathscr{R}$ by evaluating the hom-presheaves at
$1\in\mathcal{E}$. Conversely, a morphism with the ordinary lifting property
admits a lift against the second factor of its
$(\mathscr{L},\mathscr{R})$-factorisation, thus making it into a retract of
the first factor (cf. also the proof of Proposition 3.17). The conclusion
follows since $\mathscr{L}$ is closed under retracts. Part (ii) follows by
duality. ∎

We will fix a set $I$ and study a version of the small object argument that
produces an enriched weak factorisation system of $I$-cofibrations and
$I$-fibrations under suitable assumptions.

######  Definition 3.7.

Let $i\colon A\to B$ and $p\colon X\to Y$ be morphisms of $\mathcal{E}^{D}$.
Assume that we have a factorisation

${X}$${Y\text{.}}$${X^{\prime}}$$p$$p^{\prime}$

We say that $p$ satisfies the _$X^{\prime}$ -partial enriched right lifting
property_ with respect to $i$ if there is a lift in the diagram

${\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(B,X^{\prime})}$${\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p)}$${\operatorname{Prob}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(i,p^{\prime})\text{.}}$

Such partial lifting properties are a crucial ingredient of the small object
argument, but they are only tractable when $i$ is a levelwise complemented
inclusion. This is thanks to the next two lemmas, where we use the tensor
defined in (3.1).

######  Lemma 3.8.

Levelwise complemented inclusions in $\mathcal{E}^{D}$ are closed under:

  1. (i)

$E\times\mathord{\relbar}$ for all $E\in\mathcal{E}$;

  2. (ii)

countable coproducts;

  3. (iii)

pushouts along arbitrary morphisms;

  4. (iv)

sequential colimits;

  5. (v)

retracts.

Moreover, the colimits of parts (ii), (iii) and (iv) are preserved by
$E\times\mathord{\relbar}$ for all $E\in\mathcal{E}$.

###### Proof.

The functor $E\times\mathord{\relbar}$ and all the colimits mentioned are
computed levelwise in $\mathcal{E}$, so the results boil down to the fact that
complemented inclusions in $\mathcal{E}$ are stable under all these
constructions. Stability under $E\times\mathord{\relbar}$ follows from
distributivity of product over coproduct in complemented categories: if $A\to
A\sqcup B$ is a complemented inclusion, then its image under
$E\times\mathord{\relbar}$ is $E\times A\to(E\times A)\sqcup(E\times B)$ and
is a complemented inclusion. The case of a countable coproduct is also clear:
if $A_{k}\to A_{k}\sqcup B_{k}$ is a family of complemented inclusions, then
their coproduct can be written as $\coprod A_{k}\to\left(\coprod
A_{k}\right)\sqcup\left(\coprod B_{k}\right)$. Stability under pushout and
sequential composition follows from Lemma 2.9. The fact that they are
preserved by $E\times\mathord{\relbar}$ follows from Lemma 2.9. The case of
retracts can be deduced from the stability under limits proved in Lemma 2.10
as retracts can be seen as limits. ∎

######  Lemma 3.9.

Let $p\colon X\to Y$ be a map in $\mathcal{E}^{D}$ and $\mathscr{L}$ a class
of levelwise complemented inclusions in $\mathcal{E}^{D}$ that have the
enriched left lifting property with respect to $p$. Then $\mathscr{L}$ is
closed under the following operations:

  1. (i)

tensors by objects of $\mathcal{E}$,

  2. (ii)

countable coproducts,

  3. (iii)

pushouts,

  4. (iv)

colimits of sequences,

  5. (v)

retracts.

###### Proof.

For $X\in\mathcal{E}^{D}$, the functor
$\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(\mathord{\relbar},X)$
is not necessarily an adjoint. However, since split epimorphisms are closed
under limits dual to the colimits listed above, it is sufficient to verify
that it carries these colimits to limits. (In the case of tensors this means
that $\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(F\times
A,X)\mathrel{\cong}\operatorname{Hom}_{\operatorname{\mathsf{Psh}}\mathcal{E}}(A,X)^{\mathcal{E}(\mathord{\relbar},F)}$
for all $F\in\mathcal{E}$.) This follows directly from these colimits being
preserved by the tensors as recorded in Lemma 3.8. ∎

######  Definition 3.10.

Let $A\in\mathcal{E}^{D}$. We say that $A$ is _finite_ if the following hold:

  1. (i)

$\operatorname{Hom}_{\mathcal{E}}(A,X)$ exists for every
$X\in\mathcal{E}^{D}$;

  2. (ii)

$\operatorname{Hom}_{\mathcal{E}}(A,\mathord{\relbar})$ preserves colimits of
sequences of levelwise complemented inclusions;

  3. (iii)

$\operatorname{Hom}_{\mathcal{E}}(A,\mathord{\relbar})$ sends levelwise
complemented inclusions to complemented inclusions.

The next lemma provides a supply of finite objects. For its statement, recall
the functor $S\mapsto\underline{S}$ from Section 2. As Lemma 3.3, it is
formulated using $D^{{\mathord{\mathrm{missing}}{op}}}$ instead of $D$ for
convenience.

######  Lemma 3.11.

Let $D$ be a locally countable category and assume that presheaf
$A\in\operatorname{\mathsf{Psh}}D$ is a finite colimit of representables. Then
$\underline{A}\in[D^{\mathord{\mathrm{missing}}{op}},\mathcal{E}]$ is finite.

###### Proof.

First, note that since $D$ is locally countable, $A$ is levelwise countable
and thus $\underline{A}$ exists. By part (ii) of Lemma 3.3,
$\operatorname{Hom}_{\mathcal{E}}(\underline{A},\mathord{\relbar})$ exists and
is given by $\operatorname{ev}_{A}$ (evaluation at $A$). Call
$X\in\operatorname{\mathsf{Psh}}D$ _$\mathcal{E}$ -finite_ if it satisfies the
conditions of Definition 3.10 with
$\operatorname{Hom}_{\mathcal{E}}(X,\mathord{\relbar})$ replaced by
$\operatorname{ev}_{X}$. Our goal then is to show that $A$ is
$\mathcal{E}$-finite. This follows from the following observations:

  * •

Representables are $\mathcal{E}$-finite. For this, recall that evaluation at a
representable is given by evaluation at the representing object. Part (ii)
uses part (ii) of Corollary 2.12 to see that the colimit is computed
levelwise.

  * •

$\mathcal{E}$-finite presheaves are closed under finite colimits. For this, we
use that the partial two-variable functor $\operatorname{ev}$ sends colimits
in its first argument to limits. Part (i) holds since $\mathcal{E}$ has finite
limits. Part (ii) holds since finite limits preserve colimits of sequences of
complemented inclusions in $\mathcal{E}$ (Lemma 2.11). Part (iii) holds since
complemented inclusions in $\mathcal{E}$ are closed under finite limits (part
(ii) of Lemma 2.10). ∎

The hypothesis of finiteness is used in the next result, where we use the
notion of an $I$-fibration in the sense of Definition 3.2.

######  Lemma 3.12.

Assume that the domains and codomains of morphisms of $I$ are finite. Let
$Y\in\mathcal{E}^{D}$ and $(X_{k}\to X_{k+1}\ |\ k\in\mathbb{N})$ be a
sequence of morphisms in $\mathcal{E}^{D}\mathbin{\downarrow}Y$. If every
$X_{k}\to X_{k+1}$ is a levelwise complemented inclusion and each $p_{k}\colon
X_{k}\to Y$ has $X_{k+1}$-partial enriched right lifting property with respect
to $I$, then $\operatorname*{colim}_{k}X_{k}\to Y$ is an $I$-fibration.

###### Proof.

Fix a morphism $i\colon A\to B$ of $I$. Since $A$ and $B$ are finite, the
given partial enriched lifting properties are $\mathcal{E}$-enriched.
Moreover, since $X_{k}\to X_{k+1}$ is a levelwise complemented inclusion,
Lemma 2.10 implies that
$\operatorname{Prob}_{\mathcal{E}}(i,p_{k})\to\operatorname{Prob}_{\mathcal{E}}(i,p_{k+1})$
is a complemented inclusion.

