The simplicial derived category of a topos

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

The simplicial derived category $D_\bullet(T)$ of a topos $T$ is obtained from the category of simplicial objects of $T$ by inverting the quasi-isomorphisms. It plays for simplicial objects the role that the derived category plays for complexes. This lesson constructs $D_\bullet(T)$ in two steps. Inverting the homotopy equivalences gives the homotopy category $\operatorname{Hot}_\bullet(T)$, whose morphisms are homotopy classes of maps; inverting the quasi-isomorphisms of $\operatorname{Hot}_\bullet(T)$ gives $D_\bullet(T)$. When $T$ has enough points, the quasi-isomorphisms of $\operatorname{Hot}_\bullet(T)$ admit a calculus of left fractions: every morphism $X\to Y$ of $D_\bullet(T)$ is a homotopy class of maps $X\to Y'$ followed by the inverse of a quasi-isomorphism $Y\to Y'$.

At a point of $T$ the proof reduces to two gluing properties of simplicial sets: a pushout of a monomorphism that is a quasi-isomorphism is again a quasi-isomorphism, and so is the product of such a monomorphism with an interval, taken relative to the two ends. Section 2 proves these from the horn-filling arguments for Kan complexes, without topology, together with the invariance of pushouts along monomorphisms under quasi-isomorphisms. Section 6 treats simplicial objects over a fixed simplicial object.

Prerequisites. Homotopy sheaves and the free abelian Whitehead theorem, Sections 1 and 2 (Definition 1.1, Lemma 1.2, Propositions 1.4 and 1.5, Definition 2.1, Lemma 2.2); Simplicial sets, nerves and Kan complexes, Sections 1.2, 4.3, 5.2 and 5.3 (the expression (1.8), Proposition 4.3, the Kan replacement (5.5) and the comparison (5.6), anodyne extensions, Lemma 5.3a, the transports (5.15) and (5.16), Lemmas 5.3d and 5.3e, Theorem 5.3f and the factorization (5.19)); Décalage, nerves and the homotopy type of a category, Proposition 2.1 and Corollary 2.2; Fibred topoi and their total topoi, Section 5; Topoi, morphisms and points, Sections 3 to 5. From the AI-integrated Stacks project: multiplicative systems and the calculus of fractions [Stacks, Tag 04VC, Tag 04VE, Tag 04VF, Tag 04VG].

Conventions

$\mathcal C$ is a small site and $T=\operatorname{Sh}(\mathcal C)$, as in the lesson on homotopy sheaves; for $U\in\mathcal C$, $h_U$ is the sheaf associated with the presheaf represented by $U$, and $X|U$ is the restriction of $X$ to the localized topos $T_{/U}$. $\operatorname{Simpl}(T)$ is the category of simplicial objects of $T$, or simplicial sheaves. It is a topos (Section 5 of the lesson on fibred topoi, with the index category $\Delta^{\mathrm{op}}$), and its limits, colimits, monomorphisms and epimorphisms are computed degreewise.

A simplicial set $K$ is regarded as a simplicial sheaf through the constant sheaves. For a simplicial sheaf $X$, the product $K\times X$ has the terms $(K\times X)_n=\coprod_{K_n}X_n$. A simplicial set is finite if it has finitely many nondegenerate simplices. By the expression (1.8) of the lesson on Kan complexes, a finite simplicial set is the colimit of a finite diagram of standard simplices: one copy of $\Delta[\dim\sigma]$ for each nondegenerate simplex $\sigma$, glued along the maps that express the faces of $\sigma$ through nondegenerate simplices. A product of two finite simplicial sets is finite: each is a quotient of a finite coproduct of standard simplices, products commute with these colimits, a quotient of a finite simplicial set is finite, and $\Delta[n]\times\Delta[m]$ is the nerve of the poset $[n]\times[m]$, whose nondegenerate simplices are its finitely many strictly increasing chains.

Homotopies. For maps $f_0,f_1\colon X\to Y$ of simplicial sheaves, an elementary homotopy from $f_0$ to $f_1$ is a map $h\colon\Delta[1]\times X\to Y$ whose restrictions to the vertices $0$ and $1$ of $\Delta[1]$ are $f_0$ and $f_1$. Maps are homotopic if they are related by the equivalence relation generated by elementary homotopies. A zigzag interval $I$ is a chain of copies $J_1,\ldots,J_r$ of $\Delta[1]$, $r\geq1$, in which $J_q$ and $J_{q+1}$ share one vertex and are otherwise disjoint, and nonconsecutive copies are disjoint. Let $v_q$ be the vertex shared by $J_q$ and $J_{q+1}$, and let $v_0=a$ and $v_r=b$ be the remaining vertices of $J_1$ and $J_r$, the ends of $I$; put $\partial I=\{a,b\}$. A chain of elementary homotopies, each in either direction, from $f_0$ to $f_1$ is the same as a map $I\times X\to Y$ restricting to $f_0$ at $a$ and to $f_1$ at $b$.

Localization. For a class $W$ of morphisms of a category $\mathcal A$, the localization $\mathcal A[W^{-1}]$ has the objects of $\mathcal A$; its morphisms are the composable strings of morphisms of $\mathcal A$ and formal inverses $w^{-1}$ of members of $W$, modulo the relations given by composition in $\mathcal A$, identities, and $ww^{-1}=1$, $w^{-1}w=1$. The functor $Q\colon\mathcal A\to\mathcal A[W^{-1}]$ inverts $W$; every functor $F$ that inverts $W$ factors uniquely as $F=\bar FQ$; and a natural transformation $F\to G$ between two such functors is also a natural transformation $\bar F\to\bar G$, since it is natural with respect to every $Q(w)$ and hence to $Q(w)^{-1}$. The morphisms may form classes rather than sets; as usual we allow this, or pass to a larger universe.

1. Simplicial objects of a topos

1.1. Mapping objects

For $U\in\mathcal C$ put $\Delta[n]_U=\Delta[n]\times h_U$. For a sheaf $F$, regarded as a constant simplicial sheaf, a map $g\colon\Delta[n]\times F\to X$ is a family of maps $g_\alpha\colon F\to X_m$, one for each $\alpha\colon[m]\to[n]$, with $X(\beta)g_\alpha=g_{\alpha\beta}$. It is determined by $g_{\mathrm{id}}$, since $g_\alpha=X(\alpha)g_{\mathrm{id}}$, and every map $F\to X_n$ arises in this way. Hence $\operatorname{Hom}(\Delta[n]_U,X)=X_n(U).$

For simplicial sheaves $X,Y$, the mapping object $\underline{\operatorname{Homs}}(X,Y)$ is the simplicial sheaf whose sheaf of $n$-simplices is $U\mapsto\operatorname{Hom}(\Delta[n]_U\times X,Y)$, with the simplicial operators induced by those of the $\Delta[n]$. It is a sheaf because a map $\Delta[n]_U\times X\to Y$ is a map $\Delta[n]\times X|U\to Y|U$ of simplicial sheaves on $T_{/U}$, and such maps glue. Its global sections form the simplicial set $\operatorname{Hom}(X,Y)$ with $n$-simplices the maps $\Delta[n]\times X\to Y$.

