Nerves of induced and coinduced modules

Written by Claude Opus 5.5 (Anthropic) on 20 October 2026. Self-checked by the writing AI. Public domain (CC0).

Let $X$ be a pseudo-category. Its décalage $\operatorname{Dec}^1(X)$, which forgets the last face, maps to $X$ by forgetting the last vertex and to $X_0$ by the last vertex. Pull a module $L$ on $X_0$ back along the last vertex and push it down to $X$. On the first total topos the result is the nerve of the coinduced module of $L$, level by level. On the second total topos the direct image is a limit over the category of simplices of $\Delta[n]$, and a cofinality argument shows that it is the second nerve of the coinduced module. Under an injectivity hypothesis the same holds for the derived direct image: the standard resolution turns the derived direct image at each level into a Roos complex over the simplices of $\Delta[n]$, and a natural transformation between functors of the index categories gives a homotopy between Roos complexes. Dually, for two-sided pseudo-categories, the nerves of induced modules are the left adjoint images from the décalage that forgets the first face. These comparisons are the means by which the cohomology of equivariant modules is computed from the levels of $X$.

Prerequisites. Equivariant sheaves on simplicial topoi; Definition 3.2, Lemma 3.8 and Proposition 4.1 of Cohomology of total topoi; Lemma 1.1 and Proposition 1.2 of Cohomology of the second total topos and local systems; Lemma 1.2 and Theorem 3.2 of Left derived functors of left adjoints on total topoi; Proposition 2.2 of Derived left adjoints on the second total topos and local inductive limits; Sections 2 and 3 of Morphisms of fibred topoi and modules on total topoi; the simplicial replacement of Derived colimits over a small category.

Notation is that of the lesson on equivariant sheaves: $X(\alpha)\colon X_n\to X_m$ for $\alpha\colon[m]\to[n]$, vertices $v_k$, edges $e_{kl}$, source $s$, target $t$, faces $d_k$, and $\sigma\colon[1]\to[0]$. Throughout, $\mathcal O$ is a ring of $B'X$ (Section 6.1 there), so that all module inverse images along the $X(\alpha)$ are inverse images of sheaves, and modules on the décalages below are taken over the inverse images of $\mathcal O$. For a module $L$ on $X_0$, $\operatorname{coind}(L)=s_*t^*L$ carries the structure $a^{\mathrm c}$ and $\operatorname{ind}(L)=t_!s^*L$ the structure $b$, with the counit $\epsilon\colon s^*\operatorname{coind}(L)\to t^*L$ and the unit $\eta\colon s^*L\to t^*\operatorname{ind}(L)$.

1. Homotopies of Roos complexes

Let $\mathcal A$ be an additive category with products. For a small category $J$ and a functor $\Phi\colon J^{\mathrm{op}}\to\mathcal A$, the Roos complex $C^\bullet(\Phi)$ of Definition 3.2 of the lesson on cohomology of total topoi has $C^r(\Phi)=\prod_{\sigma\in N_r(J)}\Phi(\sigma_0)$, where $\sigma=(\sigma_0\to\cdots\to\sigma_r)$ runs over the strings of $r$ arrows and $\sigma_0$ is the first object. Its differential is $\sum_j(-1)^j\partial_j$, where $\partial_jc$ at $\sigma$ is $c_{d_j\sigma}$ for $j\geq1$, and $\Phi(\sigma_0\to\sigma_1)(c_{d_0\sigma})$ for $j=0$. A functor $\varphi\colon J\to J'$ and a functor $\Phi'\colon J'^{\mathrm{op}}\to\mathcal A$ give the morphism of complexes $\varphi^*\colon C^\bullet(\Phi')\to C^\bullet(\Phi'\varphi)$, $(\varphi^*c)_\sigma=c_{\varphi(\sigma)}$. A morphism $u\colon\Phi_1\to\Phi_2$ of functors gives $u_*\colon C^\bullet(\Phi_1)\to C^\bullet(\Phi_2)$, factor by factor.

Lemma 1.1. Let $\varphi_0,\varphi_1\colon J\to J'$ be functors, $\tau\colon\varphi_0\to\varphi_1$ a natural transformation and $\Phi\colon J'^{\mathrm{op}}\to\mathcal A$ a functor, so that $\Phi(\tau)\colon\Phi\varphi_1\to\Phi\varphi_0$. For $\sigma\in N_{r-1}(J)$ and $0\leq k\leq r-1$, let $\sigma^{(k)}\in N_r(J')$ be the string $\begin{gathered}\varphi_0\sigma_0\to\cdots\to\varphi_0\sigma_k\xrightarrow{\ \tau\ }\\ \varphi_1\sigma_k\to\cdots\to\varphi_1\sigma_{r-1}.\end{gathered}$ Then $h\colon C^r(\Phi)\to C^{r-1}(\Phi\varphi_0)$, $(hc)_\sigma=\sum_{k=0}^{r-1}(-1)^kc_{\sigma^{(k)}}$, with $h=0$ on $C^0$, satisfies $\delta h+h\delta=\Phi(\tau)_*\varphi_1^*-\varphi_0^* .$ So $\Phi(\tau)_*\varphi_1^*$ and $\varphi_0^*$ are homotopic.

Proof. Let $c\in C^r(\Phi)$ and $\sigma\in N_r(J)$. For $0\leq k\leq r+1$, let $\mu_k(\sigma)\in N_r(J')$ be the string $\varphi_0\sigma_0,\ldots,\varphi_0\sigma_{k-1},\varphi_1\sigma_k,\ldots,\varphi_1\sigma_r$, the arrow from $\varphi_0\sigma_{k-1}$ to $\varphi_1\sigma_k$ being the common composite given by naturality. So $\mu_0(\sigma)=\varphi_1(\sigma)$ and $\mu_{r+1}(\sigma)=\varphi_0(\sigma)$. Now $(h\delta c)_\sigma=\sum_{k=0}^{r}\sum_{j=0}^{r+1}(-1)^{k+j}(\partial_jc)_{\sigma^{(k)}} .$ The faces of $\sigma^{(k)}$ are $d_k\sigma^{(k)}=\mu_k(\sigma)$ and $d_{k+1}\sigma^{(k)}=\mu_{k+1}(\sigma)$. The first arrow of $\sigma^{(0)}$ is $\tau_{\sigma_0}$, so $(\partial_0c)_{\sigma^{(0)}}=\Phi(\tau_{\sigma_0})(c_{\varphi_1\sigma})$. Hence the terms with $j\in\{k,k+1\}$ add up to $\begin{aligned}&\Phi(\tau_{\sigma_0})(c_{\varphi_1\sigma})+\sum_{k=1}^{r}c_{\mu_k(\sigma)}-\sum_{k=0}^{r}c_{\mu_{k+1}(\sigma)}\\&\quad=\Phi(\tau_{\sigma_0})(c_{\varphi_1\sigma})-c_{\varphi_0\sigma}.\end{aligned}$ For $j<k$, $d_j\sigma^{(k)}=(d_j\sigma)^{(k-1)}$, and for $j=0<k$ the first arrow of $\sigma^{(k)}$ is $\varphi_0(\sigma_0\to\sigma_1)$. For $j>k+1$, $d_j\sigma^{(k)}=(d_{j-1}\sigma)^{(k)}$. On the other side, $(\delta hc)_\sigma$ is the sum of $\Phi\varphi_0(\sigma_0\to\sigma_1)\bigl((hc)_{d_0\sigma}\bigr)$ and the terms $(-1)^j(hc)_{d_j\sigma}$, $j\geq1$. Its terms $c_{(d_j\sigma)^{(k')}}$ with $j\leq k'$ are the terms $j<k=k'+1$ of $h\delta c$, and those with $j>k'$ are the terms $j+1>k'+1$ of $h\delta c$. In each case the two signs are opposite: $(-1)^{j+k'}$ against $(-1)^{j+k'+1}$. So everything cancels except the displayed sum. For $r=0$ the same computation gives $h\delta=\Phi(\tau)_*\varphi_1^*-\varphi_0^*$ on $C^0$. $\square$