Proceeding by induction with respect to $k$, we can pick lifts

${\operatorname{Hom}_{\mathcal{E}}(B,X_{k+1})}$${\operatorname{Prob}_{\mathcal{E}}(i,p_{k})}$${\operatorname{Prob}_{\mathcal{E}}(i,p_{k+1})}$

that are natural in $k$. Indeed, since
$\operatorname{Prob}_{\mathcal{E}}(i,p_{k-1})\to\operatorname{Prob}_{\mathcal{E}}(i,p_{k})$
is a complemented inclusion, we can construct a compatible lift by assembling
a previously constructed lift on
$\operatorname{Prob}_{\mathcal{E}}(i,p_{k-1})$ with a given lift on its
complement. Since $A$ and $B$ are finite, we have

$\displaystyle\operatorname*{colim}_{k}\operatorname{Hom}_{\mathcal{E}}(B,X_{k})$
$\displaystyle=\operatorname{Hom}_{\mathcal{E}}(B,\operatorname*{colim}_{k}X_{k})$
and
$\displaystyle\operatorname*{colim}_{k}\operatorname{Prob}_{\mathcal{E}}(i,p_{k})$
$\displaystyle=\operatorname*{colim}_{k}\left(\operatorname{Hom}_{\mathcal{E}}(A,X_{k})\times_{\operatorname{Hom}_{\mathcal{E}}(A,Y)}\operatorname{Hom}_{\mathcal{E}}(B,Y)\right)$
$\displaystyle=\left(\operatorname*{colim}_{k}\operatorname{Hom}_{\mathcal{E}}(A,X_{k})\right)\times_{\operatorname{Hom}_{\mathcal{E}}(A,Y)}\operatorname{Hom}_{\mathcal{E}}(B,Y)$
$\displaystyle=\operatorname{Hom}_{\mathcal{E}}(A,\operatorname*{colim}_{k}X_{k})\times_{\operatorname{Hom}_{\mathcal{E}}(A,Y)}\operatorname{Hom}_{\mathcal{E}}(B,Y)$
$\displaystyle=\operatorname{Prob}_{\mathcal{E}}(i,\operatorname*{colim}_{k}p_{k})\text{,}$

the latter by universality of sequential colimits of complemented inclusions
in $\mathcal{E}$ (Lemma 2.9). Thus we obtain a diagram

${\operatorname{Hom}_{\mathcal{E}}(B_{i},\operatorname*{colim}_{k}X_{k})}$${\operatorname{Prob}_{\mathcal{E}}(i,\operatorname*{colim}_{k}p_{k})}$${\operatorname{Prob}_{\mathcal{E}}(i,\operatorname*{colim}_{k}p_{k})\text{.}}$

where the bottom map is an identity, i.e., these lifts form a section that
exhibits $\operatorname*{colim}_{k}X_{k}\to~{}Y$ as an $I$-fibration. ∎

The following lemma isolates a simpler version of the inductive step in the
construction of lifts in Lemma 3.12. It is needed in Section 8.

######  Lemma 3.13.

Let

${X}$${Y}$${A}$${B\hbox to0.0pt{\text{.}\hss}}$$p$$q$

be a pullback square in $\mathcal{E}^{D}$ with $A\to B$ a levelwise
complemented inclusion. Let $i\colon U\to V$ be a map in $\mathcal{E}^{D}$
between finite objects such that
$\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,p)$ and
$\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,q)$ have sections. Then, for any
section $s$ of $\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,p)$, there is a
section $t$ of $\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,q)$ such that the
diagram

${\operatorname{Hom}_{\mathcal{E}}(V,X)}$${\operatorname{Hom}_{\mathcal{E}}(V,Y)}$${\operatorname{Prob}_{\mathcal{E}}(i,p)}$${\operatorname{Prob}_{\mathcal{E}}(i,q)\hbox
to0.0pt{\text{.}\hss}}$$\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,p)$$\widehat{\operatorname{Hom}}_{\mathcal{E}}(i,p)$

forms a morphism of retracts.

###### Proof.

The map
$\operatorname{Prob}_{\mathcal{E}}(i,p)\to\operatorname{Prob}_{\mathcal{E}}(i,q)$
is a complemented inclusion by Lemma 2.10. We construct $t$ by using $s$ on
$\operatorname{Prob}_{\mathcal{E}}(i,p)$ and a given section on its
complement. ∎

######  Theorem 3.14 (Enriched small object argument).

Let $I=(i\colon A_{i}\to B_{i}\ |\ i\in I)$ be a countable set of levelwise
complemented inclusions between finite objects of $\mathcal{E}^{D}$. Then
$I$-cofibrations and $I$-fibrations form an enriched weak factorisation system
in $\mathcal{E}^{D}$.

###### Proof.

For a morphism $p_{0}\colon X_{0}\to Y$ we form a sequence $X_{0}\to X_{1}\to
X_{2}\to\ldots$ in $\mathcal{E}\mathbin{\downarrow}Y$ by iteratively taking
pushouts

${\coprod_{i\in I}\operatorname{Prob}_{\mathcal{E}}(i,p_{k})\times
A_{i}}$${X_{k}}$${\coprod_{i\in
I}\operatorname{Prob}_{\mathcal{E}}(i,p_{k})\times
B_{i}}$${X_{k+1}}$${Y\text{.}}$$p_{k}$$p_{k+1}$

The adjoint transpose of $\operatorname{Prob}(i,p_{k})\times B_{i}\to X_{k+1}$
witnesses the $X_{k+1}$-partial enriched right lifting property of $p_{k}$
with respect to $i$. Moreover, by Lemma 3.8, $X_{k}\to X_{k+1}$ is a levelwise
complemented inclusion. Thus Lemma 3.12 applies and shows that
$\operatorname*{colim}_{k}X_{k}\to Y$ is an $I$-fibration. Using Lemma 3.9, we
show that $X_{0}\to\operatorname*{colim}_{k}X_{k}$ is an $I$-cofibration. ∎

######  Remark 3.15.

Essentially the the same argument used to prove Theorem 3.14 can be used to
prove a more general statement. Namely, instead of $\mathcal{E}^{D}$ we
consider an _$\mathcal{E}$ -module_ $\mathcal{C}$, i.e., a category equipped
with a _tensor_ functor
$\mathord{\relbar}\times\mathord{\Relbar}\colon\mathcal{E}\times\mathcal{C}\to\mathcal{C}$
that is associative in the sense that the functor
$\mathcal{E}\to\operatorname{\mathsf{End}}\mathcal{C}$, given by
$E\mapsto(E\times\mathord{\relbar})$, is monoidal (with respect to the
Cartesian product on $\mathcal{E}$ and functor composition on
$\operatorname{\mathsf{End}}\mathcal{C}$). Then $\mathcal{C}$ carries a
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enrichment defined in the same way as
the one on $\mathcal{E}^{J}$ which yields notions of an enriched lifting
property and an enriched weak factorisation system. The complication lies in
the fact that the definition of levelwise complemented inclusions is not
available in $\mathcal{C}$. However, if we assume that $\mathcal{C}$ is
equipped with a class of morphisms $\mathscr{D}$ satisfying the conclusion of
Lemma 3.8, then the proof of Theorem 3.14 applies without changes. (Note that
in this case the notion of finiteness in $\mathcal{C}$ depends on the choice
of $\mathscr{D}$.) Examples of categories that can be endowed with such
structure include the categories of internal categories in $\mathcal{E}$,
internal groupoids in $\mathcal{E}$ and marked simplicial objects in
$\mathcal{E}$.

We conclude this section by introducing the notion of a cell complex and
establish a few results that will be useful later.

######  Definition 3.16.

For a family of maps $I=(i\colon A_{i}\to B_{i}\ |\ i\in I)$, an
_$\mathcal{E}$ -enriched $I$-cell complex_ is a morphism of $\mathcal{E}^{D}$
that is a sequential colimit of maps $X\to Y$ arising as pushouts

${\coprod_{i}E_{i}\times A_{i}}$${X}$${\coprod_{i}E_{i}\times B_{i}}$${Y}$

for some family $(E_{i})_{i\in I}$ of objects of $E$.

Below, we simply speak of an $I$-cell complex for brevity.

######  Proposition 3.17.

Under the hypotheses of Theorem 3.14, a morphism of $\mathcal{E}^{D}$ is an
$I$-cofibration if and only if it is a codomain retract of an $I$-cell
complex. In particular, every $I$-cofibration is a levelwise complemented
inclusion.

###### Proof.

A retract of an $I$-cell complex is an $I$-cofibration by Lemma 3.9. It is
furthermore a levelwise complemented inclusion by Lemma 3.8. Conversely, let
$X\to Y$ be an $I$-cofibration and consider the factorisation $X\to
X^{\prime}\to Y$ defined in the proof of Theorem 3.14. Then $X\to X^{\prime}$
is an $I$-cell complex by construction. Moreover, $X\to Y$ has the
$\operatorname{\mathsf{Psh}}\mathcal{E}$-enriched left lifting property with
respect to $X^{\prime}\to Y$ and, in particular, it has the ordinary left
lifting property (by evaluating the hom-presheaves at the terminal object).
Thus there is a lift in the diagram

${X}$${X^{\prime}}$${Y}$${Y}$

which exhibits $X\to Y$ as a codomain retract of $X\to X^{\prime}$. ∎

######  Lemma 3.18.

In the setting of Theorem 3.14, the following hold.

  1. (i)

Consider a countable family of maps $f_{k}$ in the arrow category of
$\mathcal{E}^{D}$. If $f_{k}$ is an $I$-fibration for all $k$, then so is the
coproduct $\coprod_{k}f$. When $\mathcal{E}$ is $\alpha$-lextensive, the same
holds for $\alpha$-coproducts.

  2. (ii)

Consider a span $f_{0}\leftarrow f_{01}\to f_{1}$ in the arrow category of
$\mathcal{E}^{D}$. Assume that both legs form pullback squares and that
$f_{01}\to f_{0}$ is a levelwise complemented inclusion on codomains. If
$f_{k}$ is an $I$-fibration for $k=0,1,01$, then so is the pushout
$\operatorname*{colim}f$.

  3. (iii)

Consider a sequential diagram $f_{0}\to f_{1}\to\ldots$ in the arrow category
of $\mathcal{E}^{D}$. Assume that the maps $f_{k}\to f_{k+1}$ form pullback
squares and are levelwise complemented inclusions on codomains. If $f_{k}$ is
an $I$-fibration for all $i$, then so is $\operatorname*{colim}f$.

