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 $E = X(A) \times _{Y(A)} Y(B)$ ).

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.

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

2. (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.

3. (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 $\mathsf{s}\mathcal {E}$ .

4. (F4) Weak equivalences satisfy 2-out-of-6, which follows since this property holds in $\mathsf{sSet}$ .

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.

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.

Proof. The functors preserve homotopies and hence also homotopy equivalences. Thus the conclusion follows from the fact that homotopy equivalences are weak equivalences in $\mathsf{sSet}$ .

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:

$$\begin{align*}\operatorname*{colim} \colon \lim_d (\mathcal{E} \mathbin\downarrow Y_d) \leftrightarrows \mathcal{E} \mathbin\downarrow Y_\star \colon P \text{.}\end{align*}$$

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.

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 $X_c(d) \to Y_c(d)$ that is Cartesian in the C-direction, which in turn is the same as a C-indexed diagram $X_{\bullet } \in \mathcal {E}^D$ that is Cartesian over $Y_{\bullet }$ , hence proving the lemma.

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 $1 \times X \mathrel {\cong } X$ .

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.

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.

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 pseudo-functor. Since (finite or countable) coproducts are assumed van Kampen, so are the presented colimits.

Proof. The proof of [Reference Gambino, Sattler and Szumiło22, Lemma 2.1.7] applies verbatim.

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

$$\begin{align*}\mathop{\operatorname*{colim}}_{k \in \omega} A_k \times_{C_k} B_k \to \operatorname*{colim} A \times_{\operatorname*{colim} C} \operatorname*{colim} B \end{align*}$$

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).

Proof. This follows immediately from Lemmas 2.3 and 2.9.

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.

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

$$ \begin{align*} \mathop{\operatorname*{colim}}_{p \in P} A_{p \cap q} = \mathop{\operatorname*{colim}}_{p \leqslant q} A_p = A_q. \end{align*} $$

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.

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.

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: that is, ${\operatorname {ev}}_K(X)$ . This exists in our setting for K, a finite colimit of representables.

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

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.

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.

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.

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 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).

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

$$ \begin{align*} \mathop{\operatorname*{colim}}_k {\operatorname{Hom}}_{\mathcal{E}}(B, X_k) & = {\operatorname{Hom}}_{\mathcal{E}}(B, \mathop{\operatorname*{colim}}_k X_k) \end{align*} $$

and

$$ \begin{align*} \mathop{\operatorname*{colim}}_k {\operatorname{Prob}}_{\mathcal{E}}(i, p_k) & = \mathop{\operatorname*{colim}}_k \left( {\operatorname{Hom}}_{\mathcal{E}}(A, X_k) \times_{{\operatorname{Hom}}_{\mathcal{E}}(A, Y)} {\operatorname{Hom}}_{\mathcal{E}}(B, Y) \right) \\ & = \left( \mathop{\operatorname*{colim}}_k {\operatorname{Hom}}_{\mathcal{E}}(A, X_k) \right) \times_{{\operatorname{Hom}}_{\mathcal{E}}(A, Y)} {\operatorname{Hom}}_{\mathcal{E}}(B, Y) \\ & = {\operatorname{Hom}}_{\mathcal{E}}(A, \mathop{\operatorname*{colim}}_k X_k) \times_{{\operatorname{Hom}}_{\mathcal{E}}(A, Y)} {\operatorname{Hom}}_{\mathcal{E}}(B, Y) \\ & = {\operatorname{Prob}}_{\mathcal{E}}(i, \mathop{\operatorname*{colim}}_k p_k) \text{,} \end{align*} $$

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 I-fibration.

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.

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 I-fibration. Using Lemma 3.9, we show that $X_0 \to {\operatorname *{colim}}_k X_k$ is an I-cofibration.

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 $X \to X' \to Y$ defined in the proof of Theorem 3.14. Then $X \to X'$ 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 $X' \to Y$ , 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

that exhibits $X \to Y$ as a codomain retract of $X \to X'$ .

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.

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).

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.

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.

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, $I_{\mathsf{s}\mathcal {E}}$ ).