Corollary 1.2. Let $\psi\colon J\to J'$ and $\iota\colon J'\to J$ be functors with $\psi\iota=\mathrm{id}$, and let $\tau$ be a natural transformation from $\mathrm{id}_J$ to $\iota\psi$, or from $\iota\psi$ to $\mathrm{id}_J$, with $\psi(\tau)=\mathrm{id}$. Then for every $\Phi\colon J'^{\mathrm{op}}\to\mathcal A$, the map $\psi^*\colon C^\bullet(\Phi)\to C^\bullet(\Phi\psi)$ is a homotopy equivalence with homotopy inverse $\iota^*$.

Proof. First, $\iota^*\psi^*=(\psi\iota)^*=\mathrm{id}$. Next, $\psi^*\iota^*=(\iota\psi)^*$ on $C^\bullet(\Phi\psi)$. Apply Lemma 1.1 to the functor $\Phi\psi$ of $J$, to the pair $\mathrm{id}_J$, $\iota\psi$ in the appropriate order, and to $\tau$. Since $\Phi\psi(\tau)=\Phi(\psi\tau)=\mathrm{id}$, the lemma says that $(\iota\psi)^*$ is homotopic to the identity. $\square$

1.3. The category of simplices of $\Delta[n]$. Let $\Delta/[n]$ be the category of increasing maps $\alpha\colon[i]\to[n]$, a morphism from $\alpha'\colon[i']\to[n]$ to $\alpha$ being an increasing $\gamma\colon[i']\to[i]$ with $\alpha\gamma=\alpha'$. Put $\sup\alpha=\alpha(i)$ and $\inf\alpha=\alpha(0)$. A morphism $\alpha'\to\alpha$ satisfies $\sup\alpha'\leq\sup\alpha$ and $\inf\alpha'\geq\inf\alpha$. So $\sup$ is a functor $\Delta/[n]\to[n]$, and $\inf$ is a functor $(\Delta/[n])^{\mathrm{op}}\to[n]$, where $[n]$ is the ordered set viewed as a category.

Lemma 1.4. Let $\iota(j)$ be the inclusion $[j]\to[n]$, and $\iota'(j)$ the map $[n-j]\to[n]$, $k\mapsto j+k$. Then $\iota\colon[n]\to\Delta/[n]$ and $\iota'\colon[n]\to(\Delta/[n])^{\mathrm{op}}$ are functors with $\sup\iota=\mathrm{id}$ and $\inf\iota'=\mathrm{id}$. Every $\alpha$ factors uniquely through $\iota(\sup\alpha)$ and through $\iota'(\inf\alpha)$, and these factorizations are natural transformations $\mathrm{id}\to\iota\sup$ of functors on $\Delta/[n]$ and $\iota'\inf\to\mathrm{id}$ of functors on $(\Delta/[n])^{\mathrm{op}}$, whose images under $\sup$, respectively $\inf$, are identities.

Proof. The image of $\alpha$ lies in $\{0,\ldots,\sup\alpha\}$ and in $\{\inf\alpha,\ldots,n\}$, so $\alpha$ factors uniquely through $\iota(\sup\alpha)$ and $\iota'(\inf\alpha)$, by increasing maps. Uniqueness gives naturality. The factorization of $\alpha$ through $\iota(\sup\alpha)$ is a morphism $\alpha\to\iota(\sup\alpha)$ of $\Delta/[n]$, and $\sup$ carries it to the identity of $\sup\alpha$. The same holds for $\inf$. $\square$

Proposition 1.5. Let $F\colon[n]\to\mathcal A$ be a functor, with maps $F(j\leq j')\colon F(j)\to F(j')$.

Regard $F\circ\sup$ as a functor on $\Delta/[n]=\bigl((\Delta/[n])^{\mathrm{op}}\bigr)^{\mathrm{op}}$, and let $C^\bullet(F\circ\sup)$ be its Roos complex over $J=(\Delta/[n])^{\mathrm{op}}$. The augmentation from $F(0)$ to $C^0(F\circ\sup)=\prod_\alpha F(\sup\alpha)$, $x\mapsto\bigl(F(0\leq\sup\alpha)x\bigr)_\alpha$, is a homotopy equivalence from $F(0)$, placed in degree $0$, to $C^\bullet(F\circ\sup)$.
Let $\mathcal A'$ be an additive category with direct sums and $F'\colon[n]\to\mathcal A'$ a functor. Let $C_\bullet(F'\circ\inf)$ be the simplicial replacement of the functor $F'\circ\inf$ on $(\Delta/[n])^{\mathrm{op}}$, with $C_r=\bigoplus_\sigma F'(\inf\sigma_0)$ (Section 1 of the lesson on derived colimits). The augmentation $C_0=\bigoplus_\alpha F'(\inf\alpha)\to F'(n)$, given by the maps $F'(\inf\alpha\leq n)$, is a homotopy equivalence of the chain complex $C_\bullet(F'\circ\inf)$ with $F'(n)$ placed in degree $0$.

