# Local fibrations and Kan's subdivision functor *Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).* In a topos the horn-filling condition of Kan complexes becomes a local condition: a map of simplicial sheaves is a *fibration* if every horn over an object, with a simplex of the base over it, can be filled after passing to a covering of that object. This lesson develops these local fibrations and constructs, for every simplicial sheaf, a monomorphic quasi-isomorphism into a Kan simplicial sheaf. The construction is Kan's functor \(\operatorname{Ex}^\infty\), applied to sections and sheafified. It commutes with finite limits, with inverse images and with stalks, and it carries fibrations to fibrations. Most of the work concerns simplicial sets. The lesson proves Kan's theorems there: \(\operatorname{Ex}^\infty X\) is a Kan complex, the embedding \(X\to\operatorname{Ex}^\infty X\) is a quasi-isomorphism, and \(\operatorname{Ex}\) carries Kan fibrations to Kan fibrations. The proofs use the subdivision of simplices, the last vertex map and the gluing results of the lesson on the simplicial derived category; they are combinatorial. Two consequences complete the homotopy theory of simplicial sets used in this course: a Kan fibration that is a quasi-isomorphism lifts against every monomorphism, whatever its base, and a monomorphism is anodyne exactly when it is a quasi-isomorphism. **Prerequisites.** [*The simplicial derived category of a topos*](the-simplicial-derived-category-of-a-topos.md) (Conventions, Lemma 1.1, Lemma 1.2, Section 2, Corollary 3.3, and the remark at the start of Section 3 on homotopy sheaves of simplicial presheaves); [*Homotopy sheaves and the free abelian Whitehead theorem*](homotopy-sheaves-and-the-free-abelian-whitehead-theorem.md), Sections 1 and 2; [*Simplicial sets, nerves and Kan complexes*](course:AG-AS/simplicial-sets-nerves-and-kan-complexes), Sections 1.2 and 5.3 (the expression (1.8), anodyne extensions, Lemma 5.3a with the proof of (5.13), Lemmas 5.3c–5.3e, the factorization (5.19)); [*Décalage, nerves and the homotopy type of a category*](decalage-nerves-and-the-homotopy-type-of-a-category.md), Proposition 2.1 and Corollary 2.2; [*Topoi, morphisms and points*](course:ag-etale-cohomology/topoi-morphisms-and-points), Sections 4 and 5. ## Conventions As in the lesson on the simplicial derived category: \(T=\operatorname{Sh}(\mathcal C)\) for a small site \(\mathcal C\), simplicial sheaves are simplicial objects of \(T\), a simplicial set is regarded as a constant simplicial sheaf, and \(K\times X\) has the terms \(\coprod_{K_n}X_n\). For a finite simplicial set \(K\), \(\operatorname{Hom}(K,X)=\underline{\operatorname{Homs}}(K,X)_0\) is the sheaf of maps \(K\to X\); by Lemma 1.1 (3) there, its sections over \(U\in\mathcal C\) are the maps \(K\times h_U\to X\). Horns are the inclusions \(\Lambda^k[n]\subset\Delta[n]\), \(n\geq1\), \(0\leq k\leq n\). Quasi-isomorphisms are those of Definition 2.1 of the lesson on homotopy sheaves; for simplicial sets they are described in Section 2 of the lesson on the simplicial derived category. For monomorphisms \(i\colon K\to L\) and \(j\colon Y\to X\) of simplicial sets, the *pushout product* \(i\mathbin{\square}j\) is the inclusion of \((L\times Y)\cup(K\times X)\) into \(L\times X\). For a poset \(R\), \(N(R)\) is its nerve; its \(m\)-simplices are the order-preserving maps \([m]\to R\), and its nondegenerate simplices are the strictly increasing chains, each determined by its set of elements. ## 1. Strict anodyne extensions **Definition 1.1.** A class of monomorphisms of simplicial sets is *finitely saturated* if it contains the isomorphisms and is stable under pushouts, retracts in the category of arrows, finite coproducts and finite composites. The finitely saturated class *generated* by a class \(\mathcal N\) is the intersection of the finitely saturated classes containing \(\mathcal N\). A *strict anodyne extension* is a member of the finitely saturated class generated by the horns. Strict anodyne extensions are anodyne. The operation \(-\mathbin\square j\), for a fixed monomorphism \(j\), carries isomorphisms to isomorphisms, retracts to retracts and finite coproducts to finite coproducts. It carries a pushout of \(i\) to a pushout of \(i\mathbin\square j\), because products with a simplicial set preserve pushouts. For a composite \(i'i\colon K\to L\to M\), \((i'i)\mathbin\square j\) is the composite of the inclusion of \((M\times Y)\cup(K\times X)\) into \((M\times Y)\cup(L\times X)\), a pushout of \(i\mathbin\square j\), and of \(i'\mathbin\square j\). The pushout product is associative: \((i\mathbin\square i')\mathbin\square j\) and \(i\mathbin\square(i'\mathbin\square j)\) are both the inclusion of the union of the three products obtained by replacing one factor of \(L\times L'\times X\) by its subobject. **Lemma 1.2.** Let \(j\colon Y\subset X\) be a monomorphism of simplicial sets with \(X\) finite. 1. For either vertex \(\epsilon\) of \(\Delta[1]\), the map \((\{\epsilon\}\subset\Delta[1])\mathbin\square j\), the inclusion of \((\Delta[1]\times Y)\cup(\{\epsilon\}\times X)\) into \(\Delta[1]\times X\), is strict anodyne. 2. If \(i\) is strict anodyne, so is \(i\mathbin\square j\). **Proof.