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 [Reference Cisinski12, Reference Gepner and Kock23, Reference Joyal, Tierney, Davydov, Batanin, Johnson, Lack and Neeman35, Reference Lurie37, Reference Rezk42, Reference Szumiło53]. This line of research led to the reformulation of significant parts of modern homotopy theory in terms of higher category theory and the development of higher topos theory [Reference Lurie37, Reference Toën and Vezzosi54] and is of great importance for Homotopy Type Theory and the Univalent Foundations programme [Reference Awodey and Warren1, Reference van den Berg and Moerdijk4, Reference Gepner and Kock23, Reference Kapulkin and LeFanu Lumsdaine36, Reference Shulman49]. Central to these developments are model structures on categories of simplicial objects: that is, functor categories of the form $\mathsf{s}\mathcal {E} = [\Delta ^{\mathord {\mathrm {op}}}, \mathcal {E}]$ , where $\mathcal {E}$ is a category, as considered in [Reference Quillen40, Section II.4], [Reference Goerss and Jardine24, Chapter II], [Reference Christensen and Hovey11, Theorem 6.3] and [Reference Hörmann29]. In particular, the category of simplicial sets equipped with the Kan–Quillen model structure [Reference Quillen40] 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 [Reference Rezk43] and the Joyal–Jardine model structure [Reference Brown5, Reference Joyal34, Reference Jardine32] can be seen as presentations of $\infty $ toposes and their hypercompletions, respectively [Reference Dugger, Hollander and Isaksen15, Reference Lurie37].
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: that is, a category with finite limits and countable coproducts, where the latter are required to be van Kampen colimits [Reference Carboni, Lack and Walters7, Reference Rezk43]. The effective model structure is defined so that when $\mathcal {E} = \mathsf{Set}$ , we recover the Kan–Quillen model structure on simplicial sets [Reference Quillen40]. 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 [Reference Gambino and Sattler21, Reference Gambino, Sattler and Szumiło22, Reference Henry26, Reference Sattler47] could be developed further so as to obtain a new model structure on categories of simplicial sheaves. Indeed, in [Reference Gambino, Sattler and Szumiło22, Reference Henry26], 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 [Reference Riehl44, 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, has colimits that satisfy descent and is locally Cartesian closed, but it is neither a higher Grothendieck topos [Reference Lurie37] nor a higher elementary topos in the sense of [Reference Rasekh41, Reference Shulman48], since its 0truncation 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 [Reference Rezk43] 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 [Reference Carboni and Vitale8]. There, it is known that the exact completion of a (Grothendieck) topos need not be a (Grothendieck) topos [Reference Menni39]. 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 [Reference Hu and Tholen30]. 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 [Reference Hyland, Troelstra and van Dalen31], which can be described as an exact completion [Reference Carboni6].
Finally, our results may be of interest also in Homotopy Type Theory, since they help to clarify how the simplicial model of Univalent Foundations [Reference Kapulkin and LeFanu Lumsdaine36], in which types are interpreted as Kan complexes, is related to the setoid model of type theory [Reference Hofmann28], in which types are interpreted as types equipped with an equivalence relation, by showing how not only the latter [Reference Emmenegger and Palmgren18] 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
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 [Reference Gambino, Sattler and Szumiło22, Reference Henry26, Reference Sattler47] 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 by the case $\mathcal {E} = \mathsf{Set}$ in [Reference Gambino, Sattler and Szumiło22], several new ideas are necessary to implement it to the general case, as we explain below. This strategy involves 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 in the category of Kan complexes (assuming only that $\mathcal {E}$ has finite limits). Second, 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. Third, we show that weak equivalences (as determined by the two weak factorisation systems) satisfy 2outof3 by proving the socalled 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 the 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. Second, 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). Third, 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 from the one used in [Reference Gambino, Sattler and Szumiło22, Reference Henry26, Reference Sattler47]. 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 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 [Reference Gambino, Sattler and Szumiło22, Reference Henry26]. Actually, the use of enriched category theory here, especially for expressing stronger versions of the lifting properties usually phrased in terms of the mere existence of diagonal fillers, makes explicit some of the informal conventions adopted in [Reference Gambino, Sattler and Szumiło22, Reference Henry26] when treating the case $\mathcal {E} = \mathsf{Set}$ . Also, the proofs in [Reference Gambino, Sattler and Szumiło22, Reference Henry26] 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 socalled internal logic of $\mathcal {E}$ [Reference Johnstone33, Section D1.3]. Even when $\mathcal {E}$ is a Grothendieck topos, carrying out the proofs in the internal language of $\mathcal {E}$ [Reference Mac Lane and Moerdijk38, 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 into 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 properties. 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 [Reference Gambino, Sattler and Szumiło22, Reference Henry26], 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 also be developed in a constructive setting (see footnotes and Appendix A for details).
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
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
For $X, Y \in \mathsf{s}\mathcal {E}$ , the simplicial homobject is then defined by lettingFootnote ^{1}
This makes $\mathsf{s}\mathcal {E}$ into a $\mathsf{sSet}$ enriched category so that the formula in equation (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
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
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 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
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 . 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 $X \colon \Delta ^{\mathord {\mathrm {op}}} \to \mathcal {E}$ along the inclusion of $\Delta $ into the category of finite simplicial sets. Second, 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 socalled Leibniz construction [Reference Riehl and Verity45] to the twovariable functor $\operatorname {ev}$ : that is, 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
Remark 1.2. We adopt the convention of prefixing with ‘pullback’ (or ‘pushout’) the name of a twovariable 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,
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
in $\mathcal {E}$ has a section, for all $n \ge k \ge 0$ and $n> 0$ . For $Y = 1$ , this means that the morphism
has a section, for all $n \ge k \ge 0$ 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 equation (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
is a (trivial) fibration in $\mathsf{sSet}$ .
Proof. Note that the functors
and
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
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
Lemma 1.5.