Lemma 1.1.

For simplicial sheaves $Z$, $X$, $Y$ there are bijections between $\operatorname{Hom}(Z,\underline{\operatorname{Homs}}(X,Y))$ and $\operatorname{Hom}(Z\times X,Y)$, natural in all three variables.
Let $K$ be a finite simplicial set. The sheaf $\underline{\operatorname{Homs}}(K,Y)_0$ of maps $K\to Y$ is a finite limit of the terms of $Y$, and so is every term $\underline{\operatorname{Homs}}(K,Y)_n$. For a functor $\Phi$ from $T$ to a topos, or to the category of sets, that preserves finite limits, the canonical map $\Phi\underline{\operatorname{Homs}}(K,Y)\to\underline{\operatorname{Homs}}(K,\Phi Y)$ is an isomorphism. This applies to the inverse image of a morphism of topoi and to the stalk at a point. For the inverse image $f^{-1}$ of a morphism of topoi $f$, the map from $f^{-1}\underline{\operatorname{Homs}}(X,Y)$ to $\underline{\operatorname{Homs}}(f^{-1}X,f^{-1}Y)$ is an isomorphism more generally when $X$ is locally isomorphic to the constant simplicial sheaf of a finite simplicial set.
For a finite simplicial set $K$ and $U\in\mathcal C$, the maps $K\times h_U\to Y$ are the sections over $U$ of $\underline{\operatorname{Homs}}(K,Y)_0$.

Proof. (1) For $Z=\Delta[n]_U$ both sides are $\operatorname{Hom}(\Delta[n]_U\times X,Y)$, by the formula above and the definition, naturally in $(n,U)$. By Section 5 of the lesson on fibred topoi, $\operatorname{Simpl}(T)$ is the topos of sheaves on $\mathcal C\times\Delta$, in which $\Delta[n]_U$ is the sheaf associated with the presheaf represented by $(U,[n])$. So every $Z$ is the colimit of the objects $\Delta[n]_U$ mapping to it. Both sides take colimits in $Z$ to limits, because products in a topos commute with colimits.

(2) A finite simplicial set is the colimit of a finite diagram of standard simplices, and maps $\Delta[m]\times F\to Y$ are maps $F\to Y_m$. So maps $K\times F\to Y$ are maps from $F$ into a finite limit of the $Y_m$, and this finite limit is $\underline{\operatorname{Homs}}(K,Y)_0$. The same applies to the finite simplicial set $\Delta[n]\times K$, which gives the $n$-th term. A functor preserving finite limits preserves these. For $X$ locally constant, choose a covering $\{U_\alpha\}$ of the final object of $T$ with $X|U_\alpha$ constant of a finite simplicial set; the objects $f^{-1}U_\alpha$ cover the final object of the source of $f$, mapping objects and inverse images commute with these restrictions, and a map of sheaves that is an isomorphism on the members of a covering is one.

(3) The functor $F\mapsto F(U)$ preserves limits, and maps $K\times h_U\to Y$ are the elements of the corresponding finite limit of the sets $Y_m(U)$. $\square$

1.2. Homotopies and paths

By Lemma 1.1 (1), an elementary homotopy from $f_0$ to $f_1$ is the same as a map $X\to\operatorname{Path}(Y)=\underline{\operatorname{Homs}}(\Delta[1],Y)$ whose composites with the evaluations $\operatorname{ev}_0,\operatorname{ev}_1\colon\operatorname{Path}(Y)\to Y$ at the two vertices are $f_0$ and $f_1$. The vertices of the simplicial set $\operatorname{Hom}(X,Y)$ are the maps $X\to Y$ and its edges are the elementary homotopies, so the set of homotopy classes is $\operatorname{Hot}(X,Y)=\pi_0\operatorname{Hom}(X,Y).$ Composition on either side preserves elementary homotopies: $h\mapsto gh$ and $h\mapsto h\circ(\Delta[1]\times f)$. Let $\operatorname{Hot}_\bullet(T)$ be the category of simplicial sheaves with the homotopy classes of maps as morphisms. A map is a homotopy equivalence if it becomes invertible there. For simplicial sets (the topos of sets) we write $[A,B]$ for $\operatorname{Hot}(A,B)$.

The order-preserving maps $\min,\max\colon[1]\times[1]\to[1]$ give, by Proposition 2.1 of the lesson on décalage, maps $\Delta[1]\times\Delta[1]\to\Delta[1]$. With the first factor as the homotopy coordinate, $\min$ is an elementary homotopy from the constant map at $0$ to the identity of $\Delta[1]$ that fixes the vertex $0$; $\max$ is one from the identity to the constant map at $1$ that fixes the vertex $1$.

Lemma 1.2. Let $X$ and $Y$ be simplicial sheaves.

Let $\pi\colon\Delta[1]\times X\to X$ be the projection and $e_\epsilon\colon X\to\Delta[1]\times X$ the inclusion at the vertex $\epsilon$. Then $\pi e_\epsilon=1$ and $e_\epsilon\pi$ is homotopic to the identity; so $\pi$, $e_0$ and $e_1$ are homotopy equivalences.
The constant-path map $c\colon Y\to\operatorname{Path}(Y)$ is a homotopy equivalence, with homotopy inverses $\operatorname{ev}_0$ and $\operatorname{ev}_1$.
For a zigzag interval $I$, the identity of $I$ is homotopic to the constant map at $a$. Hence the projection $I\times X\to X$ and the constant-map inclusion $Y\to\underline{\operatorname{Homs}}(I,Y)$ are homotopy equivalences.

Proof. (1) The map $\min\times1_X$ from $\Delta[1]\times(\Delta[1]\times X)$ to $\Delta[1]\times X$ is an elementary homotopy from $e_0\pi$ to the identity, and $\max\times1_X$ one from the identity to $e_1\pi$.

(2) $\operatorname{ev}_\epsilon c=1$. By Lemma 1.1 (1), the map $\Delta[1]\times\operatorname{Path}(Y)\to\operatorname{Path}(Y)$ adjoint to $(s,\omega,t)\mapsto\omega(\min(s,t))$ is an elementary homotopy from $c\operatorname{ev}_0$ to the identity; $\max$ gives one from the identity to $c\operatorname{ev}_1$.