###### Proof.

In all three parts, the colimit $\operatorname*{colim}f$ exists and is
computed separately on sources and targets where they form van Kampen colimits
by Corollary 2.12. Let $C$ denote the shape of the diagram (which varies over
the parts). We check that $\operatorname*{colim}f$ is an $I$-fibration using
Proposition 3.4. For each $i\in I$, given a section of
$\widehat{\operatorname{ev}}_{i}(f_{c})$ for $c\in C$, we have to construct a
section of $\widehat{\operatorname{ev}}_{i}(\operatorname*{colim}f)$. Using
Lemma 2.15 and functoriality of colimits, it suffices to construct a family of
section of $\widehat{\operatorname{ev}}_{i}(f_{c})$ that is natural in $c\in
C$.

For part (i), the naturality is vacuous. For part (ii), we pull the section of
$\widehat{\operatorname{ev}}_{i}(f_{1})$ back to a section of
$\widehat{\operatorname{ev}}_{i}(f_{01})$ and then use Lemma 3.13 to replace
the section of $\widehat{\operatorname{ev}}_{i}(f_{0})$ by one that coheres
with the one of $\widehat{\operatorname{ev}}_{i}(f_{01})$. For part (iii), we
recurse on $k$ and use Lemma 3.13 to replace the given section of
$\widehat{\operatorname{ev}}_{i}(f_{k+1})$ by one that coheres with the one of
$\widehat{\operatorname{ev}}_{i}(f_{k})$. In all three cases, the sections
form a $D$-shaped natural transformation as required. ∎

We consider the application functor
$\operatorname{app}\colon[\mathcal{C},\mathcal{D}]\times\mathcal{C}\to\mathcal{D}$
and record some commonly used facts about pushout applications in the
following statement. We regard the pushout application of a natural
transformation $[\mathcal{C},\mathcal{D}]$ to an arrow in $\mathcal{C}$ to be
defined if the pushout in the evident commuting square exists. Recall that the
pushout application is the induced arrow from the pushout corner.

######  Lemma 3.19.

Let $u\colon X\to Y$ be a map in $[\mathcal{C},\mathcal{D}]$. Then pushout
application
$\widehat{\operatorname{app}}(u,-)\colon\mathcal{C}^{[1]}\to\mathcal{D}^{[1]}$
forms a partial functor with the following properties.

  1. (i)

Let $c\colon I\to\mathcal{C}^{{}^{[1]}}$ be a diagram of arrows with levelwise
colimit (i.e., a colimit that is computed separately on sources and targets in
$\mathcal{C}$). If $X$ and $Y$ preserve this levelwise colimit and
$\widehat{\operatorname{app}}(u,-)$ is defined on all values of $c$, then
$\widehat{\operatorname{app}}(u,-)$ preserves the levelwise colimit of $c$.

  2. (ii)

Let $f\to g$ be a morphism in $\mathcal{C}^{{}^{[1]}}$ that is a pushout
square. If $X$ and $Y$ preserve this pushout and
$\widehat{\operatorname{app}}(u,-)$ is defined on $f$ and $g$, then
$\widehat{\operatorname{app}}(u,f)\to\widehat{\operatorname{app}}(u,g)$ is a
pushout square.

  3. (iii)

For an ordinal $\alpha$, let $A_{0}\to A_{1}\to\ldots\to A_{\alpha}$ be an
$\alpha$-composition in $\mathcal{C}$. If this $\alpha$-composition is
preserved by $X$ and $Y$ and $\widehat{\operatorname{app}}(u,-)$ is defined on
$A_{\beta}\to A_{\beta^{\prime}}$ for $\beta\leq\beta^{\prime}\leq\alpha$,
then $\widehat{\operatorname{app}}(u,-)$ preserves the given the
$\alpha$-composition and the resulting step map at $\beta<\alpha$ is a pushout
of $\widehat{\operatorname{app}}(u,-)$ applied to $A_{\beta}\to A_{\beta+1}$.

###### Proof.

This is folklore technique in abstract homotopy theory. Similar proofs (in a
slightly different context) can be found in [Riehl-Verity]*Sections 4 and 5,
in particular [Riehl-Verity]*Lemma 4.8 for part (i) and [Riehl-Verity]*Lemma
5.7 for parts (ii) and (iii). ∎

######  Lemma 3.20.

Let $F,G\colon\mathcal{E}^{D}\to\mathcal{E}^{D^{\prime}}$ be two functors that
preserves levelwise complemented maps, their pushouts and their sequential
compositions. We assume that $F$ and $G$ are equipped with isomorphisms

$\displaystyle F(E\times X)\mathrel{\cong}E\times F(X)$ $\displaystyle
G(E\times X)\mathrel{\cong}E\times G(X)$

natural in $E\in\mathcal{E}$ and $X\in\mathcal{E}^{D}$ (respectively,
$X\in\mathcal{E}^{D^{\prime}}$) and let $\lambda\colon F\to G$ be a natural
transformation compatible with these isomorphisms. Let
$I_{D}\subseteq(\mathcal{E}^{D})^{[1]}$ and
$I_{D^{\prime}}\subseteq(\mathcal{E}^{D^{\prime}})^{[1]}$ be countable sets of
arrows satisfying the conditions of Theorem 3.14. If for each $i\in I_{D}$,
the pushout application $\widehat{\operatorname{app}}(\lambda,i)$ is an
$I_{D^{\prime}}$-cofibration, then for each $I_{D}$-cofibration $i$, the
pushout application $\widehat{\operatorname{app}}(\lambda,i)$ is an
$I_{D^{\prime}}$-cofibration.

###### Proof.

First, because of Lemma 3.8, all $I_{D}$-cofibrations are levelwise
complemented inclusions, so their image under $F$ are again levelwise
complemented inclusions and hence pushouts along them exist. This shows that
$\widehat{\operatorname{app}}(\lambda,i)$ always exists when $i$ is an
$I_{D}$-cofibration.

By Proposition 3.17, a general a $I_{D}$-cofibration is a retract of a
sequential composite of pushouts of countable coproducts of the form $E\times
A\to E\times B$ for a map $A\to B$ in $I_{D}$ and $E\in\mathcal{E}$. A map
$E\times i:E\times A\to E\times B$ is sent by
$\widehat{\operatorname{app}}(\lambda,\mathord{\relbar})$ to the map
$E\times\widehat{\operatorname{app}}(\lambda,i)$, so as we are assuming that
for each $i\in I_{D}$ the map $\widehat{\operatorname{app}}(\lambda,i)$ is an
$I_{D^{\prime}}$-cofibration, it follows that the map of the form $E\times i$
are also sent to $I_{D^{\prime}}$-cofibration.

Using Lemma 3.19 one concludes that any transfinite composition of pushouts of
maps of the form $E\times i$ for $i\in I_{D}$ is also sent by
$\widehat{\operatorname{app}}(\lambda,\mathord{\relbar})$ to a
$I_{D^{\prime}}$-cofibration. Finally, as
$\widehat{\operatorname{app}}(\lambda,\mathord{\relbar})$ is a functor it
preserves retract, and so retracts of such maps are also sent to
$I_{D^{\prime}}$-cofibration, and this concludes the proof as any
$I_{D}$-cofibration is a retract of such a transfinite composition of
pushouts. ∎

######  Proposition 3.21.

Let $j\colon X\to Y$ be a morphism of $\mathcal{E}^{D}$. Under the hypothesis
of Theorem 3.14, if $i\mathbin{\widehat{\times}}j$ is an $I$-cofibration for
all $i\in I$, then $f\mathbin{\widehat{\times}}j$ is an $I$-cofibration for
all $I$-cofibrations $f$.

###### Proof.

We apply Lemma 3.20 to the natural transformation $\mathord{\relbar}\times
j\colon\mathord{\relbar}\times X\to\mathord{\relbar}\times Y$ of endofunctors
on $\mathcal{E}^{D}$. Let us check the needed preservation properties of the
endofunctor $\mathord{\relbar}\times Z$ on $\mathcal{E}^{D}$ for
$Z\in\mathcal{E}$. Preservation of levelwise complemented inclusions follows
from preservation of complemented inclusions in $\mathcal{E}$ under product
with a fixed object (a consequence of lextensivity). Preservation of the
relevant colimits involving levelwise complemented inclusions is an instance
of Corollary 2.12. Preservation of tensors with objects of $\mathcal{E}$
reduces to associativity and commutativity of products in $\mathcal{E}$; this
is natural, so the map $\mathord{\relbar}\times j\colon\mathord{\relbar}\times
X\to\mathord{\relbar}\times Y$ respects the witnessing isomorphism as
appropriate. ∎

##  4 The two weak factorisation systems

In this section we consider a countably lextensive category $\mathcal{E}$. We
construct two weak factorisation systems on the category
$\mathsf{s}\mathcal{E}$ of simplicial objects in $\mathcal{E}$ that will be
proven to form a model structure in Section 9. Our main goal is to describe
the resulting cofibrations in Theorem 4.6 which relies on identification of
one of the factorisation systems as a Reedy factorisation system (Proposition
4.3). In our setting, the category $\mathsf{s}\mathcal{E}$ has relatively few
colimits and consequently much of this section is committed to discussion of
the Reedy theory under these weak hypotheses.