** (1) The finitely many nondegenerate simplices of \(X\) not in \(Y\), attached in order of dimension, write \(j\) as a finite composite of pushouts of boundary inclusions \(\partial\Delta[m]\subset\Delta[m]\) (expression (1.8) of the lesson on Kan complexes). The operation \((\{\epsilon\}\subset\Delta[1])\mathbin\square-\) carries pushouts to pushouts and composites to composites of pushouts, as above. For \(\partial\Delta[m]\subset\Delta[m]\), the proof of (5.13) in the lesson on Kan complexes attaches the \(m+1\) maximal simplices of the prism, each along a horn. (2) Let \(\mathcal A\) be the class of monomorphisms \(i\) for which \(i\mathbin\square j\) is strict anodyne. It is finitely saturated, so it suffices to show that it contains every horn \(h\colon\Lambda^k[n]\subset\Delta[n]\). If \(k0\), use \(\{1\}\subset\Delta[1]\), \(s(x)=(0,x)\) and \(r(0,x)=x\), \(r(1,x)=\max(x,k)\), in the same way; here \(r\) carries \(\{1\}\times\Delta[n]\) into the face with vertices \(k,\ldots,n\), which omits \(0\neq k\). Hence \(h\mathbin\square j\) is a retract of \(g\mathbin\square j=(\{\epsilon\}\subset\Delta[1])\mathbin\square(h\mathbin\square j)\). The latter is strict anodyne by (1), since \(h\mathbin\square j\) is a monomorphism into the finite simplicial set \(\Delta[n]\times X\). \(\square\) ## 2. Local fibrations Let \(i\colon K\to L\) be a map of simplicial sets and \(p\colon E\to B\) a map of simplicial sheaves. A *lifting problem* of \(i\) against \(p\) over \(U\in\mathcal C\) is a pair of maps \(a\colon K\times h_U\to E\) and \(b\colon L\times h_U\to B\) with \(pa=b\circ(i\times h_U)\). **Definition 2.1.** The map \(p\) has the *local lifting property* relative to \(i\) if for every lifting problem over every \(U\in\mathcal C\) there are a covering \(\{U_\alpha\to U\}\) and maps \(c_\alpha\colon L\times h_{U_\alpha}\to E\) with \(c_\alpha\circ(i\times h_{U_\alpha})=a|U_\alpha\) and \(pc_\alpha=b|U_\alpha\). For a class \(\mathcal N\) of maps, \(p\) has the local lifting property relative to \(\mathcal N\) if it has it relative to every member. A lifting problem over an arbitrary object of \(T\) restricts to lifting problems over the objects \(h_U\) mapping to it, which cover it, so the definition does not change if \(U\) runs through all objects of \(T\). **Lemma 2.2.** Let \(i\colon K\to L\) be a map of finite simplicial sets. Then \(p\) has the local lifting property relative to \(i\) if and only if the map of sheaves \[ \begin{aligned}q_i(p)\colon\ &\operatorname{Hom}(L,E)\\&\to\operatorname{Hom}(K,E)\\&\qquad\times_{\operatorname{Hom}(K,B)}\operatorname{Hom}(L,B)\end{aligned} \] is an epimorphism. **Proof.** A lifting problem over \(U\) is a section of the target of \(q_i(p)\) over \(U\), and a solution over \(U_\alpha\) is a section of the source over \(U_\alpha\) mapping to its restriction. A map of sheaves is an epimorphism exactly when sections of its target lift locally. \(\square\) **Lemma 2.3.** If \(p\) has the local lifting property relative to a class \(\mathcal N\) of monomorphisms, it has it relative to the finitely saturated class generated by \(\mathcal N\). **Proof.** Let \(\mathcal P\) be the class of monomorphisms relative to which \(p\) has the local lifting property; we show that it is finitely saturated. It contains the isomorphisms. If \(i'\colon K'\to L'\) is the pushout of \(i\colon K\to L\) along \(K\to K'\), a lifting problem of \(i'\) restricts along \(L\times h_U\to L'\times h_U\) to one of \(i\); local solutions of the latter, together with the given map on \(K'\times h_{U_\alpha}\), define local solutions of the former, because \(-\times h_{U_\alpha}\) preserves pushouts. A lifting problem of a retract of \(i\) extends, through the retraction diagram, to one of \(i\), and local solutions restrict back. A lifting problem of a finite coproduct is a finite family of lifting problems; solve each on a covering and pass to the common refinement by fibre products. For a composite \(i'i\), first solve the lifting problem of \(i\) on a covering, then on each member the induced lifting problem of \(i'\) on a covering of it. \(\square\) **Definition 2.4.** A map \(p\) of simplicial sheaves is a *fibration* if it has the local lifting property relative to all horns, equivalently (Lemma 2.3) relative to all strict anodyne extensions. A simplicial sheaf \(E\) is *Kan* if \(E\to e\) is a fibration, \(e\) the final simplicial sheaf. In the topos of sets these are the Kan fibrations and the Kan complexes. **Proposition 2.5.** 1. Isomorphisms are fibrations, and fibrations are stable under base change, retracts, finite composites and finite products. 2. If \(f\) is a morphism of topoi with inverse image \(f^{-1}\colon T\to T'\) and \(p\) is a fibration of simplicial sheaves on \(T\), then \(f^{-1}(p)\) is a fibration. 3. If \(\mathcal P\) is a conservative family of points of \(T\), a map \(p\) is a fibration if and only if its stalk \(p_x\) is a Kan fibration for every \(x\in\mathcal P\). 4. A map between constant simplicial sheaves is a fibration. **Proof.