(3) For $0\leq q\leq r$ let $\rho_q\colon I\to I$ be the identity on $J_1\cup\cdots\cup J_q$ and the constant map at $v_q$ on $J_{q+1}\cup\cdots\cup J_r$; so $\rho_r=1$ and $\rho_0$ is constant at $a$. For $1\leq q\leq r$, identify $J_q$ with $\Delta[1]$. If $v_{q-1}$ is its vertex $0$, the map $\Delta[1]\times I\to I$ that is the projection on $\Delta[1]\times(J_1\cup\cdots\cup J_{q-1})$, is $\min$ on $\Delta[1]\times J_q$, and sends $(t,x)$ to the vertex $t$ of $J_q$ on $\Delta[1]\times(J_{q+1}\cup\cdots\cup J_r)$ is an elementary homotopy from $\rho_{q-1}$ to $\rho_q$. The three definitions agree at $v_{q-1}$ and $v_q$, where the pieces meet. If $v_{q-1}$ is the vertex $1$ of $J_q$, the same construction with $\max$ is an elementary homotopy from $\rho_q$ to $\rho_{q-1}$. So $1=\rho_r$ is homotopic to $\rho_0$. Taking products with $X$, respectively applying Lemma 1.1 (1) to $\underline{\operatorname{Homs}}(I,Y)$, gives the last two assertions; the maps in the other direction are the inclusion at $a$ and the evaluation at $a$. $\square$

1.3. Mapping cylinders

Proposition 1.3. Let $g\colon E\to F$ be a map of simplicial sheaves. Let $M(g)$ be the pushout of $g$ and of the inclusion $e_0\colon E\to\Delta[1]\times E$ at the vertex $0$, the mapping cylinder of $g$. Let $i\colon E\to M(g)$ be induced by the inclusion at the vertex $1$, $j\colon F\to M(g)$ the canonical map, and $p\colon M(g)\to F$ the map that is $g\pi$ on $\Delta[1]\times E$ and the identity on $F$.

$pi=g$, $pj=1_F$, and $jp$ is homotopic to the identity by an elementary homotopy that is constant on $F$. So $p$ and $j$ are homotopy equivalences.
$i$ is a monomorphism.
If $E$ and $F$ are simplicial sheaves over a simplicial sheaf $B$ and $g$ is a map over $B$, then $M(g)$ is over $B$, the maps $i$, $j$, $p$ are maps over $B$, and the homotopy of (1) is a homotopy over $B$.

Proof. The map $p$ is well defined because $g\pi e_0=g$. In degree $n$, $(\Delta[1]\times E)_n$ is the coproduct of copies of $E_n$ indexed by $\alpha\in\Delta[1]_n$, and $e_0$ is the inclusion of the copy indexed by the constant simplex $0$. So $M(g)_n$ is the coproduct of $F_n$ and of the copies of $E_n$ indexed by $\alpha\neq0$, and $i_n$ is the inclusion of the copy indexed by the constant simplex $1$. Coproduct inclusions in a topos are monomorphisms, which proves (2).

(1) The identities $pi=g$ and $pj=1$ hold by construction. The map $\Delta[1]\times(\Delta[1]\times E)\to\Delta[1]\times E$, $(s,t,x)\mapsto(\min(s,t),x)$, followed by the map to $M(g)$, and the constant homotopy of $F$, agree on $\Delta[1]\times\{0\}\times E$: there $\min(s,0)=0$, and $(0,x)$ is identified with $g(x)$. Since $\Delta[1]\times-$ preserves the pushout, they define an elementary homotopy $\Delta[1]\times M(g)\to M(g)$. At $s=0$ it is $jp$, at $s=1$ the identity.

(3) The structure maps of $F$ and of $\Delta[1]\times E\to E\to B$ agree on $E$, so they define $M(g)\to B$. The maps $i$, $j$, $p$ and the homotopy of (1) are compatible with these maps to $B$, because $\min$ only changes the coordinate in $\Delta[1]$. $\square$

2. Quasi-isomorphisms of simplicial sets

By Definitions 1.1 and 2.1 of the lesson on homotopy sheaves, applied to the topos of sets, a map of simplicial sets $f\colon A\to B$ is a quasi-isomorphism if it bijects components and induces isomorphisms $\pi_n(A,x)\to\pi_n(B,f(x))$ for every vertex $x$ and $n\geq1$, where $\pi_n(A,x)=\pi_n(A^{\mathrm{Kan}},x)$ is computed in the Kan replacement (5.5) of the lesson on Kan complexes; the inclusion $A\subset A^{\mathrm{Kan}}$ bijects components (Lemma 1.2 (3) of the lesson on homotopy sheaves). Anodyne extensions are as in Section 5.3 of that lesson: the class of monomorphisms generated by the horn inclusions under pushouts, compositions indexed by ordinals, and retracts. Kan fibrations, and in particular maps from Kan complexes to a point, lift against them. Write $\operatorname{Ho}(\mathrm{Kan})$ for the category of Kan complexes and homotopy classes of maps.

Lemma 2.1. Let $A\subset B$ be a monomorphism of simplicial sets and $I$ a zigzag interval with ends $a$, $b$.

The inclusion $(I\times A)\cup(\{a\}\times B)\subset I\times B$ is anodyne.
If $A\subset B$ is anodyne, then $(I\times A)\cup(\partial I\times B)\subset I\times B$ and $I\times A\subset I\times B$ are anodyne.

Proof. (1) Put $I_q=J_1\cup\cdots\cup J_q$, so $I_0=\{a\}$. For $1\leq q\leq r$, the inclusion of $(I\times A)\cup(I_{q-1}\times B)$ into $(I\times A)\cup(I_q\times B)$ adds $J_q\times B$, whose intersection with the former is $(J_q\times A)\cup(\{v_{q-1}\}\times B)$. A union of two simplicial subsets is the pushout over their intersection, so this inclusion is a pushout of $(J_q\times A)\cup(\{v_{q-1}\}\times B)\subset J_q\times B$. That map is anodyne by (5.13) of the lesson on Kan complexes, for the endpoint $v_{q-1}$ of $J_q\cong\Delta[1]$. A finite composite of anodyne extensions is anodyne.

(2) For the first inclusion, add the copies $J_q\times B$ in the same order to $(I\times A)\cup(\partial I\times B)$. For $q<r$ each step is as in (1). In the last step both vertices of $J_r$ are present, so the intersection is $(J_r\times A)\cup(\partial J_r\times B)$, and the step is a pushout of the anodyne map (5.14). For the second inclusion, $I\times A\subset(I\times A)\cup(\{a\}\times B)$ is a pushout of $\{a\}\times A\subset\{a\}\times B$, and the rest is (1). $\square$

Proposition 2.2.

If $j\colon A\to B$ is anodyne and $K$ is a Kan complex, then $j^*\colon[B,K]\to[A,K]$ is bijective.
If $f_0,f_1\colon A\to B$ are homotopic, so are $f_0^{\mathrm{Kan}},f_1^{\mathrm{Kan}}\colon A^{\mathrm{Kan}}\to B^{\mathrm{Kan}}$. Hence $A\mapsto A^{\mathrm{Kan}}$, $f\mapsto[f^{\mathrm{Kan}}]$ is a functor $\Phi$ from simplicial sets to $\operatorname{Ho}(\mathrm{Kan})$ that identifies homotopic maps.
For a map $f\colon A\to B$ the following are equivalent: (a) $f$ is a quasi-isomorphism; (b) $\Phi(f)$ is an isomorphism; (c) $f^*\colon[B,K]\to[A,K]$ is bijective for every Kan complex $K$.

