Morphisms of fibred topoi and modules on total topoi

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

A morphism of fibred topoi changes the index category and maps each level to a level. It induces inverse images on the total topoi, levelwise. Their right adjoints, the direct images, are limits over comma categories of the index categories. Global sections of the total topos are the limit of the global sections of the levels. A ring in a total topos is a compatible family of rings on the levels, and so are its modules. Tensor products are computed levelwise, and internal Homs are computed levelwise from a cartesian source. This lesson proves these facts. They are the formal basis of the cohomology of diagrams of ringed topoi. Finally, diagrams of diagrams of a category have types, which are diagrams of categories, and an iterated Grothendieck construction turns a diagram of diagrams into a single diagram. For a diagram of diagrams in a topos, the second total topos, defined level by level, is the topos of diagrams of its type over it.

Prerequisites. Fibred topoi and their total topoi; modules on ringed topoi and their inverse images [Stacks, Tag 03D5]; tensor products and pullback [Stacks, Tag 03EL]; localization of topoi [Stacks, Tag 04GZ].

Notation is that of the lesson on fibred topoi. In particular, the transitivity isomorphisms of a fibred topos are not written.

1. Morphisms of fibred topoi

Definition 1.1. Let $X$ be a topos fibred over $I$ and $Y$ one fibred over $J$. A morphism $m=(u,v)\colon X\to Y$ consists of a functor $v\colon I\to J$, morphisms of topoi $u_i\colon X_i\to Y_{v(i)}$, and isomorphisms $\theta_f\colon f^*u_j^*\xrightarrow{\ \sim\ }u_i^*v(f)^*\qquad(f\colon i\to j),$ compatible with composites and identities through the transitivity isomorphisms of $X$ and $Y$. When $I=J$ and $v=\mathrm{id}$, $m$ is an $I$-morphism.

Proposition 1.2. The functors $\begin{gathered}m^*\colon\operatorname{Top}(Y)\to\operatorname{Top}(X),\\m^*(F)_i=u_i^*F_{v(i)},\\m^*(F)_f=u_i^*(F_{v(f)})\circ\theta_f,\end{gathered}$ $\begin{gathered}m^{\mathrm o*}\colon\operatorname{Top}^{\mathrm o}(Y)\to\operatorname{Top}^{\mathrm o}(X),\\m^{\mathrm o*}(G)_i=u_i^*G_{v(i)},\\m^{\mathrm o*}(G)_f=\theta_f^{-1}\circ u_i^*(G_{v(f)}),\end{gathered}$ are well defined and preserve colimits and finite limits.

Proof. The conditions on composites follow from those of $F$ and $G$ and the compatibility of $\theta$ with composites. Colimits and finite limits are computed levelwise (Proposition 2.2 of the lesson on fibred topoi), and each $u_i^*$ preserves them. $\square$

2. Direct images on the total topoi

For $j\in J$, let $I/j$ be the category of pairs $(i,f)$ with $f\colon v(i)\to j$; a morphism $(i,f)\to(i',f')$ is an arrow $g\colon i\to i'$ with $f'\circ v(g)=f$. For $F\in\operatorname{Top}(X)$ and such a $g$, let $f'_*u_{i'*}F_{i'}\longrightarrow f_*u_{i*}F_i$ be $f'_*u_{i'*}$ applied to the adjoint $F_{i'}\to g_*F_i$ of $F_g$, followed by $u_{i'*}g_*=v(g)_*u_{i*}$ (the isomorphism adjoint to $\theta_g$) and by $f'_*v(g)_*=f_*$. This makes $(i,f)\mapsto f_*u_{i*}F_i$ a contravariant functor on $I/j$ with values in $Y_j$. Put $(m_*F)_j=\varprojlim_{(i,f)\in I/j}f_*u_{i*}F_i.$ For $h\colon j\to k$, the functor $(i,f)\mapsto(i,hf)$ from $I/j$ to $I/k$ gives a map $\begin{aligned}(m_*F)_k&\longrightarrow\varprojlim_{(i,f)\in I/j}(hf)_*u_{i*}F_i\\&=h_*(m_*F)_j,\end{aligned}$ since $h_*$ preserves limits. Its adjoint is $(m_*F)_h\colon h^*(m_*F)_k\to(m_*F)_j$.