(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.

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

(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 $g f \colon X \to Z$ 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., [Reference Quillen40, Theorem II.3.3]) and transfer to $\mathsf{s}\mathcal {E}$ using Proposition 1.4.Footnote ^{2} Note that transferring (i) from $\mathsf{sSet}$ to $\mathsf{s}\mathcal {E}$ relies on the fact that 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
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 [Reference Gambino, Sattler and Szumiło22, Section 2.6].
Theorem 1.7. Let $\mathcal {E}$ be a 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.

(F1) $\mathsf{s}\mathcal {E}$ has a terminal object, and all objects are fibrant, which follows directly from the definitions.

(F2) 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.

(F3) Every morphism factors as a weak equivalence followed by a fibration. By [Reference Brown5, p. 421, Factorization lemma] it suffices to construct a path object: that is, 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 to this factorisation gives
$$ \begin{align*} {\operatorname{Hom}}_{\mathsf{sSet}}(E, X) \to \mathord{\Delta}[1] \mathop{\pitchfork} {\operatorname{Hom}}_{\mathsf{sSet}}(E, X) \to {\operatorname{Hom}}_{\mathsf{sSet}}(E, X) \times {\operatorname{Hom}}_{\mathsf{sSet}}(E, X) \end{align*} $$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, for example, [Reference Goerss and Jardine24, p. 43]. Hence $X \to \mathord {\Delta }[1] \mathop {\pitchfork } X \to X \times X$ is also such factorisation in $\mathsf{s}\mathcal {E}$ . 
(F4) Weak equivalences satisfy 2outof6, 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 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 homobject ${\operatorname {Hom}}_{\mathsf{sSet}}((A, f), (B, g))$ 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 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 $\mathsf{s}\mathcal {E}$ , that is, part (i) of Lemma 1.5, and the stability of fibrations and trivial fibration under pullback, that is, 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 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 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 [Reference Lurie37, Reference Rezk43] 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

(i) Universal, if it is preserved by pullbacks: that is, if for every map $X_\star \to Y_\star $ , $X_\star $ is the colimit of the induced diagram $X_d = X_\star {\operatorname *{\times }}_{Y_\star } Y_d$ ;

(ii) Effective, if given a Cartesian natural transformation $X \to Y$ , the diagram X has a colimit $X_\star $ , and all the squares