** (1) A lifting problem against a base change \(p'\colon E\times_BB'\to B'\) composes to one against \(p\); its local solutions, together with the given map to \(B'\), are local solutions for \(p'\). Retracts are handled as in Lemma 2.3. For \(E\xrightarrow{p}B\xrightarrow{r}C\), solve a lifting problem against \(rp\) first against \(r\) on a covering, then against \(p\) on coverings of its members. A product \(p\times p'\) is the composite of the base changes \(p\times1\) and \(1\times p'\). (2) Horns are finite. By Lemma 1.1 (2) of the lesson on the simplicial derived category, \(f^{-1}q_h(p)\) is \(q_h(f^{-1}p)\) for every horn \(h\), and \(f^{-1}\) preserves epimorphisms. Apply Lemma 2.2. (3) The stalk at a point is the inverse image along a morphism from the topos of sets, so (2) gives one direction. Conversely, \(q_h(p)_x=q_h(p_x)\) is surjective for every \(x\in\mathcal P\), so \(q_h(p)\) is an epimorphism (Proposition 5.1 of the lesson on topoi and points). (4) Since \(\Lambda^k[n]\) is connected, a map \(\Lambda^k[n]\times h_U\to F\) into a constant simplicial sheaf is one section of \(F\) over \(U\), repeated; so is a map \(\Delta[n]\times h_U\to F\). The repeated section is a solution over \(U\) itself. \(\square\) ## 3. Subdivision ### 3.1. The functors Sd and Ex For \(n\geq0\) let \(P_n\) be the poset of nonempty subsets of \([n]\), ordered by inclusion. An order-preserving map \(\alpha\colon[m]\to[n]\) gives the order-preserving map \(\alpha_*\colon P_m\to P_n\), \(S\mapsto\alpha(S)\), so \([n]\mapsto N(P_n)\) is a functor from \(\Delta\) to simplicial sets. For a simplicial set \(X\), the *subdivision* \(\operatorname{Sd}X\) is the colimit of the simplicial sets \(N(P_n)\) over the simplices \(x\in X_n\) of \(X\), indexed by the category of simplices. Its \(m\)-simplices are the classes \([x,c]\) of pairs with \(x\in X_n\) and \(c\colon[m]\to P_n\) order-preserving, with \([\alpha^*x,c']=[x,\alpha_*c']\). For a simplicial set \(Y\) put \[ \operatorname{Ex}(Y)_n=\operatorname{Hom}(N(P_n),Y), \] with the simplicial operators induced by the \(\alpha_*\). A map \(\operatorname{Sd}X\to Y\) is a family of maps \(N(P_n)\to Y\), one for each simplex \(x\in X_n\), compatible with the simplicial operators; that is, a map \(X\to\operatorname{Ex}Y\). So \(\operatorname{Sd}\) is left adjoint to \(\operatorname{Ex}\). The functor \(\operatorname{Sd}\) preserves colimits, and \(\operatorname{Sd}\Delta[n]=N(P_n)\), since the simplices of \(\Delta[n]\) have the final object \(\mathrm{id}_{[n]}\). For a set \(S\subset[n]\) write \(\Delta[S]\) for the face of \(\Delta[n]\) with vertex set \(S\). **Lemma 3.1.** 1. Let \(A\subset\Delta[n]\) be a union of faces, and \(P_A\) the set of \(S\in P_n\) with \(\Delta[S]\subset A\). Then \(\operatorname{Sd}A\to\operatorname{Sd}\Delta[n]\) is an isomorphism onto \(N(P_A)\). In particular \(\operatorname{Sd}\partial\Delta[n]\) and \(\operatorname{Sd}\Lambda^k[n]\) are the nerves of \(P_n\smallsetminus\{[n]\}\) and of \(P_n\smallsetminus\{[n],[n]\smallsetminus\{k\}\}\). 2. \(\operatorname{Sd}\) carries monomorphisms to monomorphisms. **Proof.** (1) The faces \(\Delta[S]\) contained in \(A\) and their intersections \(\Delta[S]\cap\Delta[S']=\Delta[S\cap S']\) form a diagram whose colimit is \(A\): a union of simplicial subsets is the colimit of the subsets and their pairwise intersections. Since \(\operatorname{Sd}\) preserves colimits and \(\operatorname{Sd}\Delta[S]\) is the nerve of the poset of nonempty subsets of \(S\), \(\operatorname{Sd}A\) is the colimit of these nerves. They are the nerves of subposets of \(P_n\) closed under passing to smaller sets. A chain lying in two of them lies in the nerve of their intersection, so the colimit maps isomorphically onto their union, which consists of the chains whose largest element lies in \(P_A\). (2) A monomorphism of simplicial sets is obtained by attaching the missing nondegenerate simplices along their boundaries, in order of dimension: it is a composite indexed by the natural numbers of pushouts of coproducts of boundary inclusions. \(\operatorname{Sd}\) preserves these colimits, by (1) it carries boundary inclusions to monomorphisms, and pushouts and increasing unions of monomorphisms of simplicial sets are monomorphisms. \(\square\) Let \(L\subset N(R)\) be a simplicial subset, \(\tau\) a chain of \(R\) with \(m+1\) elements, and \(\sigma=\tau\smallsetminus\{z\}\) a face of \(\tau\), \(z\) in position \(l\), such that \(\tau,\sigma\notin L\) and every other face of \(\tau\) lies in \(L\). The map \(\tau\colon\Delta[m]\to N(R)\) is injective, and the simplices of its image lying in \(L\) are those of \(\Lambda^l[m]\). So the union of \(L\) and the image of \(\tau\) is the pushout of \(\Lambda^l[m]\subset\Delta[m]\) along \(\Lambda^l[m]\to L\). We call it the *elementary expansion* of \(L\) by \((\sigma,\tau)\). **Lemma 3.2.** For \(n\geq1\) and \(0\leq k\leq n\), the inclusion \(\operatorname{Sd}\Lambda^k[n]\subset\operatorname{Sd}\Delta[n]\) is a finite composite of elementary expansions, so it is strict anodyne. Consequently \(\operatorname{Sd}\) carries anodyne extensions to anodyne extensions, and \(\operatorname{Ex}\) carries Kan fibrations to Kan fibrations. **Proof.