Proposition 2.1. $m_*$ is right adjoint to $m^*$. Hence $(m^*,m_*)$ is a morphism of topoi $\operatorname{Top}(X)\to\operatorname{Top}(Y)$.

Proof. Let $G\in\operatorname{Top}(Y)$. A morphism $\beta\colon m^*G\to F$ is a family $\beta_i\colon u_i^*G_{v(i)}\to F_i$ such that $\beta_i\circ m^*(G)_g=F_g\circ g^*(\beta_{i'})$ for every $g\colon i\to i'$. Write $\beta^\sharp_i\colon G_{v(i)}\to u_{i*}F_i$ for the adjoint map. For $j\in J$ and $(i,f)\in I/j$, let $\gamma_{j,(i,f)}\colon G_j\to f_*u_{i*}F_i$ be the adjoint of $\beta^\sharp_i\circ G_f\colon f^*G_j\to G_{v(i)}\to u_{i*}F_i$. The compatibility of $\beta$ with $g$, together with $G_{f'v(g)}=G_{v(g)}\circ v(g)^*(G_{f'})$, says that the $\gamma_{j,(i,f)}$ form a cone over the functor defining $(m_*F)_j$. So they define $\gamma_j\colon G_j\to(m_*F)_j$. The relation $G_{hf}=G_f\circ f^*(G_h)$ says that the $\gamma_j$ form a morphism $G\to m_*F$.

Conversely, given $\gamma\colon G\to m_*F$, let $\beta^\sharp_i$ be the component of $\gamma_{v(i)}$ at the object $(i,\mathrm{id}_{v(i)})$ of $I/v(i)$. The compatibility of $\gamma$ with the map $(m_*F)_f$, for $f\colon v(i)\to j$, shows that $\gamma_j$ is determined by these components, by the formula above. The cone condition at $g\colon(i,\mathrm{id})\to(i',v(g))$ gives the compatibility of the corresponding $\beta$ with $g$. The two constructions are inverse to each other. By Proposition 1.2, $m^*$ is exact, so $(m^*,m_*)$ is a morphism of topoi. $\square$

Corollary 2.2. (a) If $J$ is the category with one object and one arrow, so that $Y$ is a single topos, then $m_*F=\varprojlim_{i\in I}u_{i*}F_i$. In particular, for the punctual topos, $\Gamma(\operatorname{Top}(X),F)=\varprojlim_{i\in I}\Gamma(X_i,F_i).$ (b) If $m$ is an $I$-morphism, then $(m_*F)_j=u_{j*}F_j$: direct images commute with restriction to the levels. For the constant fibred topos $Y$ over $I$ with value the punctual topos, $\operatorname{Top}(Y)$ is the category of presheaves of sets on $I$, and $m_*F$ is the presheaf $i\mapsto\Gamma(X_i,F_i)$. (c) For a level $i$, viewed as the morphism from the fibred topos over the one-point category to $X$, one recovers $i_*$.

Proof. (a) Here $I/j=I$. (b) For an $I$-morphism, $I/j$ has the final object $(j,\mathrm{id}_j)$, and the limit of a contravariant functor is its value at a final object. The description of $\operatorname{Top}(Y)$ is Example 5.2 (c) of the lesson on fibred topoi. (c) The category $\{i\}/j$ is the discrete set $\operatorname{Hom}(i,j)$. $\square$

3. Direct images for the second total category

Let $X$ and $Y$ be good. For $g\colon i_0\to i_1$, the base change map $\mathrm{bc}_g\colon v(g)^*u_{i_1*}\longrightarrow u_{i_0*}g^*$ is the composite of the unit of $u_{i_0}$, the isomorphism $\theta_g^{-1}\colon u_{i_0}^*v(g)^*\to g^*u_{i_1}^*$, and the counit of $u_{i_1}$. Its adjoint $u_{i_0}^*v(g)^*u_{i_1*}\to g^*$ is $g^*(\text{counit})\circ\theta_g^{-1}$.