Proof. (1) Every map $A\to K$ extends along $j$, since $K$ is Kan. Let $u,v\colon B\to K$ and let $H\colon I\times A\to K$ be a chain of homotopies from $uj$ to $vj$. Then $H$, $u$ and $v$ define a map on $(I\times A)\cup(\partial I\times B)$, which extends over $I\times B$ by Lemma 2.1 (2). The extension is a chain of homotopies from $u$ to $v$.

(2) Let $H\colon I\times A\to B$ be a chain of homotopies from $f_0$ to $f_1$. Followed by $B\subset B^{\mathrm{Kan}}$, it agrees on $\partial I\times A$ with $f_0^{\mathrm{Kan}}$ and $f_1^{\mathrm{Kan}}$ on the two ends of $I\times A^{\mathrm{Kan}}$. Since $A\subset A^{\mathrm{Kan}}$ is anodyne, Lemma 2.1 (2) extends this map over $I\times A^{\mathrm{Kan}}$. Functoriality of $\Phi$ follows from that of $A\mapsto A^{\mathrm{Kan}}$.

(3) (a)$\Leftrightarrow$(b). Every vertex of $A^{\mathrm{Kan}}$ is joined by a finite chain of edges to a vertex of $A$: attaching $\Delta[1]$ along a vertex adds one vertex joined by an edge to an old one, and attaching a higher simplex along a horn adds no vertex. The transports (5.15) along edges are isomorphisms compatible with $f^{\mathrm{Kan}}$. So if $f$ is a quasi-isomorphism, $f^{\mathrm{Kan}}$ induces isomorphisms on all groups at all vertices of $A^{\mathrm{Kan}}$; it bijects components by Lemma 1.2 (3) of the lesson on homotopy sheaves. Theorem 5.3f of the lesson on Kan complexes makes $f^{\mathrm{Kan}}$ a homotopy equivalence, that is, $\Phi(f)$ an isomorphism. Conversely, if $f^{\mathrm{Kan}}$ is a homotopy equivalence, the same theorem gives the bijection on components and the isomorphisms at all vertices.

(b)$\Leftrightarrow$(c). For every Kan complex $K$, the restrictions $[A^{\mathrm{Kan}},K]\to[A,K]$ and $[B^{\mathrm{Kan}},K]\to[B,K]$ are bijective by (1), and they carry $(f^{\mathrm{Kan}})^*$ to $f^*$. So (c) says that composition with $\Phi(f)$ is bijective on morphisms of $\operatorname{Ho}(\mathrm{Kan})$ into every object, which by the Yoneda lemma is (b). $\square$

Corollary 2.3.

Homotopy equivalences and anodyne extensions are quasi-isomorphisms, and a map homotopic to a quasi-isomorphism is a quasi-isomorphism.
If two of $f$, $g$, $gf$ are quasi-isomorphisms, so is the third; if $gf$ and $hg$ are quasi-isomorphisms, so are $f$, $g$ and $h$.
A coproduct of quasi-isomorphisms is a quasi-isomorphism. If $f\colon A\to B$ is the union of an increasing sequence of quasi-isomorphisms $f_r\colon A_r\to B_r$ of simplicial subsets, then $f$ is a quasi-isomorphism.

Proof. (1) A homotopy equivalence $f$ with homotopy inverse $g$ has $\Phi(g)\Phi(f)=1$ and $\Phi(f)\Phi(g)=1$, by Proposition 2.2 (2). An anodyne extension satisfies (c) of Proposition 2.2 (3), by its part (1). Homotopic maps have the same image under $\Phi$.

(2) This holds for isomorphisms of a category, hence for quasi-isomorphisms by Proposition 2.2 (3). For instance, if $gf$ and $hg$ are isomorphisms, then $g$ has a right inverse $f(gf)^{-1}$ and a left inverse $(hg)^{-1}h$, so $g$ is an isomorphism, and then so are $f$ and $h$.

(3) A horn is connected, so a horn in a coproduct lies in one summand, and the Kan replacement of a coproduct is the coproduct of the Kan replacements. Components, cycles (5.11) and the relation (5.12) of a Kan complex at a vertex lie in the component of that vertex. Hence the components and groups of a coproduct are those of its summands. For the union, Proposition 1.5 (1) of the lesson on homotopy sheaves, in the topos of sets, gives $\pi_n(A,x)=\varinjlim_r\pi_n(A_r,x)$ for $n\geq0$, compatibly with $f$; a colimit of isomorphisms is an isomorphism. $\square$

Proposition 2.4. Let $i\colon A\to B$ be a monomorphism and a quasi-isomorphism of simplicial sets. If $B$ is a Kan complex, $i$ is anodyne.

Proof. The construction (5.19) of the lesson on Kan complexes applies to any map into a Kan complex. It factors $i=pa$, with $a\colon A\to C$ anodyne, $p\colon C\to B$ a Kan fibration and $C$ a Kan complex. By Corollary 2.3, $a$ and hence $p$ are quasi-isomorphisms. By Lemma 1.2 (2) of the lesson on homotopy sheaves, the groups of the Kan complexes $C$ and $B$ are the original ones. So $p$ satisfies the hypothesis of Lemma 5.3d of the lesson on Kan complexes, and by Lemma 5.3e it lifts against every boundary inclusion $\partial\Delta[m]\subset\Delta[m]$. A monomorphism is obtained by attaching its missing nondegenerate simplices along their boundaries, so $p$ lifts against every monomorphism. A lift $\sigma\colon B\to C$ in the square formed by $a$, $i$ and the identity of $B$ satisfies $\sigma i=a$ and $p\sigma=1$. So $i$ is a retract of $a$ in the category of arrows: the maps $(1_A,\sigma)\colon i\to a$ and $(1_A,p)\colon a\to i$ compose to the identity. Anodyne extensions are stable under retracts. $\square$

Proposition 2.5 (gluing). Let $i\colon A\to B$ be a monomorphism and a quasi-isomorphism of simplicial sets.

For every map $u\colon A\to C$, the map $C\to C\amalg_AB$ is a monomorphism and a quasi-isomorphism.
For every zigzag interval $I$, the inclusion $(I\times A)\cup(\partial I\times B)\subset I\times B$ is a quasi-isomorphism.

Proof. Let $k\colon B\to B'=B^{\mathrm{Kan}}$. The composite $ki$ is a monomorphism and a quasi-isomorphism into a Kan complex, hence anodyne by Proposition 2.4.