** Put \(P=P_n\), \(F=[n]\smallsetminus\{k\}\) and \(Q=P\smallsetminus\{[n],F\}\), so that \(\operatorname{Sd}\Lambda^k[n]=N(Q)\) (Lemma 3.1). Let \(Q_k\subset Q\) be the set of members that contain \(k\). The members of \(Q\) not containing \(k\) are the nonempty proper subsets of \(F\). Within a chain of \(Q\), the members not containing \(k\) come first, because a set containing \(k\) is not contained in one that does not. We reach \(N(P)\) from the vertex \(\{k\}\) in four stages, each a finite sequence of elementary expansions. (a) \(N(Q_k)\). The set \(\{k\}\) is the least element of \(Q_k\). For the chains \(c\) of \(Q_k\smallsetminus\{\{k\}\}\), in order of increasing length, expand by \((c,c\cup\{\{k\}\})\): the other faces of \(c\cup\{\{k\}\}\) contain \(\{k\}\) and are shorter. (b) \(N(Q)\). For a chain \(c\) of \(Q\) that contains a set not containing \(k\), let \(S_c\) be the largest such set, and \(x_c=S_c\cup\{k\}\). Then \(x_c\in Q_k\), since \(S_c\subsetneq F\). Every member of \(c\) is contained in \(S_c\) or contains \(S_c\) and \(k\), hence \(x_c\); so \(c\cup\{x_c\}\) is a chain, with the same largest set \(S_c\). The chains of \(Q\) not in \(N(Q_k)\) fall into pairs \(\{\tau\smallsetminus\{x_\tau\},\tau\}\) with \(x_\tau\in\tau\). Expand by these pairs in order of increasing length of \(\tau\), and for equal lengths in order of increasing \(|S_\tau|\). For a face \(\tau\smallsetminus\{y\}\) with \(y\neq x_\tau\): if \(y\neq S_\tau\), it still contains \(x_\tau\), with the same largest set, so it is the larger chain of a pair already used. If \(y=S_\tau\), either it lies in \(N(Q_k)\), or its largest set \(S'\) not containing \(k\) has \(|S'|<|S_\tau|\); then it is the larger chain of a shorter pair, or the smaller chain of a pair of the same length as \(\tau\) with smaller \(|S'|\). In every case it is present. (c) \(N(Q\cup\{[n]\})\). Every chain of \(Q\cup\{[n]\}\) not in \(N(Q)\) is \(c\cup\{[n]\}\) for a chain \(c\) of \(Q\), possibly empty. Expand first by \((\{[n]\},\{\{k\},[n]\})\), then by \((\sigma\cup\{[n]\},\tau\cup\{[n]\})\) for the expansions \((\sigma,\tau)\) of (a) and (b), in the same order. The faces of \(\tau\cup\{[n]\}\) other than \(\sigma\cup\{[n]\}\) are \(\tau\), and the chains \(\rho\cup\{[n]\}\) for the other faces \(\rho\) of \(\tau\); these \(\rho\) were present before \((\sigma,\tau)\), so \(\rho\cup\{[n]\}\) was added before. (d) \(N(P)\). The chains containing \(F\) are \(c\cup\{F\}\) and \(c\cup\{F,[n]\}\) for chains \(c\) of nonempty proper subsets of \(F\), possibly empty. Expand by \((c\cup\{F\},c\cup\{F,[n]\})\) in order of increasing length of \(c\). The other faces of \(c\cup\{F,[n]\}\) are \(c\cup\{[n]\}\), present by (c), and shorter chains \(c'\cup\{F,[n]\}\). By the paragraph before the lemma, each step is a pushout of a horn inclusion. Since \(\operatorname{Sd}\) preserves colimits, retracts and monomorphisms (Lemma 3.1), it carries anodyne extensions to anodyne extensions. By adjunction, a lifting problem of a horn against \(\operatorname{Ex}(p)\) is one of its subdivision against \(p\); a Kan fibration lifts against anodyne extensions. \(\square\) ### 3.2. The last vertex map The order-preserving maps \(\max\colon P_n\to[n]\) commute with the \(\alpha_*\), so they give a natural map \(h_X\colon\operatorname{Sd}X\to X\), the *last vertex map*, with \(h_X[x,c]=(\max\circ c)^*x\). On \(\operatorname{Sd}\Delta[n]=N(P_n)\) it is \(N(\max)\). Its adjunct \(v_X\colon X\to\operatorname{Ex}X\) sends \(x\in X_n\), viewed as a map \(\Delta[n]\to X\), to \(x\circ N(\max)\). The map \(N(\max)\) is surjective on simplices: \(\alpha\colon[m]\to[n]\) is the image of the chain \(\{0,\ldots,\alpha(0)\}\subset\cdots\subset\{0,\ldots,\alpha(m)\}\). Hence \(v_X\) is a monomorphism. Let \(\epsilon_Y\colon\operatorname{Sd}\operatorname{Ex}Y\to Y\) be the counit, \([y,c]\mapsto y(c)\). **Proposition 3.3.** For every simplicial set \(X\), the last vertex map \(h_X\) is a quasi-isomorphism. **Proof.** For \(X=\Delta[n]\), the posets \([n]\) and \(P_n\) have final objects, so the identities of \(\Delta[n]\) and \(N(P_n)\) are homotopic to constant maps (Corollary 2.2 of the lesson on décalage), and both maps to \(\Delta[0]\) are homotopy equivalences. So \(h_{\Delta[n]}\) is a quasi-isomorphism by two out of three (Corollary 2.3 of the lesson on the simplicial derived category). Suppose the assertion holds for simplicial sets of dimension less than \(m\), and let \(X\) have dimension at most \(m\). By the expression (1.8), \(X\) is the pushout of \(\operatorname{sk}_{m-1}X\leftarrow\coprod\partial\Delta[m]\to\coprod\Delta[m]\), over the nondegenerate \(m\)-simplices, and \(\operatorname{Sd}X\) is the pushout of the subdivided diagram. The map \(h\) is a map between these diagrams. Its components are quasi-isomorphisms by the induction hypothesis, by the case of a simplex and by Corollary 2.3 (3) there for coproducts. The maps \(\coprod\partial\Delta[m]\to\coprod\Delta[m]\) and their subdivisions are monomorphisms (Lemma 3.1). By Proposition 2.6 of the lesson on the simplicial derived category, \(h_X\) is a quasi-isomorphism. A general \(X\) is the increasing union of its skeleta, \(\operatorname{Sd}X\) is the union of their subdivisions, and Corollary 2.3 (3) there applies. \(\square\) **Proposition 3.4.** For every simplicial set \(X\), the map \(v_X\colon X\to\operatorname{Ex}X\) is a quasi-isomorphism. **Proof.