We will use the enriched small object argument of Theorem 3.14 with the
generating sets obtained by applying the partial functor of (2.2) to the sets
of boundary inclusions and horn inclusions in (1.6), i.e.,

$I_{\mathsf{s}\mathcal{E}}=\\{\underline{\mathord{\partial\mathord{\Delta}[n]}}\to\underline{\mathord{\Delta}[n]}\mid
n\geq 0\\}\text{ and
}J_{\mathsf{s}\mathcal{E}}=\\{\underline{\mathord{\Lambda}^{k}[n]}\to\underline{\mathord{\Delta}[n]}\mid
n\geq k\geq 0,n>0\\}\text{.}$

We will refer to $\underline{\mathord{\Delta}[m]}$ as a simplex in
$\mathsf{s}\mathcal{E}$ and similarly for boundaries and horns. We say that a
map in $\mathsf{s}\mathcal{E}$ is a _cofibration_ if it is a
$I_{\mathsf{s}\mathcal{E}}$-cofibration and that it is a _trivial cofibration_
if it is a $J_{\mathsf{s}\mathcal{E}}$-cofibration. Moreover, we note that
notions of (Kan) fibrations and trivial (Kan) fibrations as introduced in
Definition 1.3 coincide with the notions of
$J_{\mathsf{s}\mathcal{E}}$-fibrations and
$I_{\mathsf{s}\mathcal{E}}$-fibration.

######  Proposition 4.1.

Let $f\colon X\to Y$ be a map in $\mathsf{s}\mathcal{E}$.

  1. (i)

$f$ is a fibration if and only if it is a
$J_{\mathsf{s}\mathcal{E}}$-fibration;

  2. (ii)

$f$ is a trivial fibration if and only if it is a
$I_{\mathsf{s}\mathcal{E}}$-fibration.

###### Proof.

By Proposition 3.4, the condition of Definition 1.3 for $f$ being a (trivial)
Kan fibration is equivalent to the $\mathcal{E}$-enriched right lifting
property of $f$ with respect to $J_{\mathsf{s}\mathcal{E}}$ (respectively,
$I_{\mathsf{s}\mathcal{E}}$). ∎

The existence of weak factorisation systems linking these classes is a direct
consequence of the results of Section 3.

######  Theorem 4.2.

Let $\mathcal{E}$ be a countably lextensive category. The category
$\mathsf{s}\mathcal{E}$ of simplicial objects in $\mathcal{E}$ admits two weak
factorisation systems:

  * •

cofibrations and trivial fibrations, cofibrantly generated by
$I_{\mathsf{s}\mathcal{E}}$;

  * •

trivial cofibrations and fibrations, cofibrantly generated by
$J_{\mathsf{s}\mathcal{E}}$.

###### Proof.

All morphisms of $I_{\mathsf{s}\mathcal{E}}$ and $J_{\mathsf{s}\mathcal{E}}$
are levelwise complemented inclusions since $S\mapsto\underline{S}$ preserves
complemented inclusions. Moreover, their domains and codomains are finite
colimits of representables and thus Lemma 3.11 implies that the assumptions of
Theorem 3.14 are satisfied. ∎

Recall that $\mathcal{E}$ admits a weak factorisation system consisting of
complemented inclusions as left maps and split epimorphisms as right maps. We
now wish to characterise our cofibrations and trivial fibrations in terms of
the induced Reedy weak factorisation on $\mathsf{s}\mathcal{E}$. Traditional
treatments of Reedy theory such as [Riehl-Verity] tacitly assume that the
underlying category is bicomplete; this is not the case here. Separately,
there is the treatment [rb] of Reedy theory in the context of a (co)fibration
category, but it only considers Reedy left or right maps between Reedy left or
right objects; in our setting, not all objects are Reedy cofibrant or fibrant.
Let us thus discuss some of the details of the Reedy weak factorisation system
on $\mathsf{s}\mathcal{E}$.

Let $m\geq 0$. We write $\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$ for
$\Delta([m],-)$, i.e., the functor in $[\Delta,\mathsf{Set}]$ corepresented by
$m$. The _coboundary_ $\partial\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$
of $\Delta$ at level $m$ is the subobject of
$\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$ consisting of those maps which
are not face operators. Equivalently,
$\partial\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]_{k}\subseteq\Delta([m],[k])$
consists of those maps $[m]\to[k]$ whose degeneracy-face factorisation has
non-identity degeneracy operator.

Let $A\in\mathsf{s}\mathcal{E}$. The _latching object_ $L_{m}A$, if it exists,
is the colimit of $A$ weighted by
$\partial\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$. We have a canonical
map $L_{m}A\to A$.

Let $i\colon A\to B$ be a map in $\mathsf{s}\mathcal{E}$ and $m\geq 0$. We
wish to consider the relative latching map of $i$. Ordinarily, we would define
it as the map $A_{m}\sqcup_{L_{m}A}L_{m}B\to B_{m}$. However, its domain
depends on the existence of the latching objects $L_{m}A$ and $L_{m}B$ and a
pushout. We wish to avoid these assumptions. Consider the functor
$\mathsf{s}\mathcal{E}\to\mathsf{Set}$ sending $X$ to the set of pairs
consisting of a map $u\colon A_{m}\to X$ and a natural family $v_{f}\colon
B_{k}\to X$ for $f\colon[m]\to[k]$ not a face operator such that $u\circ
Af=v_{f}\circ i_{k}$. If this functor has a corepresenting object, we denote
it by $A_{m}\sqcup_{L_{m}A}L_{m}B$ and obtain the _relative latching map_
$A_{m}\sqcup_{L_{m}A}L_{m}B\to B_{m}$ of $i$ at level $m$. If $L_{m}A$ and
$L_{m}B$ exist, this agrees with the description in terms of the pushout
suggested by our notation.

We desire a more abstract view on the relative latching map. For this, we
introduce the notion of pushout weighted colimit. Consider the two-variable
functor

$H\colon[\Delta,\mathsf{Set}]^{\mathord{\mathrm{missing}}{op}}\times\mathsf{s}\mathcal{E}^{\mathord{\mathrm{missing}}{op}}\to[\mathcal{E},\mathsf{Set}]$
(4.1)

sending $W$ and $X$ to $I\mapsto[\Delta,\mathsf{Set}](W,\mathcal{E}(X(-),I))$
Recall that a $W$-weighted colimit of $X$, denoted
$\operatorname*{colim}^{W}X$, is by definition a representing object of
$H(W,X)$. The _pullback construction_ of $H$ is the two-variable functor

$\widehat{H}\colon([\Delta,\mathsf{Set}]^{\mathord{\mathrm{missing}}{op}})^{[1]}\times(\mathsf{s}\mathcal{E}^{\mathord{\mathrm{missing}}{op}})^{[1]}\to[\mathcal{E},\mathsf{Set}]^{[1]}$

sending $w\colon U\to V$ in $[\Delta,\mathsf{Set}]$ and $i\colon A\to B$ in
$\mathsf{s}\mathcal{E}$ to the map

$H(V,B)\to H(V,A)\times_{H(U,A)}H(U,B)$ (4.2)

in $[\mathcal{E},\mathsf{Set}]$. Assume that domain and codomain of (4.2) have
representing objects $Y$ and $X$, respectively (in particular, $Y$ is the
$V$-weighted colimit of $B$). Then under the Yoneda embedding of
$\mathcal{E}^{\mathord{\mathrm{missing}}{op}}$ into
$[\mathcal{E},\mathsf{Set}]$, (4.2) corresponds to a map $X\to Y$ in
$\mathcal{E}$. We define this to be the _pushout weighted colimit_ with
$w\colon U\to V$ of $i\colon A\to B$ and denote it by
$\widehat{\operatorname*{colim}}^{w}i$. It forms a partial two-variable
functor

${\textstyle\widehat{\operatorname*{colim}}^{(\mathord{\relbar})}(\mathord{\Relbar})}\colon[\Delta,\mathsf{Set}]^{[1]}\times\mathsf{s}\mathcal{E}^{[1]}\to[\mathcal{E},\mathsf{Set}]^{[1]}.$

Note that this is more general than a partially defined pushout construction
of the two-variable weighted colimit functor because we do not require the
individual colimits of $A$ with weight $V$ and $B$ with weights $U$ and $V$ to
exist.

Unfolding the codomain of (4.2), we see that the relative latching map of
$i\colon A\to B$ at level $m$ is precisely the pushout weighted colimit of $i$
with the coboundary inclusion
$\partial\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]\to\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$.
Each side exist when the other does. This point of view is useful because it
enables us to obtain pushout weighted colimits of $i$ with certain inclusions
as cell complexes of relative latching maps.

We call a map $i$ a _Reedy complemented inclusion_ if, for all $m$, the
relative latching map of $i$ at level $m$ exists and is a complemented
inclusion. This condition for $m<k$ suffices to guarantee the existence of the
relative latching map at level $m=k$. Thus, in the inductive verification that
a map is a Reedy complemented inclusion, the relevant latching maps always
exist. Given a map $X\to Y$ in $\mathsf{s}\mathcal{E}$, the _relative matching
map_ at level $m$ is its weighted limit, i.e., pullback evaluation, at
$\partial\mathord{\Delta}[m]\to\mathord{\Delta}[m]$, i.e., the map $X_{m}\to
Y_{m}\times_{\operatorname{ev}_{\partial\mathord{\Delta}[m]}Y}\operatorname{ev}_{\partial\mathord{\Delta}[m]}X$.
We call $X\to Y$ a _Reedy split epimorphism_ if all its relative matching maps
are split epimorphisms.