(1) Pushouts of monomorphisms of simplicial sets are monomorphisms. The map $C\to C\amalg_AB'$ is a pushout of $ki$, hence anodyne. The map $C\amalg_AB\to C\amalg_AB'=(C\amalg_AB)\amalg_BB'$ is a pushout of $k$, hence anodyne. Both are quasi-isomorphisms (Corollary 2.3 (1)), and so is $C\to C\amalg_AB$, by two out of three.

(2) Inside $I\times B'$, let $N=(I\times A)\cup(\partial I\times B)$ and $N'=(I\times A)\cup(\partial I\times B')$. The inclusion $N'\subset I\times B'$ is anodyne by Lemma 2.1 (2), applied to the anodyne $A\subset B'$. The inclusion $N\subset N'$ adds $\partial I\times B'$, whose intersection with $N$ is $\partial I\times B$; so it is a pushout of two copies of $k$, hence anodyne. The inclusion $I\times B\subset I\times B'$ is anodyne by Lemma 2.1 (2). The two composites $N\subset N'\subset I\times B'$ and $N\subset I\times B\subset I\times B'$ agree. Three of the maps are quasi-isomorphisms, so the fourth is one too. $\square$

Proposition 2.6 (pushouts along monomorphisms). Let $B\leftarrow A\to C$ and $B'\leftarrow A'\to C'$ be diagrams of simplicial sets in which $A\to B$ and $A'\to B'$ are monomorphisms. Let $A\to A'$, $B\to B'$, $C\to C'$ be quasi-isomorphisms forming a map from the first diagram to the second (the two squares commute). Then the induced map $B\amalg_AC\to B'\amalg_{A'}C'$ is a quasi-isomorphism.

Proof. Step 1: the vertical maps are monomorphisms. Put $D=B\amalg_AC$ and $D'=B'\amalg_{A'}C'$. The map $D\to B\amalg_AC'=D\amalg_CC'$ is a pushout of $C\to C'$, hence a quasi-isomorphism by Proposition 2.5 (1). Let $E=B\amalg_AA'$. The map $B\to E$ is a pushout of $A\to A'$, hence a quasi-isomorphism, and $A'\to E$ is a monomorphism, being a pushout of $A\to B$. The map $e\colon E\to B'$ induced by $B\to B'$ and $A'\to B'$ is a quasi-isomorphism by two out of three, and its restriction to $A'$ is the monomorphism $A'\to B'$. Moreover $B\amalg_AC'=E\amalg_{A'}C'$. It remains to show that $E\amalg_{A'}C'\to B'\amalg_{A'}C'$ is a quasi-isomorphism.

Let $M$ be the quotient of $B'\amalg(\Delta[1]\times E)$ that identifies $(0,x)$ with $e(x)$ for $x\in E$, and $(t,y)$ with $e(y)$ for $y\in A'$ and every $t$. In degree $n$, $M_n$ is the disjoint union of $B'_n$ and of the pairs $(\alpha,x)$ with $\alpha\in\Delta[1]_n$ not the constant simplex $0$ and $x\in E_n\smallsetminus A'_n$. The map $i\colon E\to M$, $x\mapsto(1,x)$, is injective: on $A'$ it is $e$, which is injective there, and its other values are the distinct pairs $(1,x)$. The map $p\colon M\to B'$, $(\alpha,x)\mapsto e(x)$, and the inclusion $j\colon B'\to M$ satisfy $pi=e$ and $pj=1$. The map $(s,(\alpha,x))\mapsto(\min(s,\alpha),x)$, constant on $B'$, is an elementary homotopy from $jp$ to the identity that is constant on $B'$. The maps $i$, $j$, $p$ and this homotopy are compatible with the maps from $A'$, which land in $B'$. So $p$ is a homotopy equivalence, $i$ is a monomorphism and a quasi-isomorphism by two out of three, and the map from $E\amalg_{A'}C'$ to $M\amalg_{A'}C'$, which is $M\amalg_E(E\amalg_{A'}C')$, is a quasi-isomorphism by Proposition 2.5 (1). The map $M\amalg_{A'}C'\to B'\amalg_{A'}C'$ induced by $p$ is a homotopy equivalence, with inverse induced by $j$: the homotopy above, extended by the constant homotopy of $C'$, is defined on $\begin{aligned}&\Delta[1]\times(M\amalg_{A'}C')\\&\quad=(\Delta[1]\times M)\amalg_{\Delta[1]\times A'}(\Delta[1]\times C').\end{aligned}$ The composite is the map of the claim.

Step 2: the general case. Apply the mapping cylinder of Proposition 1.3, in the topos of sets, to each vertical map. The three cylinders form a diagram with the maps $M(A\to A')\to M(B\to B')$ and $M(A\to A')\to M(C\to C')$. The first is a monomorphism: in degree $n$ it is the disjoint union of $A'_n\to B'_n$ and of copies of $A_n\to B_n$. The maps $i$ at the vertex $1$, the inclusions $j$ and the projections $p$ are maps of diagrams, with $pj=1$. The maps $i$ and $j$ are monomorphisms and quasi-isomorphisms in each position, because $p$ is a homotopy equivalence and $pi$ is the given quasi-isomorphism. By Step 1, applied to $i$ and to $j$, the maps they induce on pushouts are quasi-isomorphisms. The map induced by $p$ has the map induced by $j$ as a right inverse, so it is a quasi-isomorphism, and so is the given map, the composite of the maps induced by $i$ and by $p$. $\square$

3. Homotopy and quasi-isomorphisms of simplicial sheaves

Quasi-isomorphisms of simplicial sheaves are those of Definition 2.1 of the lesson on homotopy sheaves. Homotopy sheaves are computed from sections: by Proposition 1.4 there, $\pi_n(X|U,x)$ is the sheaf associated with the presheaf $V\mapsto\pi_n(X(V),x|V)$ on $\mathcal C_{/U}$, and the proof of that proposition applies to every simplicial presheaf in place of $X(V)$, the sheaf associated with it in place of $X$.

Proposition 3.1. Let $h\colon\Delta[1]\times X\to Y$ be an elementary homotopy from $f_0$ to $f_1$. Then $\pi_0(f_0)=\pi_0(f_1)$, and for every $U\in\mathcal C$, every base point $x$ of $X$ over $U$ and every $n\geq1$ there is an isomorphism $\tau\colon\pi_n(Y|U,f_0x)\to\pi_n(Y|U,f_1x)$, of sheaves of groups, with $\pi_n(f_1|U,x)=\tau\circ\pi_n(f_0|U,x)$.