** First, \(\epsilon_X\circ\operatorname{Sd}(v_X)=h_X\): both send \([x,c]\) to \((\max\circ c)^*x\). Second, \(h_{\operatorname{Ex}X}\) and \(v_X\epsilon_X\) are homotopic. An \(m\)-simplex of \(\Delta[1]\times\operatorname{Sd}\operatorname{Ex}X\) is \((\beta,[y,c])\), with \(\beta\colon[m]\to[1]\), \(y\in(\operatorname{Ex}X)_n\) a map \(N(P_n)\to X\), and \(c\colon[m]\to P_n\). Define the order-preserving map \(\theta_{\beta,c}\colon P_m\to P_n\) by \[ \begin{aligned}\theta_{\beta,c}(S)={}&\{\max c(i):i\in S\}\\&\cup\bigcup_{i\in S,\ \beta(i)=1}c(i),\end{aligned} \] and put \(H(\beta,[y,c])=y\circ N(\theta_{\beta,c})\), an \(m\)-simplex of \(\operatorname{Ex}X\). This is well defined: for \(\alpha\colon[n']\to[n]\), \(\alpha_*\theta_{\beta,c'}=\theta_{\beta,\alpha_*c'}\), because \(\alpha(\max S)=\max\alpha(S)\). It is a simplicial map: \(\theta_{\beta\gamma,c\gamma}=\theta_{\beta,c}\circ\gamma_*\) for \(\gamma\colon[m']\to[m]\). If \(\beta=0\), \(\theta_{\beta,c}(S)\) is \(\{\max c(i):i\in S\}=(\max\circ c)_*(S)\), and \(H\) is \(h_{\operatorname{Ex}X}\). If \(\beta=1\), \(\theta_{\beta,c}(S)=c(\max S)\), since \(\max c(i)\in c(i)\subset c(\max S)\) for \(i\in S\); and \(y\circ N(S\mapsto c(\max S))\) is \(v_X(y(c))\), which sends a chain \(S_0\subset\cdots\subset S_l\) to \(y\) applied to the chain \(c(\max S_0)\subset\cdots\subset c(\max S_l)\). So \(H\) is an elementary homotopy from \(h_{\operatorname{Ex}X}\) to \(v_X\epsilon_X\). By Proposition 3.3 and Corollary 2.3 of the lesson on the simplicial derived category, \(\epsilon_X\circ\operatorname{Sd}(v_X)\) and \(v_X\circ\epsilon_X\) are quasi-isomorphisms. By two out of six, so is \(v_X\). \(\square\) ### 3.3. Kan's theorem **Proposition 3.5.** For every horn \(\lambda\colon\Lambda^k[n]\to\operatorname{Ex}X\), the composite \(v_{\operatorname{Ex}X}\circ\lambda\colon\Lambda^k[n]\to\operatorname{Ex}^2X\) extends to \(\Delta[n]\). **Proof.** Write \(\Lambda=\Lambda^k[n]\). Under the adjunction, \(v_{\operatorname{Ex}X}\circ\lambda\) corresponds to \(h_{\operatorname{Ex}X}\circ\operatorname{Sd}(\lambda)=\lambda\circ h_\Lambda\colon\operatorname{Sd}\Lambda\to\operatorname{Ex}X\), and this in turn to \(\hat\lambda\circ\operatorname{Sd}(h_\Lambda)\colon\operatorname{Sd}^2\Lambda\to X\), where \(\hat\lambda\colon\operatorname{Sd}\Lambda\to X\) is the adjunct of \(\lambda\). By Lemma 3.1 (2), \(\operatorname{Sd}^2\Lambda\) is a simplicial subset of \(\operatorname{Sd}^2\Delta[n]\). It suffices to find \(G\colon\operatorname{Sd}^2\Delta[n]\to\operatorname{Sd}\Lambda\) restricting to \(\operatorname{Sd}(h_\Lambda)\): then \(\hat\lambda\circ G\) corresponds to the required extension. Put \(F=[n]\smallsetminus\{k\}\) and \(Q=P_n\smallsetminus\{[n],F\}\), so \(\operatorname{Sd}\Lambda=N(Q)\). The simplicial set \(\operatorname{Sd}^2\Delta[n]=\operatorname{Sd}N(P_n)\) is the colimit of the \(N(P_m)\) over the simplices \(c\colon[m]\to P_n\) of \(N(P_n)\). For such \(c\) define \(a_c\colon[m]\to[n]\) by \(a_c(i)=\max c(i)\) if \(c(i)\notin\{[n],F\}\), and \(a_c(i)=k\) otherwise, and let \(g_c\colon P_m\to P_n\), \(S\mapsto a_c(S)\). It is order-preserving, and \(a_{c\alpha}=a_c\circ\alpha\), so \(g_{c\alpha}=g_c\circ\alpha_*\); the maps \(N(g_c)\) define \(G\) on the colimit. If \(c\) takes values in \(Q\), then \(a_c=\max\circ c\) and \(N(g_c)\) is the map \(\operatorname{Sd}(h_\Lambda)\) on the copy of \(N(P_m)\) indexed by \(c\). It remains to see that \(g_c(S)\in Q\). Let \(U=\{c(i):i\in S\}\), a chain, and \(U'=U\smallsetminus\{[n],F\}\). If \(U'=U\), then \(g_c(S)\) consists of maxima of members of \(U\), so it is contained in the largest member of \(U\), which lies in \(Q\); and \(Q\) is closed under passing to smaller nonempty sets. Otherwise \(k\in g_c(S)\), so \(g_c(S)\neq F\). If \(g_c(S)=[n]\), the \(n\) elements of \(F\) are maxima of members of \(U'\), so \(U'\) has at least \(n\) members; being a strictly increasing chain of nonempty sets, its largest member \(S'\) has at least \(n\) elements. As \(S'\neq[n]\) and \(S'\neq F\), \(S'=[n]\smallsetminus\{l\}\) with \(l\neq k\). But every member of \(U'\) is contained in \(S'\), so \(l\) is not a maximum of one, and \(l\neq k\); so \(l\notin g_c(S)\), a contradiction. \(\square\) **Theorem 3.6 (Kan).** For a simplicial set \(X\), let \(\operatorname{Ex}^\infty X\) be the union of the sequence \(X\to\operatorname{Ex}X\to\operatorname{Ex}^2X\to\cdots\) of the monomorphisms \(v_{\operatorname{Ex}^rX}\), and \(e^\infty_X\colon X\to\operatorname{Ex}^\infty X\) the inclusion. 1. \(e^\infty_X\) is a monomorphism and a quasi-isomorphism, natural in \(X\). 2. \(\operatorname{Ex}^\infty X\) is a Kan complex, and \(\operatorname{Ex}^\infty\) carries Kan fibrations to Kan fibrations. 3. \(\operatorname{Ex}^\infty\) preserves finite limits, and \(\operatorname{Ex}^\infty\Delta[0]=\Delta[0]\). For each \(r\), \(\operatorname{Ex}^r(X)_n\) is a finite limit of the sets \(X_m\). **Proof.** (1) Each \(v\) is a monomorphism and a quasi-isomorphism (Proposition 3.4), and Corollary 2.3 (3) of the lesson on the simplicial derived category applies to the union. Naturality holds because \(v\) is natural. (2) A horn in \(\operatorname{Ex}^\infty X\) has finitely many nondegenerate simplices, so it lies in some \(\operatorname{Ex}^rX\) with \(r\geq1\); by Proposition 3.5, applied to \(\operatorname{Ex}^{r-1}X\), it is filled in \(\operatorname{Ex}^{r+1}X\). Let \(p\) be a Kan fibration. Each \(\operatorname{Ex}^r(p)\) is one (Lemma 3.2). A lifting problem of a horn against \(\operatorname{Ex}^\infty(p)\) lies in some \(\operatorname{Ex}^r(p)\), the square commuting there because the maps of the sequence are injective, and is solved there. (3) \(\operatorname{Ex}\) is a right adjoint, and sequential colimits of sets commute with finite limits. \(\operatorname{Ex}\Delta[0]=\Delta[0]\), as there is one map \(N(P_n)\to\Delta[0]\). \(N(P_n)\) is a finite simplicial set, so \(\operatorname{Ex}(Y)_n\) is a finite limit of the \(Y_m\). \(\square\) ### 3.4. Trivial fibrations A map of simplicial sets is a *trivial fibration* if it lifts against every monomorphism. **Theorem 3.7.