For $j\in J$, let $j/I$ be the category of pairs $(i,f)$ with $f\colon j\to v(i)$; a morphism $(i_0,f_0)\to(i_1,f_1)$ is a $g\colon i_0\to i_1$ with $v(g)f_0=f_1$. For $G\in\operatorname{Top}^{\mathrm o}(X)$, let $(m^{\mathrm o}_*G)_j$ be the equalizer of the two maps $\begin{aligned}&\prod_{(i,f)\in j/I}f^*u_{i*}G_i\\&\quad\rightrightarrows\prod_{g\colon(i_0,f_0)\to(i_1,f_1)}f_0^*u_{i_0*}g^*G_{i_1}.\end{aligned}$ The component at $g$ of the first map is the projection to $f_1^*u_{i_1*}G_{i_1}=f_0^*v(g)^*u_{i_1*}G_{i_1}$ followed by $f_0^*(\mathrm{bc}_g)$. The component of the second is the projection to $f_0^*u_{i_0*}G_{i_0}$ followed by $f_0^*u_{i_0*}(G_g)$. For $h\colon k\to j$, the functor $(i,f)\mapsto(i,fh)$ from $j/I$ to $k/I$ gives the map $(m^{\mathrm o}_*G)_h\colon(m^{\mathrm o}_*G)_k\to h^*(m^{\mathrm o}_*G)_j$. Here $h^*$ preserves the products and the equalizer, because $Y$ is good.

Proposition 3.1. If $X$ and $Y$ are good, $m^{\mathrm o}_*$ is right adjoint to $m^{\mathrm o*}$. If moreover $\operatorname{Top}^{\mathrm o}(X)$ and $\operatorname{Top}^{\mathrm o}(Y)$ are topoi, for example under the hypothesis of Theorem 4.3 of the lesson on fibred topoi, then $(m^{\mathrm o*},m^{\mathrm o}_*)$ is a morphism of topoi.

Proof. Let $H\in\operatorname{Top}^{\mathrm o}(Y)$. A morphism $\beta\colon m^{\mathrm o*}H\to G$ is a family $\beta_i\colon u_i^*H_{v(i)}\to G_i$ such that, for $g\colon i_0\to i_1$, $\begin{aligned}G_g\circ\beta_{i_0}&=g^*(\beta_{i_1})\circ\theta_g^{-1}\\&\quad\circ u_{i_0}^*(H_{v(g)})\end{aligned} \tag{3.1}$ as maps $u_{i_0}^*H_{v(i_0)}\to g^*G_{i_1}$. Let $\beta^\sharp_i\colon H_{v(i)}\to u_{i*}G_i$ be adjoint to $\beta_i$. For $(i,f)\in j/I$, put $\gamma_{j,(i,f)}=f^*(\beta^\sharp_i)\circ H_f\colon H_j\to f^*u_{i*}G_i$. For $g\colon(i_0,f_0)\to(i_1,f_1)$, we have $H_{f_1}=f_0^*(H_{v(g)})\circ H_{f_0}$. So the two maps of the equalizer agree on the family $(\gamma_{j,(i,f)})$ exactly when $\begin{aligned}&\mathrm{bc}_g\circ v(g)^*(\beta^\sharp_{i_1})\circ H_{v(g)}\\&\quad=u_{i_0*}(G_g)\circ\beta^\sharp_{i_0}\\&\quad\colon H_{v(i_0)}\to u_{i_0*}g^*G_{i_1}\end{aligned}$ holds after applying $f_0^*$ and composing with $H_{f_0}$. The adjoints of the two sides of this equation are the two sides of (3.1). So $\gamma_j$ factors through $(m^{\mathrm o}_*G)_j$, and the relation $H_{fh}=h^*(H_f)\circ H_h$ shows that the $\gamma_j$ form a morphism $H\to m^{\mathrm o}_*G$. Conversely, given $\gamma$, let $\beta^\sharp_i$ be the component of $\gamma_{v(i)}$ at $(i,\mathrm{id})$. The compatibility of $\gamma$ with the transition maps shows that $\gamma$ is recovered from these components by the formula above, and the equalizer condition at $g\colon(i_0,\mathrm{id})\to(i_1,v(g))$ gives (3.1). The last assertion follows from Proposition 1.2. $\square$

Corollary 3.2. Suppose that every base change map $\mathrm{bc}_g$ is an isomorphism on the $G_{i_1}$. Then $(i,f)\mapsto f^*u_{i*}G_i$ is a covariant functor on $j/I$, and $(m^{\mathrm o}_*G)_j=\varprojlim_{j/I}f^*u_{i*}G_i$. If moreover $m$ is an $I$-morphism, then $(m^{\mathrm o}_*G)_j=u_{j*}G_j$.