(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 pseudofunctor $\mathcal {E}^{\mathord {\mathrm {op}}} \to \mathsf{Cat}$ 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 pseudolimit $\lim _{d \in D} (\mathcal {E} \mathbin \downarrow Y_d)$ .
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:
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: that is, 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: that is, if there is a morphism $X \to 0$ , then X is initial itself. Instead, a coproduct is van Kampen if and only if it is universal and disjoint: that is, $Y_d \times _{Y_\star } Y_{d'}$ is initial for $d \neq d'$ . This can be seen inspecting the proof of [Reference Carboni, Lack and Walters7, 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 Dindexed 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 $X_c(d) \to Y_c(d)$ that is Cartesian in the Cdirection, which in turn is the same as a Cindexed diagram $X_{\bullet } \in \mathcal {E}^D$ that is Cartesian over $Y_{\bullet }$ , hence proving the lemma.
We now recall the definition of various kinds of lextensive categories [Reference Carboni, Lack and Walters7].
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

(i) Lextensive if it is $\omega $ lextensive: that is, finite coproducts exist and are van Kampen colimits,

(ii) Countably lextensive if it is $\omega _1$ lextensive: that is, countable coproducts exist and are van Kampen colimits,

(iii) Completely lextensive if it is $\alpha $ lextensive for all $\alpha $ : that is, all small coproducts exist and are van Kampen colimits.
Example 2.5. There are numerous examples of lextensive categories.

(i) Any presheaf category is completely lextensive. In particular, for any group G, the category of Gsets is countably lextensive.

(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.

(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, and so on.

(iv) The category of affine schemes is lextensive, and the category of schemes is completely lextensive.

(v) The category of countable sets is countably lextensive.

(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 Sindexed family of objects of $\mathcal {E}$ . A morphism $(S, (X_s)) \to (S', (X^{\prime }_{s'}))$ consists of a function $f \colon S \to S'$ 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
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
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 equation (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, the 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 that takes values in $\alpha $ small sets, then there is an equivalence of categories
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 $\mathcal {E}^{D \mathbin \downarrow S} \to \mathcal {E}^D \mathbin \downarrow \underline {S}$ that sends a functor $F \colon D \mathbin \downarrow S \to \mathcal {E}$ to the functor $V \colon D \to \mathcal {E}$ defined by
It comes with an obvious map to $\underline {S}$ , which was defined as
. 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:
These two adjoint functors are equivalences. Indeed, the counit of this adjunction is an isomorphism by the universality of coproducts, and the unit is an isomorphism by the effectivity of coproducts.
We now turn our attention to the class of complemented inclusions. These will be useful for the 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: that is, a morphism $j \colon C \to B$ 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 [Reference Gambino, Sattler and Szumiło22, Reference Henry26]) called decidable inclusions in reference to the notion of decidability in constructive logic.
Lemma 2.9.

(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 pseudofunctors) that preserve finite coproducts and thus are van Kampen colimits.

(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 pseudofunctors) 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 $A \sqcup D$ . 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 (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 pseudofunctor. Since (finite or countable) coproducts are assumed van Kampen, so are the presented colimits.
Lemma 2.10. Assume $\mathcal {E}$ is lextensive.

(i) Complemented subobjects in $\mathcal {E}$ are closed under finite unions.

(ii) Complemented inclusions in $\mathcal {E}$ are closed under finite limits: that is, 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 [Reference Gambino, Sattler and Szumiło22, Lemma 2.1.7] 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 that 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
is invertible. We decompose this map into three factors:
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.

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

(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 $\mathcal {E}^D$ .
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 diagram of subobjects of X closed under intersection: that is, 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:
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
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 equation (5). This statement will be needed in Section 8.
Lemma 2.15. Let D be a small category. Let $Y \colon C \to [D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ 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 Cindexed diagram of arrows in $[D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ .
Let $q \colon A \to B$ be a map in $[D^{\mathord {\mathrm {op}}}, \mathsf{Set}]$ with B representable such that $[D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ 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 $[D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ to the sequence of arrows
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 the 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 [Reference Riehl44]. 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
Given $X, Y \in \mathcal {E}^D$ , we then define the homobject ${\operatorname {Hom}}_{\operatorname {\mathsf{Psh}} \mathcal {E}}(X, Y) \in \operatorname {\mathsf{Psh}} \mathcal {E}$ by letting
This makes $\mathcal {E}^D$ into a $\operatorname {\mathsf{Psh}} \mathcal {E}$ enriched category, so that the formula in equation (9) 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 system. 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
When the relevant homobjects 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
Again, if the objects are representables, we have also an induced pullback hom in $\mathcal {E}$ , which has the form
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 equation (10) 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 equation (11) 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 homobjects 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 the 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) Ifibration is a morphism with the enriched right lifting property with respect to I.