** A Kan fibration \(p\colon E\to B\) of simplicial sets is a quasi-isomorphism if and only if it is a trivial fibration. **Proof.** A trivial fibration has a section \(s\) (lift against \(\varnothing\subset B\)), and lifting against \(\partial\Delta[1]\times E\subset\Delta[1]\times E\), with the ends \(sp\) and \(1\) over the constant homotopy of \(p\), gives an elementary homotopy from \(sp\) to \(1\). So \(p\) is a homotopy equivalence, hence a quasi-isomorphism. Conversely let \(p\) be a Kan fibration and a quasi-isomorphism. By Theorem 3.6, \(\operatorname{Ex}^\infty(p)\) is a Kan fibration between Kan complexes, and a quasi-isomorphism by two out of three, since \(e^\infty_B\circ p=\operatorname{Ex}^\infty(p)\circ e^\infty_E\). By Lemma 5.3d of the lesson on Kan complexes, its fibres over the vertices of \(\operatorname{Ex}^\infty B\) are nonempty, connected, and have zero homotopy groups. For a vertex \(b\) of \(B\), the fibre of \(\operatorname{Ex}^\infty(p)\) over \(e^\infty_B(b)\) is \(\operatorname{Ex}^\infty(F_b)\), the image of the fibre \(F_b=p^{-1}(b)\), because \(\operatorname{Ex}^\infty\) preserves fibre products and \(\operatorname{Ex}^\infty\Delta[0]=\Delta[0]\). The fibre \(F_b\) is a Kan complex, and \(F_b\to\operatorname{Ex}^\infty(F_b)\) is a quasi-isomorphism, so \(F_b\) is nonempty, connected and has zero homotopy groups. The proof of Lemma 5.3e of the lesson on Kan complexes uses only that \(p\) is a Kan fibration and that its fibres over the vertices of \(B\) have these properties; it does not use that \(B\) is a Kan complex. So \(p\) lifts against every boundary inclusion, hence against every monomorphism. \(\square\) **Corollary 3.8.** 1. The base change of a Kan fibration that is a quasi-isomorphism, along any map, is a Kan fibration and a quasi-isomorphism. 2. A monomorphism of simplicial sets is anodyne if and only if it is a quasi-isomorphism. **Proof.** (1) Lifting properties are stable under base change; apply Theorem 3.7 twice. (2) Anodyne extensions are quasi-isomorphisms (Corollary 2.3 of the lesson on the simplicial derived category). Let \(i\colon A\to B\) be a monomorphism and a quasi-isomorphism. The construction (5.19) of the lesson on Kan complexes, attaching horn fillers over \(B\), applies to any map and factors \(i=pa\) with \(a\) anodyne and \(p\) a Kan fibration. By two out of three \(p\) is a quasi-isomorphism, hence a trivial fibration by Theorem 3.7. As in the proof of Proposition 2.4 of the lesson on the simplicial derived category, \(i\) is a retract of \(a\), hence anodyne. \(\square\) ## 4. Kan's functor for simplicial sheaves For a simplicial sheaf \(X\) and \(r\geq0\), the presheaf \(U\mapsto\operatorname{Ex}^r(X(U))\) is a simplicial sheaf, since each of its terms is a finite limit of the terms of \(X\) (Theorem 3.6 (3)); write it \(\operatorname{Ex}^r(X)\). Its sections over \(U\) form the simplicial set \(\operatorname{Ex}^r(X(U))\). Let \(\operatorname{Ex}^\infty(X)\) be the colimit in \(\operatorname{Simpl}(T)\) of the sequence of the \(\operatorname{Ex}^r(X)\), the sheaf associated with the presheaf \(U\mapsto\operatorname{Ex}^\infty(X(U))\), and \(e^\infty_X\colon X\to\operatorname{Ex}^\infty(X)\) the canonical map. **Proposition 4.1.** 1. \(e^\infty_X\) is a monomorphism and a quasi-isomorphism, natural in \(X\). 2. \(\operatorname{Ex}^\infty(X)\) is a Kan simplicial sheaf. 3. \(\operatorname{Ex}^\infty\) commutes with finite limits and with inverse images by morphisms of topoi, in particular with stalks at points. 4. \(\operatorname{Ex}^\infty\) carries fibrations to fibrations. **Proof.** (1) The maps \(\operatorname{Ex}^r(X)\to\operatorname{Ex}^{r+1}(X)\) are monomorphisms, sectionwise by Section 3.2, and filtered colimits of sheaves commute with finite limits, so \(e^\infty_X\) is a monomorphism. Let \(x\) be a base point of \(X\) over \(U\). As recalled at the start of Section 3 of the lesson on the simplicial derived category, \(\pi_n(\operatorname{Ex}^\infty(X)|U,x)\) is the sheaf associated with the presheaf \(V\mapsto\pi_n(\operatorname{Ex}^\infty(X(V)),x)\) on \(\mathcal C_{/U}\). By Theorem 3.6 (1), the map to it from the presheaf \(V\mapsto\pi_n(X(V),x)\) is an isomorphism; so is the associated map of sheaves. The same holds for \(\pi_0\). (2) A horn of \(\operatorname{Ex}^\infty(X)\) over \(U\) is a section over \(U\) of \(\operatorname{Hom}(\Lambda^k[n],\operatorname{Ex}^\infty(X))\), the colimit of the sheaves \(\operatorname{Hom}(\Lambda^k[n],\operatorname{Ex}^r(X))\), since \(\operatorname{Hom}(\Lambda^k[n],-)\) is a finite limit. So on a covering \(\{U_\alpha\}\) it comes from horns of the simplicial sets \(\operatorname{Ex}^{r}(X(U_\alpha))\), \(r\geq1\). By Proposition 3.5 they are filled in \(\operatorname{Ex}^{r+1}(X(U_\alpha))\), and the images of the fillers fill the restrictions of the horn. (3) \(\operatorname{Ex}^r(X)\) is built from \(X\) by finite limits, and \(\operatorname{Ex}^\infty(X)\) by a sequential colimit. Inverse images by morphisms of topoi preserve both; so does sheafification, and filtered colimits commute with finite limits. (4) Let \(p\colon E\to B\) be a fibration. The sections of \(\operatorname{Ex}(E)\) over \(U\) form \(\operatorname{Ex}(E(U))\), so a lifting problem of a horn \(h\) against \(\operatorname{Ex}(p)\) over \(U\) is one of \(\operatorname{Sd}(h)\) against \(p\) over \(U\), by the adjunction applied to sections. By Lemma 3.2, \(\operatorname{Sd}(h)\) is strict anodyne, so by Lemma 2.3 such problems have local solutions. Hence each \(\operatorname{Ex}^r(p)\) is a fibration. A lifting problem against \(\operatorname{Ex}^\infty(p)\) over \(U\) comes, on a covering of \(U\), from lifting problems against some \(\operatorname{Ex}^r(p)\), as in (2); the squares commute there because the transition maps are monomorphisms. Solve them locally. \(\square\) **Remark 4.2.