Proof. For $V$ over $U$, the summand inclusions of $(\Delta[1]\times X)_m=\coprod_{\Delta[1]_m}X_m$ give a map $\Delta[1]\times X(V)\to(\Delta[1]\times X)(V)$, and composing with $h$ gives an elementary homotopy of simplicial sets from $f_0(V)$ to $f_1(V)$, compatible with restriction. The proof of Proposition 2.2 (2), with the interval $\Delta[1]$, extends it to an elementary homotopy from $f_0(V)^{\mathrm{Kan}}$ to $f_1(V)^{\mathrm{Kan}}$ which, at the vertex $x|V$, is the edge $u_V=h(-,x|V)$ of $Y(V)$. By (5.16) of the lesson on Kan complexes, $\pi_n(f_1(V),x)=(u_V)_*\circ\pi_n(f_0(V),x)$, and the two maps agree on components. The transports $(u_V)_*$ are isomorphisms, natural in $V$: the edges $u_V$ are compatible with restriction, and a map of Kan complexes carries a transport cylinder along an edge to one along its image. Sheafifying gives $\tau$. $\square$

Lemma 3.2. Let $f\colon X\to Y$ be a map such that $\pi_0(f)$ is an epimorphism, and let $y$ be a base point of $Y$ over $U$. There are a covering $\{U_\alpha\to U\}$ and base points $x_\alpha$ of $X$ over $U_\alpha$ with the following property. For every map $g\colon Y\to Z$ and $n\geq1$ there are isomorphisms $\sigma\colon\pi_n(Y|U_\alpha,y)\to\pi_n(Y|U_\alpha,fx_\alpha)$ and $\sigma'\colon\pi_n(Z|U_\alpha,gy)\to\pi_n(Z|U_\alpha,gfx_\alpha)$ with $\sigma'\circ\pi_n(g|U_\alpha,y)=\pi_n(g|U_\alpha,fx_\alpha)\circ\sigma$.

Proof. The section of $\pi_0(Y)$ defined by $y$ lifts locally to $\pi_0(X)$, and sections of $\pi_0(X)$ lift locally to $X_0$. So there are a covering $\{U_\alpha\to U\}$ and base points $x_\alpha$ over $U_\alpha$ such that $fx_\alpha$ and $y|U_\alpha$ define the same section of $\pi_0(Y)$. Since $\pi_0(Y)$ is the sheaf associated with the presheaf $V\mapsto\pi_0(Y(V))$, after refining the covering $fx_\alpha$ and $y|U_\alpha$ lie in the same component of the simplicial set $Y(U_\alpha)$. Then they are joined by a finite chain $\gamma$ of edges in the simplicial set $Y(U_\alpha)$. Transport along the restrictions of $\gamma$, and along their images under $g$, gives isomorphisms of presheaves of groups, natural in $V$ and compatible with $g$ as in the proof of Proposition 3.1. Sheafifying gives $\sigma$ and $\sigma'$. $\square$

Corollary 3.3.

If two of $f$, $g$, $gf$ are quasi-isomorphisms, so is the third.
Homotopy equivalences are quasi-isomorphisms, and a map homotopic to a quasi-isomorphism is a quasi-isomorphism.
If $\mathcal P$ is a conservative family of points of $T$, a map $f$ is a quasi-isomorphism if and only if $f_x$ is a quasi-isomorphism of simplicial sets for every $x\in\mathcal P$.

Proof. (1) The statements on $\pi_0$ hold for isomorphisms of sheaves. If $f$ and $g$ are quasi-isomorphisms, so is $gf$. If $gf$ and $g$ are, then $\pi_n(f|U,x)=\pi_n(g|U,fx)^{-1}\pi_n(gf|U,x)$ is an isomorphism. If $gf$ and $f$ are, then $\pi_n(g|U,fx)$ is an isomorphism at every base point of the form $fx$; by Lemma 3.2, applied to $f$, $\pi_n(g|U,y)$ is an isomorphism after restriction to a covering of $U$, for every base point $y$, and being an isomorphism of sheaves is a local property.

(3) This is Lemma 2.2 of the lesson on homotopy sheaves, for all $n$. $\square$

4. The homotopy category and the derived category

Proposition 4.1. The canonical functor $\operatorname{Simpl}(T)\to\operatorname{Hot}_\bullet(T)$ is the localization of $\operatorname{Simpl}(T)$ with respect to the homotopy equivalences.

Proof. The functor $Q\colon\operatorname{Simpl}(T)\to\operatorname{Hot}_\bullet(T)$ inverts homotopy equivalences, by definition. Let $F$ be a functor that inverts them. By Lemma 1.2 (1), $F(\pi)$ is invertible and $F(e_0)=F(\pi)^{-1}=F(e_1)$. For an elementary homotopy $h$ from $f_0$ to $f_1$ this gives $F(f_0)=F(h)F(e_0)$ and $F(f_1)=F(h)F(e_1)$, so $F(f_0)=F(f_1)$. So $F$ is constant on homotopy classes, and it factors uniquely through $Q$, which is the identity on objects and surjective on morphisms. Natural transformations factor in the same way. $\square$

Definition 4.2. The simplicial derived category $D_\bullet(T)$ is the localization of $\operatorname{Simpl}(T)$ with respect to the quasi-isomorphisms. A morphism of $\operatorname{Hot}_\bullet(T)$ is a quasi-isomorphism if one of its representatives is, equivalently all of them are (Corollary 3.3 (2)).

Corollary 4.3. The functor $\operatorname{Simpl}(T)\to D_\bullet(T)$ factors through $\operatorname{Hot}_\bullet(T)$, and $D_\bullet(T)$ is the localization of $\operatorname{Hot}_\bullet(T)$ with respect to its quasi-isomorphisms.

Proof. By Corollary 3.3 (2), homotopy equivalences are quasi-isomorphisms. So a functor out of $\operatorname{Simpl}(T)$ that inverts the quasi-isomorphisms inverts the homotopy equivalences, factors uniquely through $\operatorname{Hot}_\bullet(T)$ by Proposition 4.1, and the factorization inverts the quasi-isomorphisms of $\operatorname{Hot}_\bullet(T)$. Conversely, a functor out of $\operatorname{Hot}_\bullet(T)$ that inverts its quasi-isomorphisms gives, by composition, one out of $\operatorname{Simpl}(T)$ that inverts the quasi-isomorphisms. The two localizations have the same universal property. $\square$

5. Left fractions

Theorem 5.1. Suppose that $T$ has enough points, that is, a conservative family of points. In $\operatorname{Hot}_\bullet(T)$ the quasi-isomorphisms form a left multiplicative system [Stacks, Tag 04VC]. Consequently every morphism $X\to Y$ of $D_\bullet(T)$ is $t^{-1}u$ for a morphism $u\colon X\to Y'$ and a quasi-isomorphism $t\colon Y\to Y'$ of $\operatorname{Hot}_\bullet(T)$, and $t^{-1}u=t^{-1}v$ if and only if $su=sv$ for some quasi-isomorphism $s\colon Y'\to Y''$.

Proof. Let $\mathcal P$ be a conservative family of points. A stalk functor preserves colimits, finite limits and monomorphisms, and $(K\times X)_x=K\times X_x$ for a simplicial set $K$. Quasi-isomorphisms are detected at the points of $\mathcal P$ (Corollary 3.3 (3)).