• An (enriched) Icofibration is a morphism with the enriched left lifting property with respect to Ifibrations.
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 equation (8), we may simplify the lifting problem (assuming some technical conditions hold). Indeed, the pullback hom equation (11) reduces to a pullback evaluation. We record this in the next couple of statements, which are phrased using $D^{{\mathord {\mathrm {op}}}}$ instead of D in order to exploit the language of representable functors. We make use of the evaluation functor ${\operatorname {ev}}_K \colon [D^{\mathord {\mathrm {op}}}, \mathcal {E}] \to \mathcal {E}$ defined for finite colimits K of representables by letting
This generalises the evaluation functor defined in equation (4), which is the case $D = \Delta $ . As in Remark 1.1, we may equivalently view ${\operatorname {ev}}_K(X)$ as the Kweighted limit of X, which implies that $\operatorname {ev}$ is a (partial) twovariable functor.
Lemma 3.3. Let $K \in [D^{\mathord {\mathrm {op}}}, \mathsf{Set}]$ be levelwise countable.

(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 $d \in D$ .

(ii) Assume that K is a finite colimit of representables. Then the hompresheaf ${\operatorname {Hom}}_{\operatorname {\mathsf{Psh}}{\mathcal {E}}}(\underline {K}, X)$ is representable for $X \in [D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ , and we have an isomorphism ${\operatorname {Hom}}_{\mathcal {E}}(\underline {K}, X) \mathrel {\cong } {\operatorname {ev}}_K(X)$ , natural in K and $X \in [D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ .
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 Kweighted limit of X: that is, ${\operatorname {ev}}_K(X)$ . 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 $[D^{\mathord {\mathrm {op}}}, \mathsf{Set}]$ between objects that are levelwise countable and finite colimits of representables, and let $p \colon X \to Y$ be a map in $[D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ . Then the following are equivalent:

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

(ii) the pullback evaluation $\widehat {\operatorname {ev}}_i(p)$ is a split epimorphism in $\mathcal {E}$ .
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 .
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 $\mathcal {E}^D$ :

• 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.

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

(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 hompresheaves 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 Icofibrations and Ifibrations 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
We say that p satisfies the $X'$ partial enriched right lifting property with respect to i if there is a lift in the diagram
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 equation (9).
Lemma 3.8. Levelwise complemented inclusions in $\mathcal {E}^D$ are closed under:

(i) for all $E \in \mathcal {E}$ ;

(ii) Countable coproducts;

(iii) Pushouts along arbitrary morphisms;

(iv) Sequential colimits;

(v) Retracts.
Moreover, the colimits of parts (ii), (iii) and (iv) are preserved by for all $E \in \mathcal {E}$ .
Proof. The functor 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 follows from distributivity of product over coproduct in complemented categories: if $A \to A \sqcup B$ is a complemented inclusion, then its image under 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 . Stability under pushout and sequential composition follows from Lemma 2.9. The fact that they are preserved by 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:

(i) Tensors by objects of $\mathcal {E}$ ,

(ii) Countable coproducts,

(iii) Pushouts,

(iv) Colimits of sequences,

(v) Retracts.
Proof. For $X \in \mathcal {E}^D$ , the functor 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 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:

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

(ii) preserves colimits of sequences of levelwise complemented inclusions;