Following standard Reedy theory, Reedy complemented inclusions and Reedy split
epimorphisms form a weak factorisation system. For this, we observe that
instantiating the treatment of [Riehl-Verity] and making use of Lemma 3.19,
the use of (co)limits in $\mathsf{s}\mathcal{E}$ may be reduced to pushouts
along complemented inclusions and pullbacks along split epimorphisms. We now
relate this weak factorisation system to that of cofibrations and trivial
fibrations, given in Theorem 4.2 (cf. also Proposition 4.1).

######  Proposition 4.3.

The weak factorisation system of cofibrations and trivial fibrations of
Theorems 4.2 and 4.1 coincides with the weak factorisation system of Reedy
complemented inclusions and Reedy split epimorphisms.

###### Proof.

Two weak factorisation systems coincide as soon as their right classes do.
But, by inspecting the definition of a trivial fibration in Definition 1.3, a
map in $\mathsf{s}\mathcal{E}$ is a Reedy split epimorphism if and only if it
is a trivial Kan fibration. ∎

The next lemma will be useful to simplify some saturation arguments in Section
6, as it allows us to avoid considering retracts, cf. the notion of a cell
complex in Definition 3.16.

######  Lemma 4.4.

Every cofibration in $\mathsf{s}\mathcal{E}$ is an
$I_{\mathsf{s}\mathcal{E}}$-cell complex.

###### Proof.

If $A\to B$ is a cofibration, then $B$ can be written as the colimit of its
skeleta relative to $A$:

${\operatorname{Sk}^{-1}_{A}B}$${\operatorname{Sk}^{0}_{A}B}$${\operatorname{Sk}^{1}_{A}B}$${\ldots}$

where $\operatorname{Sk}^{-1}_{A}B=A$ and for $k\geq 0$ the square

${B_{k}\times\mathord{\partial\mathord{\Delta}[k]}\cup(A_{m}\sqcup_{L_{m}A}L_{m}B)\times\mathord{\Delta}[k]}$${\operatorname{Sk}^{k-1}_{A}B}$${B_{k}\times\mathord{\Delta}[k]}$${\operatorname{Sk}^{k}_{A}B}$

is a pushout. These statements are justified analogously to the proofs of
[GSS]*Lemma 2.3.1, Corollary 2.3.3. The colimits used in the construction
exist by Corollary 2.12 since they are colimits of sequences of levelwise
complemented inclusions and pushouts along levelwise complemented inclusions
which is ensured by the assumption that $A\to B$ is a cofibration. ∎

Our next goal is to provide a characterisation of cofibrations in terms of
actions of degeneracy operators, stated in Theorem 4.6 below. This is a
generalisation of [Hwms]*Proposition 5.1.4 or [GSS]*Proposition 1.4.4 to a
setting without arbitrary colimits. The proof is made significantly more
complex by the fact that $\mathcal{E}$ is not assumed to be a Grothendieck
topos. Instead, the required exactness properties are substituted by Lemma
2.14. We also need the following statement. For this, we observe that our
discussion of Reedy theory and latching objects for the case of $\Delta$
applies just as well to arbitrary countable Reedy categories of countable
height. Note that the assumption of a Reedy cofibrant diagram includes the
hypothesis that all latching objects exist.

######  Lemma 4.5.

Let $D$ be a finite direct category. Let $F\colon D\to\mathsf{s}\mathcal{E}$
be a Reedy cofibrant diagram. Then the colimit of $F$ exists and is van
Kampen.

###### Proof.

We proceed by induction on the height of $D$. For height $0$, note that $D$ is
the empty and the claim holds because initial objects are van Kampen since
$\mathsf{s}\mathcal{E}$ is lextensive.

Now assume the claim for height $n$ and let $D$ have height $n+1$. Let
$D^{\prime}$ of height $n$ denote the restriction of $D$ to objects of degree
below $n$. Let $I$ be the collection of objects of $D$ of degree $n$. As per
usual Reedy theory, we may compute the colimit of $F$ as the following
pushout:

${\coprod_{i\in
I}L_{i}F}$${\operatorname*{colim}_{D^{\prime}}F|_{D^{\prime}}}$${\coprod_{i\in
I}F(i)}$${\operatorname*{colim}_{D}F\hbox to0.0pt{.\hss}}$

Here, the left map is a cofibration because it is a finite coproduct of
cofibrations, and hence the pushout exists and is van Kampen by Lemma 2.9. By
the inductive hypothesis, the colimit computing the latching object $L_{i}F$
for $i\in I$ is van Kampen, and so is the colimit of $F|_{D^{\prime}}$. The
finite coproducts are van Kampen since $\mathsf{s}\mathcal{E}$ is lextensive.
Using the characterisation of van Kampen colimits given by Lemma 2.2, one sees
that $\operatorname*{colim}_{D}F$ is van Kampen. ∎

######  Theorem 4.6 (Characterisation of cofibrations).

Let $i\colon A\to B$ be a map in $\mathsf{s}\mathcal{E}$. Then the following
are equivalent:

  1. (i)

the map $i$ is a cofibration;

  2. (ii)

the map $i$ is a levelwise complemented inclusion and the map
$A_{m}\sqcup_{A_{n}}B_{n}\to B_{m}$ is a complemented inclusion for every
degeneracy operator $[m]\rightarrowtriangle[n]$.

###### Proof.

We use from Proposition 4.3 that cofibrations are the same as Reedy
complemented inclusions. As in [Riehl-Verity], we work freely with pushout
weighted colimits in $\mathcal{E}$, with index category both $\Delta$ and its
wide subcategory $\Delta_{-}$ of degeneracy operators. As explained above (in
the case of $\Delta$), these are partial two-variable functors in our
situation. Mirroring our notation for $\Delta$, we write
$\partial\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$ for the subobject
of $\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]=\Delta([m],-)$ in
$[\Delta_{-},\mathsf{Set}]$ consisting of the non-identity maps. Recall that
the coboundary inclusion
$\partial\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]\to\Delta^{\mathord{\mathrm{missing}}{op}}[{m}]$
arises as left Kan extension along $\Delta_{-}\to\Delta$ of the coboundary
inclusion
$\partial\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]\to\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$.
For working with weighted colimits, we recall that left Kan extension on the
side of the weight corresponds to restriction on the side of the diagram.

We start with the direction from (i) to (ii). Let $i$ be a Reedy complemented
inclusion. Then the pushout weighted colimit of $i$ with any finite cell
complex (finite composite of pushouts) of coboundary inclusions is a
complemented inclusion. In particular, the pushout weighted colimit of the
restriction $i|_{\Delta_{-}}$ of $i$ to $\Delta_{-}$ with any finite cell
complex of coboundary inclusions
$\partial\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{k}]\to\Delta^{\mathord{\mathrm{missing}}{op}}[{k}]$
of $\Delta_{-}$ is a complemented inclusion. For $m\geq 0$, the map $A_{m}\to
B_{m}$ is the pushout weighted colimit of $i|_{\Delta_{-}}$ with such a finite
cell complex $\varnothing\to\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$,
hence a complemented inclusion. Every degeneracy operator
$[m]\rightarrowtriangle[n]$ is a split epimorphism. It follows that
$\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{n}]\to\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$
is an inclusion with levelwise finite complement, thus we can write it as a
finite cell complex of coboundary inclusions of $\Delta_{-}$. Therefore, the
pushout weighted colimit of $i$ with
$\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{n}]\to\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$
is a complemented inclusion. But this is the map $A_{m}\sqcup_{A_{n}}B_{n}\to
B_{m}$.

We finish with the direction from (ii) to (i). We show that the relative
latching map $A_{m}\sqcup_{L_{m}A}L_{m}B\to B_{m}$ of $i$ is a complemented
inclusion by induction on $m$. Recall that this is the pushout weighted limit
of $i|_{\Delta_{-}}$ with
$\partial\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]\to\Delta_{-}^{\mathord{\mathrm{missing}}{op}}[{m}]$.
Let $\partial(\Delta_{-}^{\mathord{\mathrm{missing}}{op}}\downarrow[m])$
denote the opposite of the poset of non-identity degeneracy operators with
source $[m]$. Consider the diagram
$F\colon\partial(\Delta_{-}^{\mathord{\mathrm{missing}}{op}}\downarrow[m])\to\mathcal{E}\downarrow
B_{m}$ sending a degeneracy operator $[m]\rightarrowtriangle[n]$ to the object
$A_{m}\sqcup_{A_{n}}B_{n}$. It lives canonically under the object $A_{m}$ over
$B_{m}$. By switching from the weighted colimit to the conical colimit point
of view, the object $A_{m}\sqcup_{L_{m}A}L_{m}B$ is the colimit of $F$ in the
category of factorisations of $A_{m}\to B_{m}$. Equivalently, in the slice
over $B_{m}$, the object $A_{m}\sqcup_{L_{m}A}L_{m}B$ is the colimit of the
diagram $F_{*}$ that is $F$ with shape adjoined with an initial object sent to
$A_{m}$.

Note that, using our assumptions, we can regard $F$ as a diagram of
complemented subobjects of $B_{m}$ that are bounded from below by the
complemented subobject $A_{m}$. It remains to show that the colimit of $F_{*}$
in the slice over $B_{m}$ has a complemented inclusion as underlying map. It
will suffice to show that this colimit is subterminal. For then, it is given
by the non-empty finite union of the subobjects that constitute the values of
$F_{*}$, and complemented subobjects are closed under finite unions by part
(i) of Lemma 2.10.