** For a quasi-isomorphism \(f\colon X\to Y\), \(\operatorname{Ex}^\infty(f)\) is a quasi-isomorphism between Kan simplicial sheaves, by Proposition 4.1 (1) and two out of three. Because \(\operatorname{Ex}^\infty\) preserves finite products and \(e^\infty_{\Delta[1]}\) maps \(\Delta[1]\) into \(\operatorname{Ex}^\infty\Delta[1]\), it carries an elementary homotopy \(\Delta[1]\times X\to Y\) to the elementary homotopy \[ \begin{aligned}&\Delta[1]\times\operatorname{Ex}^\infty X\\&\quad\to\operatorname{Ex}^\infty(\Delta[1])\times\operatorname{Ex}^\infty X\\&\quad\to\operatorname{Ex}^\infty Y.\end{aligned} \] So \(\operatorname{Ex}^\infty\) is a functor on homotopy categories. **Proposition 4.3.** Suppose that \(T\) has a conservative family of points. Let \(p\colon E\to B\) be a fibration of simplicial sheaves which is a quasi-isomorphism. Then \(p\) has the local lifting property relative to every monomorphism of finite simplicial sets, and every base change of \(p\) is a fibration and a quasi-isomorphism. **Proof.** Let \(x\) be a point of the family. The stalk \(p_x\) is a Kan fibration (Proposition 2.5 (3)) and a quasi-isomorphism (Corollary 3.3 (3) of the lesson on the simplicial derived category), hence a trivial fibration by Theorem 3.7. For a monomorphism \(i\) of finite simplicial sets, \(q_i(p)_x=q_i(p_x)\) is therefore surjective, so \(q_i(p)\) is an epimorphism and Lemma 2.2 applies. Stalks commute with fibre products, so the stalk of a base change \(p'\) of \(p\) is a base change of \(p_x\), a trivial fibration, hence a Kan fibration and a quasi-isomorphism (Theorem 3.7). By Proposition 2.5 (3) and Corollary 3.3 (3) there, \(p'\) is a fibration and a quasi-isomorphism. \(\square\) **Remark 4.4.** Illusie leaves open whether the base change assertion of Proposition 4.3 holds in every topos. Proposition 4.1 reduces it to Kan simplicial sheaves. Let \(p'\colon E'\to B'\) be the base change of \(p\) along \(u\colon B'\to B\). Since \(\operatorname{Ex}^\infty\) commutes with fibre products, \(\operatorname{Ex}^\infty(p')\) is the base change of \(\operatorname{Ex}^\infty(p)\) along \(\operatorname{Ex}^\infty(u)\); and \(\operatorname{Ex}^\infty(p)\) is a fibration and a quasi-isomorphism between Kan simplicial sheaves. If base changes of such maps between Kan simplicial sheaves are quasi-isomorphisms, then \(\operatorname{Ex}^\infty(p')\) is one, and so is \(p'\), by two out of three with the quasi-isomorphisms \(e^\infty\). For locally fibrant simplicial presheaves on any site, Jardine proved that a local fibration that is a local weak equivalence has the local lifting property relative to all boundary inclusions, and that this property implies a local weak equivalence ([*Boolean localization, in practice*](https://www.math.uni-bielefeld.de/documenta/vol-01/13.html), Documenta Math. 1 (1996), Lemmas 7 and 11), by Boolean localization. That method rests on Barr's theorem, which this course does not prove; the general case is not settled here. ## 5. Exercises **Exercise 5.1** (easy). Show that the inclusion of the vertex \(0\) into \(\Delta[n]\) is strict anodyne. **Exercise 5.2** (medium). Let \(X\) be a simplicial set. Show that \(e^\infty_X\colon X\to\operatorname{Ex}^\infty X\) is anodyne, and that every map from \(X\) to a Kan complex \(K\) extends along \(e^\infty_X\), uniquely up to a homotopy that is constant on \(X\). **Exercise 5.3** (medium). Let \(p\colon E\to B\) be a Kan fibration of simplicial sets, \(b\) a vertex of \(B\), \(F_b\) its fibre and \(e\) a vertex of \(F_b\). Without assuming that \(B\) is a Kan complex, show that for \(n\geq1\) the sequence \[ \begin{aligned}&\pi_{n+1}(E,e)\to\pi_{n+1}(B,b)\to\pi_n(F_b,e)\\&\quad\to\pi_n(E,e)\to\pi_n(B,b)\end{aligned} \] of the groups of the lesson on homotopy sheaves, with a suitable boundary map, is exact. **Exercise 5.4** (medium). Let \(p\colon E\to B\) be a Kan fibration of simplicial sets. 1. Show that \(p\) is a quasi-isomorphism if and only if its fibres over all vertices of \(B\) are nonempty, connected and have zero homotopy groups. 2. Show that the inclusion of the vertex \(0\) into \(\Delta[1]\) is a quasi-isomorphism whose fibre over the vertex \(1\) is empty. Which hypothesis of (1) fails? **Exercise 5.5** (hard). Let \(X\) be a connected topological space with an open covering \(\{U_1,U_2\}\) such that \(U_1\neq X\neq U_2\), let \(T\) be the topos of sheaves on \(X\), and let \(U=h_{U_1}\amalg h_{U_2}\). Let \(E\) be the simplicial sheaf with \(E_n=U^{n+1}\), whose faces omit and whose degeneracies repeat a factor. Show that \(E\to e\) is a fibration and a quasi-isomorphism with the local lifting property relative to every monomorphism of finite simplicial sets, but that \(E\) has no vertex over \(X\). Conclude that this local lifting property cannot be replaced by lifting over the given object. ## 6. Complete solutions **Solution 5.1.