Proof. (1) Apply Corollary 1.2 to $\psi=\sup^{\mathrm{op}}\colon J\to J'=[n]^{\mathrm{op}}$, $\iota^{\mathrm{op}}$, the transformation of Lemma 1.4, and $\Phi=F$. It shows that $\sup^*\colon C^\bullet(F)\to C^\bullet(F\circ\sup)$ is a homotopy equivalence, where $C^\bullet(F)$ is the Roos complex over $[n]^{\mathrm{op}}$. The object $0$ is final in $[n]^{\mathrm{op}}$. By Lemma 3.8 of the lesson on cohomology of total topoi, the augmentation $F(0)\to C^\bullet(F)$, $x\mapsto(F(0\leq j)x)_j$, is a homotopy equivalence: the map $h$ of that lemma, extended by the projection $C^0(F)\to F(0)$, contracts the augmented complex. The composite of the two augmentations is the augmentation of the statement.

(2) Apply (1) in the opposite category $\mathcal A'^{\mathrm{op}}$, which has products, with the order of $[n]$ reversed. In $\mathcal A'^{\mathrm{op}}$, the simplicial replacement of a functor $Z\colon I\to\mathcal A'$ is the Roos complex of $Z$, viewed as a contravariant functor on $I$ with values in $\mathcal A'^{\mathrm{op}}$: its terms are the products in $\mathcal A'^{\mathrm{op}}$, that is, the sums in $\mathcal A'$, over the strings of $I$, indexed by their first objects, and its differentials have the same formulas. Reversing the order of $[n]$, $j\mapsto n-j$, carries $\inf$ to $\sup$, $\iota'$ to $\iota$, and the final object $n$ of $[n]$ to the object $0$. So (1) gives that $C_\bullet(F'\circ\inf)\to F'(n)$ is a homotopy equivalence. $\square$

2. The two décalages

2.1. Setting. Let $\operatorname{Dec}^1(X)$ be the simplicial topos with levels $X_{k+1}$ and morphisms $X(\beta\sqcup\mathrm{id}_{[0]})$, where $\beta\sqcup\mathrm{id}_{[0]}\colon[k+1]\to[l+1]$ extends $\beta\colon[k]\to[l]$ by $k+1\mapsto l+1$ (Definition 4.3 of the lesson on derived left adjoints). The faces $d_{k+1}=X(\delta^{k+1})\colon X_{k+1}\to X_k$ form a morphism $q\colon\operatorname{Dec}^1(X)\to X$ of simplicial topoi over $\Delta^{\mathrm{op}}$, because $(\beta\sqcup\mathrm{id})\delta^{k+1}=\delta^{l+1}\beta$. The last vertices $X(v_{k+1})\colon X_{k+1}\to X_0$ form a morphism $\lambda\colon\operatorname{Dec}^1(X)\to X_0$, because $(\beta\sqcup\mathrm{id})v_{k+1}=v_{l+1}$. Likewise $\operatorname{Dec}_1(X)$ has levels $X_{k+1}$ and morphisms $X(\mathrm{id}_{[0]}\sqcup\beta)$ (Definition 4.1 of the lesson on simplicial topoi). The faces $d_0\colon X_{k+1}\to X_k$ form $q'\colon\operatorname{Dec}_1(X)\to X$, and the first vertices $X(v_0)\colon X_{k+1}\to X_0$ form $\pi\colon\operatorname{Dec}_1(X)\to X_0$. All four morphisms, with the inverse images of $\mathcal O$, are morphisms of ringed topoi on the levels, and $q$, $q'$ induce morphisms of both total topoi as in the lesson on morphisms of fibred topoi.

For a module $L$ on $X_0$, the modules $\lambda^*L$ on $\operatorname{Top}(\operatorname{Dec}^1X)$ and $\lambda^{\mathrm o*}L$ on $\operatorname{Top}^{\mathrm o}(\operatorname{Dec}^1X)$ have the levels $X(v_{k+1})^*L$, and all their maps are the transitivity isomorphisms; they are cartesian. The same holds for $\pi^*L$ and $\pi^{\mathrm o*}L$, with the levels $X(v_0)^*L$.

2.2. Direct images along $q$. For a morphism $u$ of fibred topoi over $I$ and an object $E$ of the first total topos, the levels of $u_*E$ are $u_{i*}E_i$ (Corollary 2.2 (b) of the lesson on morphisms of fibred topoi). For an arrow $f\colon i\to j$ of $I$, the map $f^*u_{j*}E_j\to u_{i*}E_i$ of $u_*E$ is the base change map $f^*u_{j*}E_j\to u_{i*}f^*E_j$ of the square formed by $u_i$, $u_j$ and the two morphisms for $f$, followed by $u_{i*}(E_f)$. Indeed both maps are adjoint to the map $u_{j*}E_j\to f_*u_{i*}E_i$ obtained from the adjoint of $E_f$ and $u_{j*}f_*\cong f_*u_{i*}$.

Lemma 2.3 (base change maps are transports). Let $X$ be a pseudo-category and $L$ a module on $X_0$. For $k\geq0$, let $\begin{gathered}\psi_k\colon X(v_k)^*\operatorname{coind}(L)\\ \xrightarrow{\ \sim\ }d_{k+1*}X(v_{k+1})^*L\end{gathered}$ be the base change isomorphism of Definition 3.2 of the lesson on equivariant sheaves; for $k=0$ it is the identity of $s_*t^*L$. For $\beta\colon[m]\to[i]$, let $\begin{gathered}b_\beta\colon X(\beta)^*d_{m+1*}X(v_{m+1})^*L\\ \longrightarrow d_{i+1*}X(v_{i+1})^*L\end{gathered}$ be the base change map of the square formed by $d_{i+1}$, $d_{m+1}$, $X(\beta)$ and $X(\beta\sqcup\mathrm{id})$. Then $\psi_i^{-1}\circ b_\beta\circ X(\beta)^*(\psi_m)=X(e_{\beta(m)\,i})^*(a^{\mathrm c}).$

Proof. Both sides are maps from $X(v_{\beta(m)})^*\operatorname{coind}(L)$ to $X(v_i)^*\operatorname{coind}(L)$. By Lemma 3.5 of the lesson on equivariant sheaves, a map $\varphi$ into $X(v_i)^*\operatorname{coind}(L)$ is determined by $X(e_{i,i+1})^*(\epsilon)\circ d_{i+1}^*(\varphi)$, and $X(e_{i,i+1})^*(\epsilon)$ is the adjoint of $\psi_i$. So for the left side this composite is the adjoint of $b_\beta\circ X(\beta)^*(\psi_m)$. The adjoint of $b_\beta$ is $X(\beta\sqcup\mathrm{id})^*$ of the counit of $d_{m+1}$, and $d_{i+1}^*X(\beta)^*=X(\beta\sqcup\mathrm{id})^*d_{m+1}^*$. Hence the composite is $X(\beta\sqcup\mathrm{id})^*$ of the adjoint of $\psi_m$, that is, $\begin{gathered}X(\beta\sqcup\mathrm{id})^*X(e_{m,m+1})^*(\epsilon)\\=X(e_{\beta(m),\,i+1})^*(\epsilon).\end{gathered}$ For the right side, the composite is $X(e_{i,i+1})^*(\epsilon)\circ X(e_{\beta(m)\,i})^*(a^{\mathrm c})$ on $X_{i+1}$. By identity (4.1) of the lesson on equivariant sheaves, pulled back along the map $[2]\to[i+1]$ with values $\beta(m),i,i+1$, this is also $X(e_{\beta(m),\,i+1})^*(\epsilon)$. $\square$