The indexing category of $F_{*}$ is a finite direct category. The latching map
of $F_{*}$ at the initial object is $0\to A_{m}$, a complemented inclusion.
The latching map of $F_{*}$ at an object $[m]\rightarrowtriangle[n]$ is a
pushout of the relative latching map of $A\to B$ at $[m]$, a complemented
inclusion by induction hypothesis. Thus, the diagram $F_{*}$ is Reedy
cofibrant. By Lemma 4.5, the colimit of $F_{*}$ is van Kampen. All of this
holds both in $\mathcal{E}$ as well as its slice over $B_{m}$.

Given a complemented subobject $U\to B_{m}$ and an arbitrary subobject $V\to
B_{m}$, the pushout corner map in the pullback of $U\to B_{m}$ and $V\to
B_{m}$ exists. If it is a monomorphism, it computes the union $U\cup V\to
B_{m}$ of the given subobjects. Since degeneracy operators are split
epimorphisms, the natural transformation $i|_{\Delta_{-}}$ is Cartesian. This
makes the value of $F$ at an object $[m]\rightarrowtriangle[n]$ the union of
the subobjects $A_{m}\to B_{m}$ and $B_{n}\to B_{m}$.

Since $\Delta$ is elegant [Bergner-Rezk], given non-identity degeneracy
operators $[m]\rightarrowtriangle[n_{i}]$ for $i=1,2$, we have an absolute
pushout

${[m]}$${[n_{1}]}$${[n_{2}]}$${[k]}$

in $\Delta$ with $[n_{1}]\rightarrowtriangle[k]$ and
$[n_{2}]\rightarrowtriangle[k]$ degeneracy operators. Note that
$[m]\rightarrowtriangle[k]$ is distinct from the identity. By absoluteness, we
obtain a pullback

${B_{k}}$${B_{n_{1}}}$${B_{n_{2}}}$${B_{m}\hbox to0.0pt{.\hss}}$

We now work in subobjects of $B_{m}$. From the above pullback, we have
$B_{k}=B_{n_{1}}\cap B_{n_{2}}$. Using from Lemma 2.9 twice that pushouts
along complemented inclusions are stable under pullback, we compute

$\displaystyle(A_{m}\cup B_{n_{1}})\cap(A_{m}\cup B_{n_{2}})$
$\displaystyle=((A_{m}\cup B_{n_{1}})\cap A_{m})\cup((A_{m}\cup B_{n_{1}})\cap
B_{n_{2}})$ $\displaystyle=A_{m}\cup((A_{m}\cup B_{n_{1}})\cap B_{n_{2}})$
$\displaystyle=A_{m}\cup(B_{n_{1}}\cap B_{n_{2}})$ $\displaystyle=A_{m}\cup
B_{k}.$

We obtain, in subobjects of $B_{m}$, that $F$ at $[m]\rightarrowtriangle[n]$
is the intersection (computed as pullback) of $F$ at $[m]\to[n_{1}]$ and
$[m]\to[n_{2}]$. Thus, in subobjects of $B_{m}$, the diagram $F$ (and then
also $F_{*}$) preserves binary meets. Recollecting from above that the colimit
of $F_{*}$ in the slice over $B_{m}$ is van Kampen, Lemma 2.14 shows that it
is subterminal. ∎

##  5 Closure properties of cofibrations

This section is devoted to further study of weak factorisation systems
constructed in Section 4, in preparation for the proof of the existence of the
effective model structure. We begin with a simple verification.

######  Lemma 5.1.

If $A\to B$ is a (trivial) cofibration between levelwise countable simplicial
sets, then $\underline{A}\to\underline{B}$ is a (trivial) cofibration in
$\mathsf{s}\mathcal{E}$.

###### Proof.

Recall that the partial functor $X\mapsto\underline{X}$ is a partial left
adjoint to the levelwise global sections functor. This is equivalently the
functor $\operatorname{Hom}_{\mathsf{sSet}}(1,-)$ with
$1\in\mathsf{s}\mathcal{E}$ from Section 1. By adjointness using the weak
factorisation systems of Theorem 4.2 and Proposition 4.1, it suffices to show
that $\operatorname{Hom}_{\mathsf{sSet}}(1,-)$ preserves (trivial) fibrations.
This holds by Proposition 1.4. ∎

######  Proposition 5.2.

  1. (i)

Trivial fibrations are fibrations.

  2. (ii)

Trivial cofibrations are cofibrations.

###### Proof.

The first part is immediate since trivial Kan fibrations are Kan fibrations in
simplicial sets. The second parts follows by adjointness using the weak
factorisation systems of Theorem 4.2. ∎

We now establish some formal properties of the two enriched weak factorisation
systems, regarding the pushout-product, pushout-tensor and pullback-cotensor
functors (cf. Remark 1.2).

######  Proposition 5.3 (Pushout-product properties).

  1. (i)

In $\mathsf{s}\mathcal{E}$, cofibrations are closed under pushout product.

  2. (ii)

In $\mathsf{s}\mathcal{E}$, the pushout product of a cofibration and a trivial
cofibration is a trivial cofibration.

###### Proof.

For part (i), recall that cofibrations in $\mathsf{sSet}$ are closed under
pushout product.333See [Hwms]*Proposition 5.1.5 or [GSS]*Proposition 1.3.1 for
the constructive version of this fact. Since $S\mapsto\underline{S}$ preserves
pushouts and products, it follows that the pushout product of generating
cofibrations in $\mathsf{s}\mathcal{E}$ is a cofibration. The same follows for
general cofibrations in $\mathsf{s}\mathcal{E}$ by Proposition 3.21. These
pushout products exist by Lemma 2.13.

For part (ii), The result holds in $\mathsf{sSet}$ by444See [Hwms]*Corollary
5.2.3 or [GSS]*Proposition 1.3.1 for the constructive version of this fact.
[GZ]*Proposition IV.2.2 and thus it carries over to $\mathsf{s}\mathcal{E}$ by
the argument of part (i). ∎

######  Lemma 5.4.

Let $X\in\mathsf{s}\mathcal{E}$. For every finite simplicial set $K$, the
tensor $K\mathbin{\cdot}X$ exists and is given by $\underline{K}\times X$.

###### Proof.

Given $Y\in\mathsf{s}\mathcal{E}$, a morphism $X\to K\mathop{\pitchfork}Y$
consists of a family of morphisms $X_{m}\to
Y_{n}^{(K\times\mathord{\Delta}[m])_{n}}$, natural in $m$ and dinatural in
$n$. This corresponds to a family of morphisms
$\underline{K\times\mathord{\Delta}[m]}_{n}\times X_{m}\to Y_{n}$, dinatural
in $m$ and natural in $n$. Moreover:

$\underline{K\times\mathord{\Delta}[m]}_{n}\times
X_{m}=\underline{K_{n}}\times\underline{\operatorname{Hom}([m],[n])}\times
X_{m}\text{.}$

Since $\textstyle\int^{[m]}\underline{\operatorname{Hom}([m],[n])}\times
X_{m}=X_{n}$, such family of maps corresponds to a morphism
$\underline{K}_{n}\times X_{n}\to Y_{n}$ natural in $n$, i.e., a morphism
$\underline{K}\times X\to Y$ in $\mathsf{s}\mathcal{E}$. ∎

######  Proposition 5.5 (Pushout tensor properties).

Let $A\to B$ be a cofibration between finite simplicial sets. Then, the
pushout tensor with $A\to B$ exists. Furthermore,

  1. (i)

it preserves trivial cofibrations,

  2. (ii)

it preserves cofibrations,

  3. (iii)

if $A\to B$ is a trivial cofibration, then it sends cofibrations to trivial
cofibrations.

###### Proof.

The existence follows from Corollary 2.12 and Lemma 5.4. These other
statements are dual to the ones of part (i) of Lemma 1.5 under the tensor-
cotensor adjunction of Lemma 5.4. Note that for this conclusion it suffices to
consider the underlying ordinary weak factorisation system of Lemma 3.6 so
that we do not need to verify that the adjunction is enriched over
$\operatorname{\mathsf{Psh}}\mathcal{E}$. ∎

We now turn our attention to the cofibrations and the cofibrant objects in
$\mathsf{s}\mathcal{E}$. From Section 3 and Proposition 4.1 these are exactly
the maps with the left lifting property with respect to Kan fibrations. The
next lemma provides us with a stock of cofibrant objects.

######  Lemma 5.6.

  1. (i)

Let $E\in\mathcal{E}$. The constant simplicial object
$E\in\mathsf{s}\mathcal{E}$ is cofibrant.

  2. (ii)

The domains and codomains of all morphisms of $I_{\mathsf{s}\mathcal{E}}$ and
$J_{\mathsf{s}\mathcal{E}}$ are cofibrant.

  3. (iii)

Let $X\in\mathsf{s}\mathcal{E}$ and $K$ be a finite simplicial set. If $X$ is
cofibrant, then so is $K\mathop{\pitchfork}X$.