** The poset \([n]\) has the least element \(0\). As in step (a) of the proof of Lemma 3.2, \(\Delta[n]=N([n])\) is obtained from the vertex \(0\) by the elementary expansions \((c,c\cup\{0\})\), for the nonempty subsets \(c\) of \(\{1,\ldots,n\}\) in order of increasing size: the other faces of \(c\cup\{0\}\) are \(c'\cup\{0\}\) for smaller \(c'\), or the vertex \(0\). Each is a pushout of a horn inclusion, so the inclusion is a finite composite of such pushouts. **Solution 5.2.** By Theorem 3.6, \(e^\infty_X\) is a monomorphism and a quasi-isomorphism, hence anodyne by Corollary 3.8 (2). A Kan complex lifts against anodyne extensions, which gives the extension. For two extensions \(u,v\), the inclusion of \((\Delta[1]\times X)\cup(\partial\Delta[1]\times\operatorname{Ex}^\infty X)\) into \(\Delta[1]\times\operatorname{Ex}^\infty X\) is anodyne by Lemma 2.1 (2) of the lesson on the simplicial derived category; the map that is \(u\), \(v\) on the ends and the given map composed with the projection on \(\Delta[1]\times X\) extends over it, and the extension is the required homotopy. **Solution 5.3.** By Theorem 3.6, \(\operatorname{Ex}^\infty(p)\) is a Kan fibration between Kan complexes. Its fibre over the vertex \(b\) is \(\operatorname{Ex}^\infty(F_b)\), because \(\operatorname{Ex}^\infty\) commutes with fibre products and \(\operatorname{Ex}^\infty\Delta[0]=\Delta[0]\). The exact sequence (5.17) of the lesson on Kan complexes, for \(\operatorname{Ex}^\infty(p)\), reads \[ \begin{aligned}&\pi_{n+1}(\operatorname{Ex}^\infty E,e)\to\pi_{n+1}(\operatorname{Ex}^\infty B,b)\\&\quad\to\pi_n(\operatorname{Ex}^\infty F_b,e)\to\pi_n(\operatorname{Ex}^\infty E,e)\\&\quad\to\pi_n(\operatorname{Ex}^\infty B,b).\end{aligned} \] The maps \(e^\infty\) are quasi-isomorphisms, natural in the simplicial set (Theorem 3.6 (1)), so they identify each term with the corresponding group of \(E\), \(B\) or \(F_b\), compatibly with the maps induced by \(p\) and by the inclusion \(F_b\subset E\). Define the boundary map of the sequence for \(p\) by transporting that of \(\operatorname{Ex}^\infty(p)\) through these isomorphisms. Exactness is preserved. **Solution 5.4.** (1) If \(p\) is a quasi-isomorphism, it is a trivial fibration by Theorem 3.7, and so is its base change \(F_b\to\Delta[0]\) for every vertex \(b\). By Theorem 3.7 again, \(F_b\to\Delta[0]\) is a quasi-isomorphism: \(F_b\) is nonempty, connected, and has zero groups. Conversely, if the fibres have these properties, the proof of Lemma 5.3e of the lesson on Kan complexes, which does not use that \(B\) is Kan, shows that \(p\) lifts against every boundary inclusion. So \(p\) is a trivial fibration, hence a quasi-isomorphism by Theorem 3.7. (2) The poset \([1]\) has a final object, so \(\Delta[1]\) is contractible, and the inclusion of the vertex \(0\) is a homotopy equivalence, hence a quasi-isomorphism. Its fibre over the vertex \(1\) is empty. It is not a Kan fibration: in the lifting problem of \(\Lambda^0[1]\subset\Delta[1]\) against it, with the vertex \(0\) of \(\Delta[0]\) on top and the identity of \(\Delta[1]\) below, a lift would be a map \(\Delta[1]\to\Delta[0]\) whose composite with the inclusion is the identity of \(\Delta[1]\). **Solution 5.5.** A map \(K\times h_V\to E\), for a simplicial set \(K\), is the same as a family of sections of \(U\) over \(V\) indexed by the vertices of \(K\): a simplex of \(E\) is the tuple of its vertices. For \(n\geq2\) the horn \(\Lambda^k[n]\) contains every vertex of \(\Delta[n]\), so a horn of \(E\) over \(V\) determines a unique filler; for \(n=1\) the horn is one vertex, and the degenerate edge on it fills it. So \(E\to e\) is a fibration, with fillers over \(V\) itself. For a monomorphism \(K\subset L\) of finite simplicial sets, a lifting problem over \(V\) asks for sections of \(U\) over \(V\) at the vertices of \(L\) not in \(K\). The map \(U\to e\) is an epimorphism, since the \(U_i\) cover \(X\), so such sections exist locally. In particular \(E\to e\) has the local lifting property relative to all boundary inclusions; at the points of \(X\) its stalks are trivial fibrations, hence quasi-isomorphisms (Theorem 3.7), and \(T\) has enough points, so \(E\to e\) is a quasi-isomorphism (Corollary 3.3 (3) of the lesson on the simplicial derived category). A vertex of \(E\) over \(X\) is a section of \(U\) over \(X\). It would decompose \(X\) into two disjoint open sets \(W_i\subset U_i\), the loci where it lies in \(h_{U_i}\); as \(X\) is connected, one of them is \(X\), so \(X\subset U_1\) or \(X\subset U_2\), which is excluded. So the lifting problem of \(\varnothing\subset\Delta[0]\) over \(X\) has no solution over \(X\), although it has local solutions. ## Sources Local lifting properties, fibrations of simplicial sheaves, Kan simplicial sheaves and the functor \(\operatorname{Ex}^\infty\) for simplicial sheaves are due to Illusie, in his work on the cotangent complex. The functors \(\operatorname{Sd}\), \(\operatorname{Ex}\) and \(\operatorname{Ex}^\infty\), the theorem that \(\operatorname{Ex}^\infty X\) is a Kan complex quasi-isomorphic to \(X\), and the extension of Proposition 3.5 are due to Kan. Theorem 3.7 and Corollary 3.8 (2) are due to Gabriel and Zisman and to Quillen. Here the comparison of \(X\) with \(\operatorname{Ex}X\) is proved through an explicit homotopy between \(h_{\operatorname{Ex}X}\) and \(v_X\epsilon_X\), and the subdivided horns through explicit expansions. Kerodon, [Section 3.3](https://kerodon.net/tag/00XF), treats \(\operatorname{Ex}^\infty\) with complete proofs and a different comparison argument.