LMS1. Identities are quasi-isomorphisms, and so are composites of quasi-isomorphisms.

LMS2. Let $t\colon X\to Z$ be a quasi-isomorphism and $g\colon X\to Y$ a map, both chosen as representatives. By Proposition 1.3, $t=pi$ with $i\colon X\to M(t)$ a monomorphism and $p$ a homotopy equivalence with homotopy inverse $j$. By Corollary 3.3, $i$ is a quasi-isomorphism. Let $s\colon Y\to W=Y\amalg_XM(t)$ and $w\colon M(t)\to W$ be the maps to the pushout of $g$ and $i$, so that $wi=sg$. At every point $x\in\mathcal P$, $s_x$ is the pushout of the monomorphism and quasi-isomorphism $i_x$, hence a quasi-isomorphism by Proposition 2.5 (1). So $s$ is a quasi-isomorphism. Put $f=wj$. Then $ft=wjpi$ is homotopic to $wi=sg$.

LMS3. Let $f,g\colon X\to Y$ and a quasi-isomorphism $t\colon Z\to X$ satisfy $ft\simeq gt$. Write $t=pi$ as above, with $p\colon M(t)\to X$ and homotopy inverse $j$. If a quasi-isomorphism $s$ satisfies $sfp\simeq sgp$, then $sf\simeq sfpj\simeq sgpj\simeq sg$. Since $fpi=ft\simeq gt=gpi$, we may replace $X$, $f$, $g$, $t$ by $M(t)$, $fp$, $gp$, $i$, and assume that $t$ is a monomorphism.

Choose a chain of homotopies $H\colon I\times Z\to Y$ from $ft$ to $gt$, for a zigzag interval $I$. Let $N=(I\times Z)\cup(\partial I\times X)\subset I\times X$. The maps $H$ and $(f,g)\colon\partial I\times X\to Y$ agree on $\partial I\times Z$, so they define $k\colon N\to Y$. Let $s\colon Y\to W=Y\amalg_N(I\times X)$ and $K\colon I\times X\to W$ be the maps to the pushout, so that $K|_N=sk$. At a point $x\in\mathcal P$, $N_x\to I\times X_x$ is the inclusion $(I\times Z_x)\cup(\partial I\times X_x)\subset I\times X_x$, a quasi-isomorphism by Proposition 2.5 (2), and a monomorphism; so $s_x$, its pushout, is a quasi-isomorphism by Proposition 2.5 (1). Hence $s$ is a quasi-isomorphism. The map $K$ is a chain of homotopies from $sf$ to $sg$.

The description of the morphisms of $D_\bullet(T)$, which by Corollary 4.3 is the localization of $\operatorname{Hot}_\bullet(T)$, follows from [Stacks, Tag 04VE, Tag 04VF, Tag 04VG]: the category of fractions constructed there has the universal property of the localization. $\square$

Remark 5.2. The points of $T$ are used only to bring the two gluing properties of Proposition 2.5 from simplicial sets to simplicial sheaves. Illusie remarks that the hypothesis is probably unnecessary; Theorem 5.1 is proved here under it.

6. Over a fixed simplicial sheaf

Fix a simplicial sheaf $B$. A map in the category $\operatorname{Simpl}(T)_{/B}$ of simplicial sheaves over $B$ is a quasi-isomorphism if the underlying map is one. For $X,Y$ over $B$, an elementary $B$-homotopy is an elementary homotopy $\Delta[1]\times X\to Y$ which is a map over $B$, where $\Delta[1]\times X$ lies over $B$ through the projection to $X$. Let $\operatorname{Hot}_\bullet(T,B)$ be the category of simplicial sheaves over $B$ and classes of maps over $B$ for the equivalence relation generated by elementary $B$-homotopies; its invertible maps are the $B$-homotopy equivalences. As in Section 1.2, these classes form $\pi_0$ of the simplicial set whose $n$-simplices are the maps $\Delta[n]\times X\to Y$ over $B$. Let $D_\bullet(T,B)$ be the localization of $\operatorname{Simpl}(T)_{/B}$ with respect to the quasi-isomorphisms. For $B$ the final simplicial sheaf these are $\operatorname{Hot}_\bullet(T)$ and $D_\bullet(T)$.

Proposition 6.1. The canonical functor $\operatorname{Simpl}(T)_{/B}\to\operatorname{Hot}_\bullet(T,B)$ is the localization with respect to the $B$-homotopy equivalences, and $D_\bullet(T,B)$ is the localization of $\operatorname{Hot}_\bullet(T,B)$ with respect to the classes of quasi-isomorphisms over $B$. Every map over $B$ is the composite of a monomorphism over $B$ and a $B$-homotopy equivalence.

Proof. The homotopies of Lemma 1.2 (1) only change the coordinate in $\Delta[1]$, so they are $B$-homotopies, and the projection $\Delta[1]\times X\to X$ is a $B$-homotopy equivalence. The proofs of Proposition 4.1 and Corollary 4.3 then apply, since a $B$-homotopy equivalence is a homotopy equivalence and hence a quasi-isomorphism. The last assertion is Proposition 1.3 (3). $\square$

Theorem 6.2. If $T$ has enough points, the quasi-isomorphisms of $\operatorname{Hot}_\bullet(T,B)$ form a left multiplicative system.

Proof. The proof of Theorem 5.1 applies over $B$. Colimits in $\operatorname{Simpl}(T)_{/B}$ are computed in $\operatorname{Simpl}(T)$. In LMS2, the mapping cylinder lies over $B$ and $p$ is a $B$-homotopy equivalence (Proposition 1.3 (3)). In LMS3, a chain of $B$-homotopies is a map $H\colon I\times Z\to Y$ over $B$; the maps $N\to Y\to B$ and $N\subset I\times X\to X\to B$ agree, so the pushout $W$ lies over $B$, and $K\colon I\times X\to W$ is a chain of $B$-homotopies from $sf$ to $sg$. $\square$

7. Exercises

Exercise 7.1 (easy). Let $A\subset B$ be a monomorphism and a quasi-isomorphism of simplicial sets, and $K$ a Kan complex. Show that every map $A\to K$ extends to $B$, and that two extensions are homotopic by an elementary homotopy that is constant on $A$.

Exercise 7.2 (medium). Let $S=\Delta[1]/\partial\Delta[1]$ be the simplicial set obtained from $\Delta[1]$ by identifying its two vertices, and $q\colon\Delta[1]\to S$ the quotient map. Show that the pushout of the quasi-isomorphism $\Delta[1]\to\Delta[0]$ along $q$ is $S\to\Delta[0]$, and that it is not a quasi-isomorphism. Which hypothesis of Proposition 2.5 (1) fails?

Exercise 7.3 (medium). Show that $\operatorname{Hom}_{D_\bullet(\mathrm{Sets})}(\Delta[0],Y)=\pi_0(Y)$ for every simplicial set $Y$.