3. Nerves of coinduced modules

Proposition 3.1 (first total topos). Let $X$ be a pseudo-category and $L$ a module on $X_0$. The maps $\psi_n$ form an isomorphism of modules on $\operatorname{Top}(X)$ $\operatorname{ner}(\operatorname{coind}L)\xrightarrow{\ \sim\ }q_*\lambda^*L .$ If, for every $n$, the higher direct images $R^jd_{n+1*}\bigl(X(v_{n+1})^*L\bigr)$ vanish, then $Rq_*\lambda^*L\cong\operatorname{ner}(\operatorname{coind}L)$. This holds when $L$ is injective and every inverse image $X(v_k)^*$ along a last vertex carries injective modules to injective modules.

Proof. By 2.2, $q_*\lambda^*L$ has the levels $d_{n+1*}X(v_{n+1})^*L$, and since the maps of $\lambda^*L$ are transitivity isomorphisms, its map for $\alpha\colon[m]\to[n]$ is the base change map $b_\alpha$. By Lemma 2.3 it corresponds under the $\psi$ to $X(e_{\alpha(m)\,n})^*(a^{\mathrm c})$, which is the map of $\operatorname{ner}(\operatorname{coind}L)$ for $\alpha$. The levels of $Rq_*\lambda^*L$ are the $Rd_{n+1*}X(v_{n+1})^*L$, by Proposition 4.1 of the lesson on cohomology of total topoi, with Remark 4.2 there for modules. Under the vanishing hypothesis they are concentrated in degree $0$, so $q_*\lambda^*L\to Rq_*\lambda^*L$ is an isomorphism. An injective module has vanishing higher direct images. $\square$

Proposition 3.2 (second total topos). Let $X$ be a good pseudo-category and $L$ a module on $X_0$. There is an isomorphism of modules on $\operatorname{Top}^{\mathrm o}(X)$ $\operatorname{ner}^{\mathrm o}(\operatorname{coind}L)\xrightarrow{\ \sim\ }q^{\mathrm o}_*\lambda^{\mathrm o*}L .$ At the level $n$, let $F_n\colon[n]\to\operatorname{Mod}(\mathcal O_n)$ be the functor $j\mapsto X(v_j)^*\operatorname{coind}(L)$, with the maps $X(e_{jj'})^*(a^{\mathrm c})$ (Lemma 1.2 of the lesson on equivariant sheaves). Then $(q^{\mathrm o}_*\lambda^{\mathrm o*}L)_n$ is the limit over $\Delta/[n]$ of $F_n\circ\sup$, and the isomorphism is the projection of this limit onto the factor of the vertex $v_0\colon[0]\to[n]$, at which $F_n\circ\sup$ takes the value $F_n(0)$.

Proof. Write $G=\lambda^{\mathrm o*}L$. By Proposition 3.1 of the lesson on morphisms of fibred topoi, with the arrows $n\to i$ of $\Delta^{\mathrm{op}}$ written as maps $\alpha\colon[i]\to[n]$, the level $n$ of $q^{\mathrm o}_*G$ is the equalizer of two maps out of $\prod_\alpha X(\alpha)^*d_{i+1*}G_i$. They correspond to the morphisms $\gamma\colon\alpha'\to\alpha$ of $\Delta/[n]$, with $\alpha'\colon[i']\to[n]$ and $\alpha\gamma=\alpha'$, and have values in $X(\alpha)^*d_{i+1*}G_i$. One is the projection to the factor $\alpha'$ followed by $X(\alpha)^*$ of the base change map $b_\gamma$; the other is the projection to the factor $\alpha$ followed by $X(\alpha)^*d_{i+1*}$ of the map of $G$ for $\gamma$, an identity. By Lemma 2.3, under the isomorphisms $\begin{gathered}X(\alpha)^*(\psi_i)\colon F_n(\sup\alpha)\\=X(v_{\alpha(i)})^*\operatorname{coind}(L)\\ \cong X(\alpha)^*d_{i+1*}G_i,\end{gathered}$ the map $X(\alpha)^*(b_\gamma)$ becomes $F_n(\sup\alpha'\leq\sup\alpha)$. So the equalizer is the limit of $F_n\circ\sup$: the families $(x_\alpha)$ with $x_\alpha=F_n(\sup\alpha'\leq\sup\alpha)(x_{\alpha'})$ for every morphism $\alpha'\to\alpha$.

Such a family is determined by $x_{v_0}\in F_n(0)$. Indeed, $v_0$ maps to every $\iota(j)$ by $0\mapsto0$, so $x_{\iota(j)}=F_n(0\leq j)(x_{v_0})$; and the factorization $\alpha\to\iota(\sup\alpha)$ of Lemma 1.4 gives $x_{\iota(\sup\alpha)}=x_\alpha$. Conversely, for $x\in F_n(0)$, the family $x_\alpha=F_n(0\leq\sup\alpha)(x)$ is compatible, because $F_n$ is a functor. So the projection to $v_0$ is an isomorphism from the limit onto $F_n(0)=\operatorname{ner}^{\mathrm o}(\operatorname{coind}L)_n$, and its inverse is $x\mapsto(F_n(0\leq\sup\alpha)x)_\alpha$.

It remains to compare the maps for $\beta\colon[m]\to[n]$. By Proposition 3.1 of the lesson on morphisms of fibred topoi, the map $(q^{\mathrm o}_*G)_n\to X(\beta)^*(q^{\mathrm o}_*G)_m$ sends the factor $\beta\alpha''$ to the factor $\alpha''\colon[i]\to[m]$ by the identity of $X(\beta\alpha'')^*d_{i+1*}G_i$. Composing with the projection to the factor $v_0$ of the level $m$, we obtain the projection of $(q^{\mathrm o}_*G)_n$ to its factor $\beta v_0=v_{\beta(0)}$. On the limit, that projection is $F_n(0\leq\beta(0))$ of the projection to $v_0$. So the map is $X(e_{0\,\beta(0)})^*(a^{\mathrm c})$, which is the map of $\operatorname{ner}^{\mathrm o}(\operatorname{coind}L)$ for $\beta$. $\square$

The direct image along $q$ on the second total topos is therefore not computed level by level: at the level $n$ it involves all the simplices of $\Delta[n]$. Its derived functor is computed by a standard resolution, whose shape we describe first.