###### Proof.

For part (i), by Lemma 3.9, the tensor of
$\underline{\partial\mathord{\Delta}[0]}\to\underline{\mathord{\Delta}[0]}$
with $E$ is a cofibration. By Lemma 5.4, this map is the tensor of
$E\in\mathsf{s}\mathcal{E}$ with
$\partial\mathord{\Delta}[0]\to\mathord{\Delta}[0]$, i.e., the map
$\varnothing\to E$ in $\mathsf{s}\mathcal{E}$. Part (ii) holds since
$S\mapsto\underline{S}$ preserves cofibrations by Lemma 5.1.555Constructively,
for part (ii) one needs to check also that the relevant objects are cofibrant
in $\mathsf{sSet}$. The simplices and their boundaries are cofibrant in
$\mathsf{sSet}$ by [GSS]*Lemma 1.3.5 and the horns by [GSS]*Lemma 1.4.9.
Finally, for part (iii), if $[m]\rightarrowtriangle[n]$ is a degeneracy
operator, then the map
$(K\mathop{\pitchfork}X)_{n}\to(K\mathop{\pitchfork}X)_{m}$ can be identified
with the map $X(K\times\mathord{\Delta}[n])\to X(K\times\mathord{\Delta}[m])$.
It follows from [H]*Proposition 3.1.11 that when $K$ is a finite simplicial
set, the map $K\times\mathord{\Delta}[n]\to K\times\mathord{\Delta}[m]$ is a
finite composite of pushouts of degeneracy operators. This implies that the
map $(K\mathop{\pitchfork}X)_{n}\to(K\mathop{\pitchfork}X)_{m}$ is a finite
composite of pullbacks of degeneracy operator $X_{a}\to X_{b}$. As $X$ is
cofibrant these maps are all complemented inclusions, hence as complemented
inclusions are closed under pullback and composition, this implies that
$(K\mathop{\pitchfork}X)_{n}\to(K\mathop{\pitchfork}X)_{m}$ is a complemented
inclusion as well. ∎

######  Lemma 5.7.

Cofibrations are closed under pullback along a monomorphism.

###### Proof.

Consider a pullback square of simplicial objects:

${S^{\prime}}$${S}$${A}$${B}$

We check that $S^{\prime}\to S$ is a cofibration using characterisation (ii)
of Theorem 4.6. In an lextensive category, a pullback of a complemented
inclusion is a complemented inclusion, hence the map $S^{\prime}\to
S^{\prime}$ is a levelwise complemented inclusion. Given any degeneracy
operator $[m]\rightarrowtriangle[n]$, as it is a split epimorphism and $S\to
B$ is a monomorphism, the naturality square:

${S_{n}}$${S_{m}}$${B_{n}}$${B_{m}}$

is a pullback. The pushout $B_{m}\sqcup_{A_{m}}A_{n}$ is a van Kampen colimit
because the map $A_{m}\to B_{m}$ is a complemented inclusion, it hence follows
that we have a pullback square:

${S_{n}\sqcup_{S^{\prime}_{n}}S^{\prime}_{m}}$${S_{m}}$${B_{n}\sqcup_{A_{n}}A_{m}}$${B_{m}}$

and hence as the bottom map is a complemented inclusion by assumption, the top
map is also a complemented inclusion. This shows that $S^{\prime}\to S$ is a
cofibration. ∎

As discussed just before Lemma 1.8, the slice
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ is enriched over simplicial sets
and has cotensors by finite simplicial sets. Under the present hypotheses, it
also has tensors by finite (and even countable) simplicial sets, which are
simply tensors in the underlying category $\mathsf{s}\mathcal{E}$.

Part (iii) of the next Proposition extends the pullback cotensor properties of
part (i) of Lemma 1.5 to slice categories.

######  Proposition 5.8.

Let $X\in\mathsf{s}\mathcal{E}$.

  1. (i)

Pushout products of cofibrations in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ exist. Moreover, cofibrations in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ are closed under pushout product.

  2. (ii)

The pushout tensor properties of Proposition 5.5 hold also in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$.

  3. (iii)

The pullback cotensor in $\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ of a
cofibration between finite simplicial sets and a fibration is a fibration. If
the given cofibration or fibration is trivial, then the result is a trivial
fibration.

###### Proof.

For part (i), recall that pushout products in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$ are computed from pushout
products in $\mathsf{s}\mathcal{E}$ by pulling back along the diagonal $X\to
X\times X$. Since the latter is a monomorphism, the conclusion follows from
Proposition 5.3 and Lemma 5.7. For part (ii), note that the forgetful functor
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}X\to\mathsf{s}\mathcal{E}$ preserves
tensors and pushouts and thus the pushout tensor properties follow directly
from Proposition 5.5. Part (iii) was already established as Lemma 1.8, but now
it also follows by the tensor-cotensor adjunction. ∎

######  Proposition 5.9.

  1. (i)

Let $f\colon X\to Y$ be a morphism in $\mathsf{s}\mathcal{E}$. If $X$ is
cofibrant, then the pullback functor
$f^{*}\colon\mathsf{s}\mathcal{E}\mathbin{\downarrow}Y\to\mathsf{s}\mathcal{E}\mathbin{\downarrow}X$
preserves cofibrations.

  2. (ii)

Let $A\to X$ and $B\to X$ be morphisms in $\mathsf{s}\mathcal{E}$. If $A$ and
$B$ are cofibrant, then so is $A\times_{X}B$.

  3. (iii)

Cofibrant objects in $\mathsf{s}\mathcal{E}$ are closed under finite limits.

###### Proof.

For (i), if $A\to B$ is a cofibration over $Y$, then its pullback along
$f\colon X\to Y$ coincides with the pushout product of $A\to B$ and
${\mathord{\varnothing}}\to X$ in
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}Y$, which is a cofibration by part
(i) of Proposition 5.3. Part (ii) is a special case of part (i). Finally, for
part (iii), it suffices to check that cofibrant objects are closed under
pullback and that the terminal object is cofibrant. The former follows from
part (ii). The latter follows by definition since $0\to 1$ is a generating
cofibration. ∎

##  6 Pushforward along cofibrations

This section and Sections 7, 8 and 9 constitute the third part of the paper,
in which we show how the two weak factorisation systems of Section 4 give rise
to the effective model structure (Theorem 9.9). For this, we shall work with a
fixed countably lextensive category $\mathcal{E}$. We do not assume that the
category $\mathcal{E}$ is (locally) Cartesian closed, but we establish the
existence of certain exponentials and pushforwards required by our argument.
We also provide a criterion for the cofibrancy of some of these constructions.
We begin with a few remarks on exponentiable maps.

######  Proposition 6.1.

Let $f\colon X\to Y$ in $\mathcal{E}$. Then, the following are equivalent:

  1. (i)

the pullback functor
$f^{*}\colon\mathcal{E}\mathbin{\downarrow}Y\to\mathcal{E}\mathbin{\downarrow}X$
has a right adjoint
$f_{*}\colon\mathcal{E}\mathbin{\downarrow}X\to\mathcal{E}\mathbin{\downarrow}Y$,

  2. (ii)

$X$ is exponentiable as an object of $\mathcal{E}\mathbin{\downarrow}Y$.

###### Proof.

This follows from [Elephant]*Lemma A1.5.2 (i) and (the proof of)
[Elephant]*Corollary A1.5.3. ∎

When the equivalent conditions of Proposition 6.1 hold, we say that $f$ is
_exponentiable_ and refer to the right adjoint $f_{*}$ as the _pushforward
along $f$_. (It is also known as the _dependent product along $f$_.)

######  Example 6.2.

Let $S$ be a finite set. Then, $\underline{S}\in\mathcal{E}$ defined in (2.2)
is exponentiable in $\mathcal{E}$ and the exponential of $X$ by
$\underline{S}$ is the product $X^{S}$. Indeed, as finite coproducts in
$\mathcal{E}$ are universal, $\underline{S}\times
X\mathrel{\cong}\coprod_{s\in S}X$. Hence, a map $\underline{S}\times A\to X$
is the same as an $S$-indexed collection of maps $A\to X$, that is the same as
a map $A\to X^{S}$.

######  Proposition 6.3.

Let

${W}$${X}$${Y}$${Z}$$v$$u$$g$$f$

be a pullback square in $\mathcal{E}$. If $f$ is exponentiable, then so is $g$
and the canonical natural transformation $u^{*}f_{*}\to g_{*}v^{*}$ is an
isomorphism.

###### Proof.

This follows from [Elephant]*Lemma A1.5.2 (ii) applied in the slice category
over $Z$. If $K$ is an object over $W$, the pushforward $g_{*}K$ is
constructed explicitly as the pullback:

${g_{*}K}$${f_{*}K}$${Y}$${f_{*}W}$

where the bottom arrow is the unit of adjunction $Y\to f_{*}f^{*}Y=f_{*}W$. ∎

######  Proposition 6.4.

Let $D$ be a small category and $f_{\bullet}\colon X_{\bullet}\to Y_{\bullet}$
a natural transformation between two $D$-diagrams in $\mathcal{E}$ such that
$f_{\bullet}$ is Cartesian, $f_{d}$ is exponentiable for every $d\in D$, and
$Y_{\bullet}$ has a van Kampen colimit in $\mathcal{E}$. Then the colimit map

$f\colon\operatorname*{colim}_{d\in D}X_{d}\to\operatorname*{colim}_{d\in
D}Y_{d}$

is exponentiable, and up to the equivalences

$\mathcal{E}\mathbin{\downarrow}\operatorname*{colim}_{D}X_{d}\simeq\lim_{D}\left(\mathcal{E}\mathbin{\downarrow}X_{d}\right)\text{,
}\qquad\mathcal{E}\mathbin{\downarrow}\operatorname*{colim}_{D}Y_{d}\simeq\lim_{D}\left(\mathcal{E}\mathbin{\downarrow}Y_{d}\right),$

the functor $f_{*}$ coincides with the collection of functors $(f_{d})_{*}$.