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.

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.

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

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

is a pushout. These statements are justified analogously to the proofs of [Reference Gambino, Sattler and Szumiło22, Lemma 3.3.1, Corollary 3.3.3]. 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.

Proof. We proceed by induction on the height of D. For height $0$ , note that D is 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 $D'$ 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:

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 $F|_{D'}$ . 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.

Proof. We use from Proposition 4.3 that cofibrations are the same as Reedy complemented inclusions. As in [Reference Riehl and Verity45], 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 $\partial \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ for the subobject of $\Delta _{-}^{\mathord {\mathrm {op}}}[{m}] = \Delta ([m], -)$ in $[\Delta _-, \mathsf{Set}]$ consisting of the non-identity maps. Recall that the coboundary inclusion $\partial \Delta ^{\mathord {\mathrm {op}}}[m] \to \Delta ^{\mathord {\mathrm {op}}}[{m}]$ arises as left Kan extension along $\Delta _- \to \Delta $ of the coboundary inclusion $\partial \Delta _{-}^{\mathord {\mathrm {op}}}[{m}] \to \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ . For working with weighted colimits, we recall that the left Kan extension on the side of the weight corresponds to the 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 $\partial \Delta _-^{\mathord {\mathrm {op}}}[{k}] \to \Delta ^{\mathord {\mathrm {op}}}[{k}]$ 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 $\varnothing \to \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ and hence a complemented inclusion. Every degeneracy operator is a split epimorphism. It follows that $\Delta _{-}^{\mathord {\mathrm {op}}}[{n}] \to \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ 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 $\Delta _{-}^{\mathord {\mathrm {op}}}[{n}] \to \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ 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 $\partial \Delta _{-}^{\mathord {\mathrm {op}}}[{m}] \to \Delta _{-}^{\mathord {\mathrm {op}}}[{m}]$ . Let $\partial (\Delta _{-}^{\mathord {\mathrm {op}}} \downarrow [m])$ denote the opposite of the poset of non-identity degeneracy operators with source $[m]$ . Consider the diagram $F \colon \partial (\Delta _-^{\mathord {\mathrm {op}}} \downarrow [m]) \to \mathcal {E} \downarrow B_m$ sending a degeneracy operator 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 an underlying map. It will suffice to show that this colimit is subterminal. 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 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 the union of the subobjects $A_m \to B_m$ and $B_n \to B_m$ .

Since $\Delta $ is elegant [Reference Bergner and Rezk2], given non-identity degeneracy operators

for $i = 1, 2$ , we have an absolute pushout

in $\Delta $ with

and

degeneracy operators. Note that

is distinct from the identity. By absoluteness, we obtain a pullback

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

$$ \begin{align*} (A_m \cup B_{n_1}) \cap (A_m \cup B_{n_2}) &= ((A_m \cup B_{n_1}) \cap A_m) \cup ((A_m \cup B_{n_1}) \cap B_{n_2}) \\&= A_m \cup ((A_m \cup B_{n_1}) \cap B_{n_2}) \\&= A_m \cup (B_{n_1} \cap B_{n_2}) \\&= A_m \cup B_k. \end{align*} $$

We obtain, in subobjects of $B_m$ , that F at 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.

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.

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

Proof. For part (i), recall that cofibrations in $\mathsf{sSet}$ are closed under pushout product.Footnote ³ 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 $\mathsf{sSet}$ byFootnote ⁴ [Reference Gabriel and Zisman20, Proposition IV.2.2], and thus it carries over to $\mathsf{s}\mathcal {E}$ by the argument of part (i).

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:

$$\begin{align*}\underline{K \times \mathord{\Delta}[m]}_n \times X_m = \underline{K_n} \times \underline{\operatorname{Hom}([m],[n])} \times X_m \text{.} \end{align*}$$

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: that is, a morphism $\underline {K} \times X \to Y$ in $\mathsf{s}\mathcal {E}$ .

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 $\operatorname {\mathsf{Psh}} \mathcal {E}$ .