Lemma 3.3 (standard resolutions of cartesian modules). Let $Y$ be a good fibred topos over a small category $I$, with a cartesian ring, and $G$ a cartesian module on $\operatorname{Top}^{\mathrm o}(Y)$. For an object $k$ and $r\geq0$, let $S_r(k)$ be the set of strings $k=z_0\to z_1\to\cdots\to z_r$ of $r$ arrows of $I$ starting at $k$. Let $R^r=(e^{\mathrm o}_*e^{\mathrm o*})^{r+1}G$ be the standard resolution (Proposition 1.2 of the lesson on the second total topos), with $R^{-1}=G$ and the cofaces $d^j\colon R^{r-1}\to R^r$, $0\leq j\leq r$, where $d^j$ inserts the unit of $e^{\mathrm o}$ after $j$ applications of $e^{\mathrm o}_*e^{\mathrm o*}$. Then $R^r=e^{\mathrm o}_*(N^r)$ with $N^r_l=\prod_{y\in S_r(l)}G_l ,$ and $R^r_k\cong\prod_{z\in S_{r+1}(k)}G_k$. Under these isomorphisms, $d^j$ sends the factor of $d_{j+1}z$ to the factor of $z$ by the identity, where $d_{j+1}z\in S_r(k)$ omits the object $z_{j+1}$: it composes the arrows into and out of $z_{j+1}$, or drops the last arrow when $j=r$. The terms with $j=0$ are the units: $d^0$ is the unit of $e^{\mathrm o}$ at $R^{r-1}$.

Proof. The isomorphisms are those of Proposition 1.2 (2) of the lesson on the second total topos, $R^r_k\cong\prod_{z\in S_{r+1}(k)}c(z)^*G_{z_{r+1}}$, composed with the inverses of the isomorphisms $G_{c(z)}\colon G_k\to c(z)^*G_{z_{r+1}}$ of the cartesian module $G$. By Lemma 1.1 there, the unit of $e^{\mathrm o}$ at an object $H$ has at the level $k$ the components $H_f\colon H_k\to f^*H_l$, and the maps of $e^{\mathrm o}_*N$ for $f\colon k\to l$ are the projections onto the factors indexed by the arrows $g\circ f$. So the unit at $R^{r-1}=e^{\mathrm o}_*N^{r-1}$ sends the factor of $(gf,y')$ to the factor of $(f,g,y')$: this is $d^0$, omitting $z_1$. For $j\geq1$, $d^j=e^{\mathrm o}_*e^{\mathrm o*}(d^{j-1})$ acts on the strings with the first arrow fixed, and induction on $j$ gives the description. For $j=r$, the unit at $G$ has the components $G_f$, which become identities under the chosen isomorphisms, so $d^r$ omits the last object. $\square$

Theorem 3.4 (derived direct images of coinduced nerves). Let $X$ be a good pseudo-category, $L$ a module on $X_0$, $G=\lambda^{\mathrm o*}L$, and $G\to R^\bullet$ the standard resolution on $\operatorname{Top}^{\mathrm o}(\operatorname{Dec}^1X)$.

For every $n$, the level $n$ of the augmented complex $q^{\mathrm o}_*G\to q^{\mathrm o}_*R^\bullet$ is homotopy equivalent to zero. In particular $q^{\mathrm o}_*R^\bullet$ is a resolution of $q^{\mathrm o}_*G\cong\operatorname{ner}^{\mathrm o}(\operatorname{coind}L)$.
If $L$ is injective and every inverse image $X(v_k)^*$ along a last vertex carries injective modules to injective modules, then the $R^r$ are injective, and $Rq^{\mathrm o}_*\bigl(\lambda^{\mathrm o*}L\bigr)\cong\operatorname{ner}^{\mathrm o}(\operatorname{coind}L).$

Proof. (1) Let $u\colon Y\to X$ be any morphism of good fibred topoi over $I$, here $q$. Since $e^{\mathrm o*}u^{\mathrm o*}=(u^{\mathrm{dis}})^*e^{\mathrm o*}$, with $u^{\mathrm{dis}}$ the product of the $u_i$, the right adjoints satisfy $u^{\mathrm o}_*e^{\mathrm o}_*=e^{\mathrm o}_*u^{\mathrm{dis}}_*$. With Lemma 1.1 of the lesson on the second total topos this gives $(u^{\mathrm o}_*e^{\mathrm o}_*N)_n=\prod_{f\colon n\to i}f^*u_{i*}N_i .$ Under this isomorphism, $u^{\mathrm o}_*e^{\mathrm o}_*(\chi)$ acts factor by factor for every morphism $\chi$ of modules on $Y^{\mathrm{dis}}$. For a module $H$ on $\operatorname{Top}^{\mathrm o}(Y)$, the component of $u^{\mathrm o}_*$ of the unit $H\to e^{\mathrm o}_*e^{\mathrm o*}H$ at the factor $f$ is the projection of the equalizer $(u^{\mathrm o}_*H)_n$ onto its factor $f^*u_{i*}H_i$. Indeed, write $(u^{\mathrm o}_*e^{\mathrm o}_*N)_n$ as the equalizer inside $\prod_ff^*u_{i*}(e^{\mathrm o}_*N)_i=\prod_{(f,g)}f^*u_{i*}g^*N_m$. The equalizer condition for the morphism of $n/I$ given by $g$ says that the component at $(f,g)$ is the image of the component at $(gf,\mathrm{id})$ under $f^*$ of the base change map for $g$. So the isomorphism with $\prod_ff^*u_{i*}N_i$ is the projection onto the components $(f,\mathrm{id})$. The unit at $H$ has, at the level $i$, the components $H_g$ (Lemma 1.1 there), and $H_{\mathrm{id}}=\mathrm{id}$; so its image under $u^{\mathrm o}_*$, followed by this projection, is the inclusion of the equalizer.