(iii) 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 $D^{{\mathord {\mathrm {op}}}}$ 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 $\underline {A} \in [D^{\mathord {\mathrm {op}}}, \mathcal {E}]$ 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, 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 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 twovariable 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 Ifibration 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 Ifibration.
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
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
and
the latter by universality of sequential colimits of complemented inclusions in $\mathcal {E}$ (Lemma 2.9). Thus we obtain a diagram
where the bottom map is an identity: that is, these lifts form a section that exhibits ${\operatorname *{colim}}_k X_k \to ~Y$ as an Ifibration.
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
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
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 Icofibrations and Ifibrations 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
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 Ifibration. Using Lemma 3.9, we show that $X_0 \to {\operatorname *{colim}}_k X_k$ is an Icofibration.
Remark 3.15. Essentially 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 $\mathcal {C}$ : that is, a category equipped with a tensor functor that is associative in the sense that the functor $\mathcal {E} \to \operatorname {\mathsf{End}} \mathcal {C}$ , given by , 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 $\mathcal {E}^J$ that 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 Icell complex is a morphism of $\mathcal {E}^D$ that is a sequential colimit of maps $X \to Y$ arising as pushouts
for some family $(E_i)_{i \in I}$ of objects of E.
Below, we simply speak of an Icell complex for brevity.
Proposition 3.17. Under the hypotheses of Theorem 3.14, a morphism of $\mathcal {E}^D$ is an Icofibration if and only if it is a codomain retract of an Icell complex. In particular, every Icofibration is a levelwise complemented inclusion.
Proof. A retract of an Icell complex is an Icofibration by Lemma 3.9. It is furthermore a levelwise complemented inclusion by Lemma 3.8. Conversely, let $X \to Y$ be an Icofibration, and consider the factorisation $X \to X' \to Y$ defined in the proof of Theorem 3.14. Then $X \to X'$ is an Icell complex by construction. Moreover, $X \to Y$ has the $\operatorname {\mathsf{Psh}} \mathcal {E}$ enriched left lifting property with respect to $X' \to Y$ , and, in particular, it has the ordinary left lifting property (by evaluating the hompresheaves at the terminal object). Thus there is a lift in the diagram
that exhibits $X \to Y$ as a codomain retract of $X \to X'$ .
Lemma 3.18. In the setting of Theorem 3.14, the following hold.

(i) Consider a countable family of maps $f_k$ in the arrow category of $\mathcal {E}^D$ . If $f_k$ is an Ifibration for all k, then so is the coproduct . When $\mathcal {E}$ is $\alpha $ lextensive, the same holds for $\alpha $ coproducts.

(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 Ifibration for $k = 0, 1, 01$ , then so is the pushout $\operatorname *{colim} f$ .

(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 Ifibration 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 Ifibration 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 Dshaped 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.

(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.

(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.

(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 $A_\beta \to A_{\beta '}$ for $\beta \leq \beta ' \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 [Reference Riehl and Verity45, Sections 4 and 5], in particular [Reference Riehl and Verity45, Lemma 4.8] for part (i) and [Reference Riehl and Verity45, Lemma 5.7] for parts (ii) and (iii).
Lemma 3.20. Let $F, G \colon \mathcal {E}^{D} \to \mathcal {E}^{D'}$ be two functors that preserves levelwise complemented maps, their pushouts and their sequential compositions. We assume that F and G are equipped with isomorphisms
natural in $E \in \mathcal {E}$ and $X \in \mathcal {E}^{D}$ (respectively, $X \in \mathcal {E}^{D'}$ ), and let $\lambda \colon F \to G$ be a natural transformation compatible with these isomorphisms. Let $I_D \subseteq (\mathcal {E}^{D})^{[1]}$ and $I_{D'} \subseteq (\mathcal {E}^{D'})^{[1]}$ 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 $I_{D'}$ cofibration, then for each $I_{D}$ cofibration i, the pushout application $\widehat {\operatorname {app}}(\lambda , i)$ is an $I_{D'}$ 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 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 $I_{D'}$ cofibration, it follows that the map of the form $E \times i$ are also sent to $I_{D'}$ 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 to a $I_{D'}$ cofibration. Finally, as is a functor, it preserves retract, and so retracts of such maps are also sent to $I_{D'}$ 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 Icofibration for all $i \in I$ , then $f \mathbin {\widehat {\times }} j$ is an Icofibration for all Icofibrations f.
Proof. We apply Lemma 3.20 to the natural transformation of endofunctors on $\mathcal {E}^D$ . Let us check the needed preservation properties of the endofunctor on $\mathcal {E}^D$ for $Z \in \mathcal {E}$ . Preservation of levelwise complemented inclusions follows from the preservation of complemented inclusions in $\mathcal {E}$ under the 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 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 the 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 equation (8) to the sets of boundary inclusions and horn inclusions in equation (6): that is,
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 $\mathsf{s}\mathcal {E}$ .

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

(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