Exercise 7.4 (hard). Show that for every simplicial set $K$ the functor $K\times-$ carries quasi-isomorphisms of simplicial sets to quasi-isomorphisms. Deduce that if $A\subset B$ is a monomorphism and a quasi-isomorphism and $K\subset L$ is any monomorphism of simplicial sets, then $(L\times A)\cup(K\times B)\subset L\times B$ is a quasi-isomorphism.

8. Complete solutions

Solution 7.1. Let $k\colon B\to B^{\mathrm{Kan}}$. The composite $A\subset B\subset B^{\mathrm{Kan}}$ is a monomorphism and a quasi-isomorphism into a Kan complex, hence anodyne by Proposition 2.4. So a map $A\to K$ extends to $B^{\mathrm{Kan}}$, and its restriction to $B$ extends it to $B$. Let $u,v\colon B\to K$ be two extensions. The inclusion $(\Delta[1]\times A)\cup(\partial\Delta[1]\times B)\subset\Delta[1]\times B$ is a monomorphism and a quasi-isomorphism by Proposition 2.5 (2), with the zigzag interval $\Delta[1]$. The map that is $u$ and $v$ on the two ends and the given map composed with the projection on $\Delta[1]\times A$ is well defined, and by the first part it extends to $\Delta[1]\times B$. The extension is the required homotopy.

Solution 7.2. The image of $q$ is all of $S$, so the pushout $S\amalg_{\Delta[1]}\Delta[0]$ is the quotient of $S$ identifying all its simplices with the corresponding degenerate simplices of one vertex, that is, $\Delta[0]$. The map $\Delta[1]\to\Delta[0]$ is a homotopy equivalence, because $\Delta[1]$ is the nerve of $[1]$, which has a final object (Corollary 2.2 of the lesson on décalage); it is a quasi-isomorphism by Corollary 2.3. The simplicial set $S$ has one vertex $x$ and one nondegenerate edge $\gamma$, and no nondegenerate simplex of higher dimension. Its groupoid $P_1(S)$, from Section 5.1 of the lesson on Kan complexes, has one object and one generator $\gamma$; degenerate triangles impose only identity laws, so it is the free group on $\gamma$, isomorphic to $\mathbb Z$. By (5.6) there, $P_1(S)$ is equivalent to $\Pi_1(S^{\mathrm{Kan}})$, whose vertex group at $x$ is $\pi_1(S,x)$ by Proposition 4.3 there. So $\pi_1(S,x)\cong\mathbb Z$, while $\pi_1(\Delta[0])=0$, and $S\to\Delta[0]$ is not a quasi-isomorphism. The map $q$ along which the pushout is taken is not a monomorphism, and neither is $\Delta[1]\to\Delta[0]$; Proposition 2.5 (1) needs the quasi-isomorphism to be a monomorphism, and Proposition 2.6 needs one of the two maps out of $A$ to be one.

Solution 7.3. For a simplicial set $Y'$, $[\Delta[0],Y']$ is the set of vertices of $Y'$ modulo the relation generated by edges, since an elementary homotopy $\Delta[1]\times\Delta[0]\to Y'$ is an edge; that is, $[\Delta[0],Y']=\pi_0(Y')$. The functor $\pi_0$ from $\operatorname{Hot}_\bullet(\mathrm{Sets})$ to sets inverts quasi-isomorphisms, so it factors through $D_\bullet(\mathrm{Sets})$ (Corollary 4.3), and gives a map $\Psi\colon\operatorname{Hom}_{D_\bullet}(\Delta[0],Y)\to\pi_0(Y),$ with $\Psi(t^{-1}u)=\pi_0(t)^{-1}(u)$. By Theorem 5.1 every morphism $\Delta[0]\to Y$ has the form $t^{-1}u$, with $u\in\pi_0(Y')$ and $t\colon Y\to Y'$ a quasi-isomorphism, and two such morphisms can be written with a common denominator [Stacks, Tag 04VE]. If $\Psi(t^{-1}u)=\Psi(t^{-1}v)$, then $u=v$ in $\pi_0(Y')$, so $u$ and $v$ are homotopic and $t^{-1}u=t^{-1}v$. So $\Psi$ is injective. It is surjective, since the class of a vertex $y$ of $Y$ is $\Psi(1^{-1}y)$.

Solution 7.4. Let $f\colon A\to B$ be a quasi-isomorphism. For $K=\Delta[n]$: the poset $[n]$ has a final object, so the identity of $\Delta[n]$ is homotopic to the constant map at $n$ (Corollary 2.2 of the lesson on décalage). Hence the projections $\Delta[n]\times A\to A$ and $\Delta[n]\times B\to B$ are homotopy equivalences, with the inclusions at the vertex $n$ as inverses, and $\Delta[n]\times f$ is a quasi-isomorphism by two out of three.

Next let $K$ have dimension at most $m$, and assume the assertion for simplicial sets of dimension less than $m$. By the expression (1.8) of the lesson on Kan complexes, $K$ is the pushout of $\operatorname{sk}_{m-1}K\leftarrow\coprod\partial\Delta[m]\to\coprod\Delta[m]$, the coproducts running over the nondegenerate $m$-simplices, and products with $A$ and with $B$ preserve this pushout. The maps $\operatorname{sk}_{m-1}K\times f$, $\coprod\partial\Delta[m]\times f$ and $\coprod\Delta[m]\times f$ are quasi-isomorphisms, by the induction hypothesis, the case of a simplex, and Corollary 2.3 (3) for coproducts. The maps $\coprod\partial\Delta[m]\times A\to\coprod\Delta[m]\times A$ and its analogue for $B$ are monomorphisms. By Proposition 2.6, $K\times f$ is a quasi-isomorphism. A general $K$ is the increasing union of its skeleta, and Corollary 2.3 (3) applies.

For the second assertion, $(L\times A)\cup(K\times B)$ is the pushout of $L\times A\leftarrow K\times A\to K\times B$, whose first map is a monomorphism, and $L\times B$ is the pushout of $L\times B\leftarrow K\times B\to K\times B$. The maps $L\times f$, $K\times f$ and the identity of $K\times B$ form a map between these diagrams, and they are quasi-isomorphisms by the first part. Proposition 2.6 gives the assertion.

Sources

The simplicial derived category of a topos, its description as a localization of the homotopy category, the mapping cylinder argument and the calculus of left fractions in a topos with enough points are due to Illusie, in his work on the cotangent complex. Proposition 2.4 is a special case of the theorem of Gabriel and Zisman and of Quillen that every monomorphic quasi-isomorphism of simplicial sets is anodyne; here it is deduced from the horn-filling lemmas of the lesson on Kan complexes. Kerodon, Section 3.1, treats Kan complexes, anodyne morphisms and weak homotopy equivalences with complete proofs. Multiplicative systems and the calculus of fractions are taken from the AI-integrated Stacks project [Stacks, Tag 04VB].