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 $\partial \mathord {\Delta }[0] \to \mathord {\Delta }[0]$ : that is, the map $\varnothing \to E$ in $\mathsf{s}\mathcal {E}$ . Part (ii) holds since $S \mapsto \underline {S}$ preserves cofibrations by Lemma 5.1.Footnote ⁵ Finally, for part (iii), if 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 [Reference Henry26, 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.

Proof. Consider a pullback square of simplicial objects:

We check that $S' \to S$ 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 $S' \to S'$ is a levelwise complemented inclusion. Given any degeneracy operator

, as it is a split epimorphism and $S \to B$ is a monomorphism, the naturality square

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, and it hence follows that we have a pullback square

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

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.

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.

Proof. This follows from [Reference Johnstone33, Lemma A1.5.2 (i)] and (the proof of) [Reference Johnstone33, Corollary A1.5.3].

Proof. This follows from [Reference Johnstone33, 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

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

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 $F_d \dashv R_d$ .

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 (14) above.

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

Then the two squares are pullbacks (because i is a monomorphism for the one on the right), and 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

$$\begin{align*}j: X \rightarrow X \sqcup_A B \text{,} \end{align*}$$

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

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

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

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.

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

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

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.

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

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

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

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

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

obtained by reindexing along the sieve inclusion: $\Delta ^{{\mathord {\mathrm {op}}}} \mathbin \downarrow \mathord {\partial \mathord {\Delta }[n]} \to \Delta ^{\mathord {\mathrm {op}}} \mathbin \downarrow \mathord {\Delta }[n]$ ; 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

This is a limit over an infinite category, so it is not guaranteed to exist, but the category P has a finite reflective category given by the objects such that the map $\mathord {\Delta }[a] \rightarrow \mathord {\Delta }[k]$ is injective, with the reflection given by the image factorisation of this map, and hence this limit coincides with

which is a finite limit, and hence exists, which proves the existence of $i_*$ .

Next, we assume that $\mathcal {F}$ is cofibrant, and we will show that $i_* \mathcal {F}$ is cofibrant. That is, given a degeneracy

, the action $i_* \mathcal {F} ([k']) \rightarrow i_* \mathcal {F}( [k])$ is a complemented inclusion (by Theorem 4.6). The map $i_* \mathcal {F}([k]) \rightarrow \mathord {\Delta }[n] ([k])$ gives a decomposition of the map above into a coproduct indexed by all the map $\alpha \colon [k] \to [n]$ , so it is enough to show that the fiber above each such map is a complemented inclusion. The fiber over such a map $\alpha $ of $i_* \mathcal {F}( [k])$ is by definition of $i_*$ the object classifying maps $P \rightarrow \mathcal {F}$ over $\mathord {\partial \mathord {\Delta }[n]}$ , where P is the pullback square

The fiber of $i_* \mathcal {F} ([k'])$ over $\alpha $ is described similarly with $P'$ the pullback of $\mathord {\Delta }[k'] \rightarrow \mathord {\Delta }[n]$ , and the map we are interested in is induced by the map $P' \rightarrow P$ obtained as the pullback of $\mathord {\Delta }[k'] \rightarrow \mathord {\Delta }[k]$ . But it follows from [Reference Henry26, Proposition 3.1.11] that a pullback of a degeneracy operator is an iterated pushout of degeneracy operators, in this case a finite such iterated pushout as $P'$ is finite. As $\mathcal {F}$ is cofibrant, this decomposes $\mathcal {F}(P) \rightarrow \mathcal {F}(P')$ as a composite of complemented inclusions and hence concludes the proof.

Proof of Theorem 6.5

We show that all cofibrations with cofibrant domain are in $\mathcal {G}$ . By Lemma 4.4, it suffices to show that the generating cofibrations are in $\mathcal {G}$ and that $\mathcal {G}$ is closed under operations appearing in a cell complex. The case of generators is Proposition 6.10. Closure under tensoring by objects of $\mathcal {E}$ is Proposition 6.9, closure under pushout (along maps with cofibrant target) is Proposition 6.7 and closure under sequential composition is Proposition 6.8.