###### Proof.

The claim follows from a general fact. If $F\colon\mathcal{A}\to\mathcal{B}$
is a pseudo-natural transformation between two diagrams
$\mathcal{A}\,,\mathcal{B}\colon D\to\mathsf{Cat}$ of categories such that
each $F_{d}$ has a right adjoint $R_{d}$ and for each naturality square of
$F_{d}$ the Beck–Chevalley conditions are satisfied, then the isomorphisms
given by the Beck–Chevalley condition exhibit
$R_{d}\colon\mathcal{B}_{d}\to\mathcal{A}_{d}$ as a pseudo-natural
transformation, and $\lim R_{d}$ is a right adjoint to $\lim F_{d}$, with the
unit and counit of this adjunction being levelwise the unit and counit of the
adjunction $F_{d}\dashv R_{d}$. ∎

We now move on to discuss how exponentiability interacts with cofibrancy. In
particular, the aim of the rest of the section is to prove the following
result.

######  Theorem 6.5.

Let $i\colon A\to B$ be a cofibration between cofibrant object in
$\mathsf{s}\mathcal{E}$. Then:

  1. (i)

$i$ is exponentiable,

  2. (ii)

$i_{*}$ sends cofibrant objects to cofibrant objects.

We will prove this theorem by a saturation argument. For this purpose, we
introduce now the class $\mathcal{G}$ of cofibrations between cofibrant
objects that satisfy properties (i) and (ii) of the theorem.

Assume $i\colon A\to B$ an exponentiable monomorphism in $\mathcal{E}$. Then,
for any $X\in\mathcal{E}\mathbin{\downarrow}A$, the unit of the adjunction
$i^{*}\dashv i_{*}$ induces a pullback square

${X}$${i_{*}X}$${A}$${B\hbox to0.0pt{.\hss}}$$i$ (6.1)

Indeed, since $i$ is a monomorphism, the counit
$i^{*}i_{!}\to\operatorname{id}$ of the adjunction $i_{!}\dashv i^{*}$ is
invertible, and therefore so is the unit $\operatorname{id}\to i^{*}i_{*}$.

######  Lemma 6.6.

Let $i\colon A\to B$ be a map in $\mathcal{G}$. For cofibrant
$X\in\mathcal{E}\mathbin{\downarrow}A$, the map $X\rightarrow i_{*}X$ is a
cofibration.

###### Proof.

The claim follows from part (i) of Proposition 5.9, since the map
$X\rightarrow i_{*}X$ is a pullback of a cofibration between cofibrant objects
by (6.1) above. ∎

######  Proposition 6.7.

The class $\mathcal{G}$ is closed under pushouts along maps with cofibrant
target.

###### Proof.

If $i\colon A\to B$ is in $\mathcal{G}$ and $f\colon A\to X$ is an arbitrary
arrow in $\mathsf{s}\mathcal{E}$ with $X$ cofibrant, we consider the diagram

${X}$${A}$${A}$${X}$${A}$${B\hbox to0.0pt{\text{.}\hss}}$$i$$f$$i$

Then the two squares are pullbacks (because $i$ is a monomorphism for the one
on the right) the vertical maps are all exponentiable by assumption, so by
Proposition 6.4, the map between the colimit of the first row to the colimit
of the second row, that is the map

$j:X\rightarrow X\sqcup_{A}B\text{,}$

is indeed exponentiable. Moreover, still by Proposition 6.4, if $K$ is a
cofibrant object over $X$, it corresponds with respect to the van Kampen
pushout of the first row to the Cartesian natural transformation

${K}$${f^{*}K}$${f^{*}K}$${X}$${A}$${A\hbox to0.0pt{\text{.}\hss}}$$f$

Hence its image by $j_{*}$ corresponds to the Cartesian natural transformation

${K}$${f^{*}K}$${i^{*}f^{*}K}$${X}$${A}$${B\hbox to0.0pt{\text{.}\hss}}$$f$$i$

So, by gluing along the bottom van Kampen colimit, we have a pushout square

${f^{*}K}$${i_{*}f^{*}K}$${K}$${j_{*}K}$

where the top arrow is a cofibration by Lemma 6.6 and the assumption that
$i\in\mathcal{G}$ applied to the cofibrant object $f^{*}K$. It follows that
$j_{*}K$ is cofibrant.∎

######  Proposition 6.8.

The class $\mathcal{G}$ is closed under sequential composition.

###### Proof.

The class $\mathcal{G}$ is clearly closed under finite composition. Given an
$\omega$-chain
$A_{0}\overset{i_{0}}{\rightarrowtail}A_{1}\overset{i_{1}}{\rightarrowtail}A_{2}\overset{i_{2}}{\rightarrowtail}\dots$
of arrows in $\mathcal{G}$, we consider the diagram:

${A_{0}}$${A_{0}}$${A_{0}}$${\ldots}$${A_{0}}$${A_{1}}$${A_{2}}$${\ldots\hbox
to0.0pt{.\hss}}$

Each vertical map is in $\mathcal{G}$ as a composite of maps in $\mathcal{G}$;
each square is a pullback as all these maps are monomorphisms, so by
Proposition 6.4, the comparison map $j\colon
A_{0}\to\operatorname*{colim}A_{i}$ between the two colimit is exponentiable.
If $K$ is a cofibrant object over $A_{0}$, then again by Proposition 6.4 its
image by $j_{*}$ corresponds to the Cartesian natural transformation:

${K_{0}}$${K_{1}}$${K_{2}}$${\ldots}$${A_{0}}$${A_{1}}$${A_{2}}$${\ldots}$

where $K_{0}=K$ and $K_{n+1}=(i_{n})_{*}K_{n}$, hence all the maps in the top
row are cofibrations, and so $j_{*}K=\operatorname*{colim}K_{i}$ is cofibrant.
∎

######  Proposition 6.9.

The class $\mathcal{G}$ is closed under tensors by objects of $\mathcal{E}$.

###### Proof.

Let $i\colon A\rightarrowtail B$ an arrow in $\mathcal{G}$, and let $X$ an
object of $\mathcal{E}$. The square

${A\times X}$${B\times X}$${A}$${B}$$j$$i$

is a pullback, so $j$ is exponentiable by Proposition 6.3. Moreover, the
formula for $j_{*}$ given in the proof of Proposition 6.3 gives that $K$ over
$A\times X$ we have a pullback square

${j_{*}K}$${i_{*}K}$${B\times X}$${i_{*}(A\times X)}$

Since $i\in\mathcal{G}$ and $B\times X$ is cofibrant, $i_{*}K$ is cofibrant,
and so $j_{*}K$ is cofibrant, as required. ∎

In order to conclude the proof of Theorem 6.5, it remains to show that the
generating cofibrations
$i\colon\mathord{\partial\mathord{\Delta}[n]}\rightarrowtail\mathord{\Delta}[n]$
are in $\mathcal{G}$. This is based on an explicit description of $i_{*}$
using the characterisation of
$\mathsf{s}\mathcal{E}\mathbin{\downarrow}\mathord{\partial\mathord{\Delta}[n]}$
and $\mathsf{s}\mathcal{E}\mathbin{\downarrow}\mathord{\Delta}[n]$ of Lemma
2.8.

######  Proposition 6.10.

The generating cofibrations
$i\colon\mathord{\partial\mathord{\Delta}[n]}\rightarrowtail\mathord{\Delta}[n]$
are in $\mathcal{G}$.

###### Proof.

Under the equivalence of Lemma 2.8, the pullback functor
$i^{*}\colon\mathsf{s}\mathcal{E}\mathbin{\downarrow}\mathord{\Delta}[n]\to\mathsf{s}\mathcal{E}\mathbin{\downarrow}\mathord{\partial\mathord{\Delta}[n]}$
coincides with the functor

$\mathsf{s}\mathcal{E}^{\Delta^{{\mathord{\mathrm{missing}}{op}}}\mathbin{\downarrow}\mathord{\Delta}[n]}\to\mathsf{s}\mathcal{E}^{\Delta^{{\mathord{\mathrm{missing}}{op}}}\mathbin{\downarrow}\mathord{\partial\mathord{\Delta}[n]}}$

obtained by reindexing along the sieve inclusion:
$\Delta^{{\mathord{\mathrm{missing}}{op}}}\mathbin{\downarrow}\mathord{\partial\mathord{\Delta}[n]}\to\Delta^{\mathord{\mathrm{missing}}{op}}\mathbin{\downarrow}\mathord{\Delta}[n]$,
hence its right adjoint, if it exists, is the right Kan extension along this
sieve inclusion. So if we prove that the pointwise right Kan extension along
this sieve inclusion exists, it will coincide with $i_{*}$. If
$\mathcal{F}\in\mathsf{s}\mathcal{E}\mathbin{\downarrow}\mathord{\partial\mathord{\Delta}[n]}$,
then this pointwise right Kan extension evaluated at
$\mathord{\Delta}[k]\to\mathord{\Delta}[n]\in\Delta\mathbin{\downarrow}\mathord{\Delta}[n]$
is given by the limit

$(i_{*}\mathcal{F})([k])=\lim_{p\in P}\mathcal{F}(p)\text{,}\quad\text{ where
}P=\left\\{\leavevmode\hbox to80.29pt{\vbox