Now take $u=q$ and the cofaces of Lemma 3.3. In the level $n$ of $q^{\mathrm o}_*R^r$, the factors are indexed by the strings $z\in S_{r+1}(n)$ of $\Delta^{\mathrm{op}}$. Write $z=(f,y)$, with $f\colon n\to i$ the first arrow, corresponding to $\alpha\colon[i]\to[n]$, and $y\in S_r(i)$. The factor is $X(\alpha)^*d_{i+1*}G_i$, which is $F_n(\sup\alpha)$ under $X(\alpha)^*(\psi_i)$. A string $z$ is the same as a string $\alpha_0\to\alpha_1\to\cdots\to\alpha_r$ in the category $J=(\Delta/[n])^{\mathrm{op}}$, where $\alpha_k$ is the composite of the first $k+1$ arrows of $z$, read as a map into $[n]$; its first object is $\alpha_0=\alpha$. The cofaces $d^j$ with $j\geq1$ act factor by factor, omit $z_{j+1}$, that is $\alpha_j$, and keep the first arrow; so they are the cofaces $\partial_j$ of the Roos complex $C^\bullet(F_n\circ\sup)$ over $J$. The coface $d^0$ omits $z_1$, that is $\alpha_0$, and by the previous paragraph it is the projection of the equalizer $(q^{\mathrm o}_*R^{r-1})_n$ onto its factors. On the factor of $z=(f,g,y')$, it is the map $f^*(b_g)$ from the factor of $d_1z=(gf,y')$. By Lemma 2.3 this is $F_n(\sup\alpha_1\leq\sup\alpha_0)$, which is the coface $\partial_0$ of the Roos complex. In degree $-1$, the augmentation $(q^{\mathrm o}_*G)_n\to(q^{\mathrm o}_*R^0)_n$ is the inclusion of the equalizer. By Proposition 3.2, it is the map $x\mapsto(F_n(0\leq\sup\alpha)x)_\alpha$ from $F_n(0)$. So the level $n$ of the augmented complex is the augmented Roos complex of Proposition 1.5 (1), which is homotopy equivalent to zero.

(2) The levels of $G$ are the injective modules $X(v_{k+1})^*L$. By Lemma 3.3, $R^r=e^{\mathrm o}_*(N^r)$, where every $N^r_l$ is a product of injective modules, hence injective, and $e^{\mathrm o}_*$ carries injectives to injectives (Lemma 1.1 of the lesson on the second total topos). The standard resolution is exact (Proposition 1.2 (1) there; its proof uses only the adjunction between $e^{\mathrm o*}$ and $e^{\mathrm o}_*$ and the conservativity of $e^{\mathrm o*}$). So $G\to R^\bullet$ is an injective resolution, $Rq^{\mathrm o}_*G$ is represented by $q^{\mathrm o}_*R^\bullet$, and (1) and Proposition 3.2 give the claim. $\square$

4. Nerves of induced modules

For induced modules the roles of the first and the last vertex are exchanged, and direct images are replaced by left adjoints. Let $X$ be a good two-sided pseudo-category (Definition 3.2 of the lesson on equivariant sheaves). For $k\geq1$, the left base change map of the second square of that definition is an isomorphism, by Lemma 3.1 there; together with the identity for $k=0$ it gives isomorphisms $\psi'_k\colon d_{0!}X(v_0)^*L\xrightarrow{\ \sim\ }X(v_0)^*\operatorname{ind}(L)$ on $X_k$, where on the left $X(v_0)\colon X_{k+1}\to X_0$ and $d_{0!}$ is the left adjoint of the module inverse image $d_0^*$. By the argument of Lemma 3.6 of the lesson on equivariant sheaves, the adjoint of $\psi'_k$ is $X(e_{01})^*(\eta)$, and a map $\varphi$ out of $X(v_0)^*\operatorname{ind}(L)$ on $X_k$ is determined by $d_0^*(\varphi)\circ X(e_{01})^*(\eta)$ on $X_{k+1}$.

Lemma 4.1. For $\beta\colon[m]\to[i]$, the faces $d_0$ of $X_{i+1}$ and $X_{m+1}$, $X(\beta)$ and $X(\mathrm{id}_{[0]}\sqcup\beta)$ form a commutative square, since $(\mathrm{id}\sqcup\beta)\delta^0=\delta^0\beta$. Let $b'_\beta\colon d_{0!}X(\mathrm{id}\sqcup\beta)^*\to X(\beta)^*d_{0!}$ be the map adjoint to $X(\mathrm{id}\sqcup\beta)^*$ of the unit of $d_{0!}$, followed by $X(\mathrm{id}\sqcup\beta)^*d_0^*=d_0^*X(\beta)^*$. Then, as maps from $X(v_0)^*\operatorname{ind}(L)$ to $X(v_{\beta(0)})^*\operatorname{ind}(L)$ on $X_i$, $\begin{gathered}X(\beta)^*(\psi'_m)\circ b'_\beta\circ(\psi'_i)^{-1}\\=X(e_{0\,\beta(0)})^*(b).\end{gathered}$

Proof. The proof of Lemma 2.3 applies with the arrows reversed. Both sides are maps out of $X(v_0)^*\operatorname{ind}(L)$, so they are determined by their images under $\varphi\mapsto d_0^*(\varphi)\circ X(e_{01})^*(\eta)$. The adjoint of $\psi'_i$ is $X(e_{01})^*(\eta)$, so $d_0^*\bigl((\psi'_i)^{-1}\bigr)\circ X(e_{01})^*(\eta)$ is the unit of $d_{0!}$. Hence the image of the left side is the adjoint of $X(\beta)^*(\psi'_m)\circ b'_\beta$, which is $X(\mathrm{id}\sqcup\beta)^*$ of the adjoint $X(e_{01})^*(\eta)$ of $\psi'_m$, that is, $X(e_{0,\beta(0)+1})^*(\eta)$. The image of the right side is $X(e_{1,\beta(0)+1})^*(b)\circ X(e_{01})^*(\eta)$, since $d_0^*X(e_{kl})^*=X(e_{k+1,l+1})^*$. Identity (5.1) of the lesson on equivariant sheaves, pulled back along the map $[2]\to[i+1]$ with values $0,1,\beta(0)+1$, shows that it is also $X(e_{0,\beta(0)+1})^*(\eta)$. $\square$

Proposition 4.2 (second total topos). Let $X$ be a good two-sided pseudo-category and $L$ a module on $X_0$. The maps $\psi'_n$ form an isomorphism of modules on $\operatorname{Top}^{\mathrm o}(X)$ $q'^{\mathrm o}_!\,\pi^{\mathrm o*}L\xrightarrow{\ \sim\ }\operatorname{ner}^{\mathrm o}(\operatorname{ind}L).$

Proof. The morphism $q'$ is a morphism over $\Delta^{\mathrm{op}}$, so the levels of $q'^{\mathrm o}_!\pi^{\mathrm o*}L$ are $d_{0!}X(v_0)^*L$ (Proposition 2.2 (2) of the lesson on derived left adjoints). Its map for $\beta$ is built from $b'_\beta$ and the maps of $\pi^{\mathrm o*}L$, which are transitivity isomorphisms. By Lemma 4.1 it becomes $X(e_{0\,\beta(0)})^*(b)$, the map of $\operatorname{ner}^{\mathrm o}(\operatorname{ind}L)$ for $\beta$. $\square$

Theorem 4.3 (first total topos). Let $X$ be a good two-sided pseudo-category, $L$ a module on $X_0$, $H=\pi^*L$ on $\operatorname{Top}(\operatorname{Dec}_1X)$, and $S_\bullet=(e_!e^*)^{\bullet+1}H\to H$ its standard simplicial resolution.

For every $n$, the level $n$ of the augmented complex $q'_!S_\bullet\to\operatorname{ner}(\operatorname{ind}L)$ is homotopy equivalent to zero. Here the augmentation is $q'_!$ of the augmentation of $S_\bullet$, followed by an isomorphism $q'_!H\cong\operatorname{ner}(\operatorname{ind}L)$, whose level $n$ is the colimit over $(\Delta/[n])^{\mathrm{op}}$ of $F'_n\circ\inf$, with $F'_n(j)=X(v_j)^*\operatorname{ind}(L)$.
If $L$ is flat and every $d_{0!}\colon\operatorname{Mod}(\mathcal O_{k+1})\to\operatorname{Mod}(\mathcal O_k)$ carries short exact sequences of flat modules to exact sequences, then $Lq'_!(\pi^*L)\cong\operatorname{ner}(\operatorname{ind}L)$.

Proof. (1) The proof of Theorem 3.4 (1) applies with the following changes. Products are replaced by direct sums, and the standard resolution by $S_\bullet$, whose faces are built from the counit of the adjunction between $e_!$ and $e^*$. The formula $(u^{\mathrm o}_*e^{\mathrm o}_*N)_n=\prod_ff^*u_{i*}N_i$ is replaced by $(q'_!e_!N)_n=\bigoplus_{f\colon n\to i}f^*d_{0!}N_i$ (Lemma 1.2 (2) of the lesson on left derived functors). Lemma 3.3 holds for $S_\bullet$ with sums and faces in place of products and cofaces, by the description of $e_!$ in Lemma 1.1 of the lesson on cohomology of total topoi. The base change maps $b_\gamma$ are replaced by the left base change maps $b'_\gamma$, the transports of Lemma 2.3 by those of Lemma 4.1, and $F_n\circ\sup$ by $F'_n\circ\inf$, a functor on $(\Delta/[n])^{\mathrm{op}}$. The level $n$ of $q'_!S_\bullet$ becomes the simplicial replacement $C_\bullet(F'_n\circ\inf)$, and Proposition 1.5 (2) shows that it is homotopy equivalent to $F'_n(n)=\operatorname{ner}_n(\operatorname{ind}L)$ through its augmentation. Since $q'_!$ is right exact and $S_1\to S_0\to H\to0$ is exact, $q'_!H$ is the cokernel of $q'_!S_1\to q'_!S_0$. At the level $n$ this is $H_0$ of the simplicial replacement, that is, $F'_n(n)$. For $\beta\colon[m]\to[n]$, the map $X(\beta)^*(q'_!e_!N)_m\to(q'_!e_!N)_n$ sends the summand of $\alpha''\colon[i]\to[m]$ to the summand of $\beta\alpha''$ by the identity (Lemma 1.2 of the lesson on left derived functors). Hence the induced map $X(\beta)^*F'_m(m)=F'_n(\beta(m))\to F'_n(n)$ composed with the image of the summand $F'_m(\inf\alpha'')$ is $F'_n(\beta(\inf\alpha'')\leq n)$. For $\alpha''=v_m$ this shows that the map is $F'_n(\beta(m)\leq n)$, the map of $\operatorname{ner}(\operatorname{ind}L)$ for $\beta$.

Formally, (1) is Theorem 3.4 (1) for the opposite module categories, with the order of the vertices reversed: under this exchange, inverse images keep their role, direct images become left adjoints, products become sums, the last vertex becomes the first, and the pseudo-category condition becomes the left base change form of the second condition of two-sidedness.

(2) The levels of $H$ are $X(v_0)^*L$, which are flat, so $H$ is levelwise flat. The morphism $q'$ satisfies condition (L) of the lesson on left derived functors, because $X$ is good, and condition (H) by the hypothesis on $d_{0!}$, the inverse images of $X$ being exact. By Theorem 3.2 (2) of that lesson, $Lq'_!H$ is represented by the total complex of $q'_!S_\bullet$. By (1) it is quasi-isomorphic to $\operatorname{ner}(\operatorname{ind}L)$. $\square$

The hypothesis on $d_{0!}$ holds for the nerve of an internal category of a topos: there $d_{0!}$ is extension by zero along a localization, which is exact.

5. Nerves and décalage

Proposition 5.1. Let $(M,m)$ be a module of $BX$.

The inverse image $q^{\mathrm o*}\operatorname{ner}^{\mathrm o}(M)$ is the décalage $\operatorname{Dec}^1_+\bigl(\operatorname{ner}^{\mathrm o}(M)\bigr)$, with levels $\operatorname{ner}^{\mathrm o}_{k+1}(M)$. It is the second nerve, relative to the simplicial topos $\operatorname{Dec}^1(X)$, of the equivariant sheaf $\bigl(s^*M,\ X(e_{01})^*(m)\bigr)$.
The inverse image $q'^*\operatorname{ner}(M)$ is the décalage $\operatorname{Dec}^+_1\bigl(\operatorname{ner}(M)\bigr)$, with levels $\operatorname{ner}_{k+1}(M)$. It is the nerve, relative to $\operatorname{Dec}_1(X)$, of the equivariant sheaf $\bigl(t^*M,\ X(e_{12})^*(m)\bigr)$.

Proof. (1) The level $k$ of $q^{\mathrm o*}\operatorname{ner}^{\mathrm o}(M)$ is $d_{k+1}^*X(v_0)^*M=X(v_0)^*M$ on $X_{k+1}$, and its map for $\beta\colon[k]\to[l]$ is $d_{l+1}^*$ of $a_{0\,\beta(0)}$, that is $a_{0\,\beta(0)}$ on $X_{l+1}$. These are the level $k+1$ of $\operatorname{ner}^{\mathrm o}(M)$ and its map for $\beta\sqcup\mathrm{id}$, because $(\beta\sqcup\mathrm{id})(0)=\beta(0)$. The simplicial topos $\operatorname{Dec}^1(X)$ has level $0$ equal to $X_1$, source $X(e_{02})$ and target $X(e_{12})$ on $X_2$, and its vertex $0$ at the level $k$ is $X(e_{0,k+1})$. The map $X(e_{01})^*(m)\colon X(v_0)^*M\to X(v_1)^*M$ on $X_2$ is an equivariant structure on $s^*M$ relative to $\operatorname{Dec}^1(X)$: conditions 1 and 2 are pullbacks of those of $m$. The second nerve of this sheaf has the level $X(e_{0,k+1})^*s^*M=X(v_0)^*M$ and the maps $m_{0\,\beta(0)}$, as computed.

(2) The same computation with $q'$: the level $k$ is $d_0^*X(v_k)^*M=X(v_{k+1})^*M$, and the maps are $m_{\beta(k)+1,\,l+1}$, those of $\operatorname{ner}(M)$ for $\mathrm{id}\sqcup\beta$. The simplicial topos $\operatorname{Dec}_1(X)$ has source $X(e_{01})$ and target $X(e_{02})$ on $X_2$, and its last vertex at the level $k$ is $X(e_{0,k+1})$. The map $X(e_{12})^*(m)\colon X(v_1)^*M\to X(v_2)^*M$ is an equivariant structure on $t^*M$ relative to $\operatorname{Dec}_1(X)$, and the nerve of this sheaf has the levels $X(e_{0,k+1})^*t^*M=X(v_{k+1})^*M$ and the maps $m_{\beta(k)+1,\,l+1}$. $\square$

6. Exercises

Exercise 6.1. List the objects $\alpha\colon[i]\to[1]$ of $\Delta/[1]$ with $i\leq2$, with their values $\sup\alpha$ and $\inf\alpha$, and the factorizations of Lemma 1.4. Which of them are isomorphisms?

Exercise 6.2. Let $X$ be the simplicial topos of the nerve of the ordered set $\{0<1\}$ in the topos of sets (Exercise 7.3 of the lesson on equivariant sheaves), and let $L=(L_0,L_1)$ be a pair of abelian groups, that is, an abelian sheaf on $X_0$. Compute the level $1$ of $q_*\lambda^*L$ over each of the three $1$-simplices, and compare with $\operatorname{ner}_1(\operatorname{coind}L)$ as in Proposition 3.1.

Exercise 6.3. In the situation of Exercise 6.2, take $\beta=v_0\colon[0]\to[1]$. Show that the base change map $b_\beta$ of Lemma 2.3 is not injective when $L_0\neq0$. Deduce that the direct image along $q$ on the second total topos is not computed level by level by the maps $d_{n+1*}$.

7. Complete solutions

6.1. The objects with $i\leq2$ are the increasing sequences $(\alpha(0),\ldots,\alpha(i))$ in $\{0,1\}$: $(0)$, $(1)$, $(0,0)$, $(0,1)$, $(1,1)$, $(0,0,0)$, $(0,0,1)$, $(0,1,1)$, $(1,1,1)$. Here $\sup\alpha$ is the last entry and $\inf\alpha$ the first. The functor $\iota$ sends $0$ to $(0)$ and $1$ to $(0,1)$; the functor $\iota'$ sends $0$ to $(0,1)$ and $1$ to $(1)$. The factorization $\alpha\to\iota(\sup\alpha)$ is the unique increasing map from $[i]$ to $[\sup\alpha]$ with the same values as $\alpha$. For example, $(0,0,1)\to(0,1)$ is the map $[2]\to[1]$ with values $0,0,1$, and $(0,0)\to(0)$ is $[1]\to[0]$. It is an isomorphism exactly when $\alpha$ is $\iota(\sup\alpha)$ itself, that is, for $(0)$ and $(0,1)$. Likewise $\alpha\to\iota'(\inf\alpha)$ is an isomorphism exactly for $(0,1)$ and $(1)$.

6.2. The $1$-simplices are $\mathrm{id}_0=(0,0)$, $f=(0,1)$ and $\mathrm{id}_1=(1,1)$, and the $2$-simplices are $(0,0,0)$, $(0,0,1)$, $(0,1,1)$, $(1,1,1)$. The face $d_2$ drops the last vertex, and $X(v_2)^*L$ has the fibre $L_{x_2}$ over a $2$-simplex $(x_0,x_1,x_2)$. Direct images along maps of sets take products over the fibres (Solution 7.1 of the lesson on equivariant sheaves), so the level $1$ of $q_*\lambda^*L$ has the fibres $L_0\times L_1$ over $(0,0)$, $L_1$ over $(0,1)$ and $L_1$ over $(1,1)$. On the other hand, $\operatorname{ner}_1(\operatorname{coind}L)$ has over $(x_0,x_1)$ the fibre $\operatorname{coind}(L)_{x_1}$, which is $L_0\times L_1$ for $x_1=0$ and $L_1$ for $x_1=1$ (Solution 7.3 there). The fibres agree, as Proposition 3.1 asserts.

6.3. Here $X(\beta)=X(v_0)\colon X_1\to X_0$ is the source, and $b_\beta\colon X(v_0)^*d_{1*}X(v_1)^*L\to d_{2*}X(v_2)^*L$ is a map on $X_1$. By Lemma 2.3 it is the transport $X(e_{01})^*(a^{\mathrm c})\colon s^*\operatorname{coind}(L)\to t^*\operatorname{coind}(L)$. Over the $1$-simplex $f$, it is the action of $f$ on $\operatorname{coind}(L)$, the projection $L_0\times L_1\to L_1$, which kills $L_0\times0$. In the formula for the level $n$ of a direct image on the second total topos (Proposition 3.1 of the lesson on morphisms of fibred topoi), the factor of $\mathrm{id}$ is $d_{n+1*}$ of the level, and the other factors are tied to it by such base change maps. Since these are not isomorphisms, the projection to the factor of $\mathrm{id}$ is not an isomorphism in general. Indeed, by Proposition 3.2, the level $1$ of $q^{\mathrm o}_*\lambda^{\mathrm o*}L$ is $\operatorname{ner}^{\mathrm o}_1(\operatorname{coind}L)$. Over $f$ it is $\operatorname{coind}(L)_0=L_0\times L_1$, while $d_{2*}X(v_2)^*L$ is $L_1$ there.

Sources

The nerves of induced and coinduced modules and their identification with direct images and left adjoints along the décalages are due to Illusie, in his work on the cotangent complex, where the identification on the second total topos is derived from a homotopy equivalence between a cosimplicial object and its cosimplicial replacement. The argument here uses instead the cofinality of the functors that take a simplex of $\Delta[n]$ to its last or its first vertex. The homotopy of Lemma 1.1 is the cochain version of Segal's observation that a natural transformation induces a homotopy of nerves (G. Segal, Classifying spaces and spectral sequences, Publ. Math. IHÉS 34 (1968), 105–112, Numdam).