Proof. With $\mathrm{bc}_g$ invertible, the maps $\mathrm{bc}_g^{-1}\circ u_{i_0*}(G_g)$ make the family a functor, and the equalizer is its limit. For an $I$-morphism, $j/I$ has the initial object $(j,\mathrm{id}_j)$, and the limit of a covariant functor is its value at an initial object. $\square$

4. Rings and modules

4.1. Rings and modules in $\operatorname{Top}(X)$. Inverse images preserve finite products. So a ring $A$ of $\operatorname{Top}(X)$ is a family of rings $A_i$ of $X_i$ with ring maps $f^{-1}A_j\to A_i$, where $f^{-1}$ denotes the inverse image of sheaves. Then $X$, with the rings $A_i$, is a ringed fibred topos. For an $A_i$-module $L$ and $f\colon k\to i$, write $f^*L=A_k\otimes_{f^{-1}A_i}f^{-1}L$ for the inverse image of modules.

Proposition 4.2. (a) A left $A$-module $M$ of $\operatorname{Top}(X)$ is the same as a family of left $A_i$-modules $M_i$ with $A_i$-linear maps $M_f\colon f^*M_j\to M_i$ satisfying $M_{gf}=M_f\circ f^*(M_g)$ and $M_{\mathrm{id}}=\mathrm{id}$. (b) The restriction $i^*\colon\operatorname{Mod}(A)\to\operatorname{Mod}(A_i)$ has the left adjoint $i_!$, with $i_!(L)_j=\bigoplus_{f\colon j\to i}f^*L$. (c) For a right $A$-module $L$ and a left $A$-module $M$, $i^*(L\otimes_AM)\cong L_i\otimes_{A_i}M_i$. (d) For left $A$-modules $L,M$ there is a canonical map $i^*\,\mathcal{H}om_A(L,M)\longrightarrow\mathcal{H}om_{A_i}(L_i,M_i),$ which is an isomorphism when $L$ is cartesian, that is, when every $L_f$ is an isomorphism.

Proof. (a) A module structure $A\times M\to M$ in $\operatorname{Top}(X)$ is a family of module structures $A_i\times M_i\to M_i$, compatible with the maps $f^{-1}M_j\to M_i$ and $f^{-1}A_j\to A_i$. Such a compatible map $f^{-1}M_j\to M_i$ of $f^{-1}A_j$-modules is the same as an $A_i$-linear map $f^*M_j\to M_i$.

(b) This is the proof of Proposition 3.1 of the lesson on fibred topoi, with module inverse images and direct sums of modules.

(c) The pair $(i^*,i_*)$ is a morphism of topoi $X_i\to\operatorname{Top}(X)$ (Theorem 4.2 there), and $i^*A=A_i$. So it is a morphism of ringed topoi whose module inverse image is $i^*$ itself. Inverse images of ringed topoi commute with tensor products [Stacks, Tag 03EL].

(d) For an $A_i$-module $N$, the adjunctions give $\begin{aligned}&\operatorname{Hom}_{A_i}\bigl(N,i^*\mathcal Hom_A(L,M)\bigr)\\&\quad=\operatorname{Hom}_A(i_!N\otimes_AL,M).\end{aligned}$ There is a canonical map $i_!(N\otimes_{A_i}L_i)\to i_!N\otimes_AL$: on the summand $f^*(N\otimes L_i)=f^*N\otimes f^*L_i$ of level $j$, it is $1\otimes L_f$. It induces the canonical map of (d), through the identifications $\begin{aligned}&\operatorname{Hom}_A(i_!(N\otimes L_i),M)\\&\quad=\operatorname{Hom}_{A_i}(N\otimes L_i,M_i)\\&\quad=\operatorname{Hom}_{A_i}(N,\mathcal Hom_{A_i}(L_i,M_i))\end{aligned}$ and the Yoneda lemma. If $L$ is cartesian, every $L_f$ is an isomorphism, so this map is an isomorphism for every $N$, and hence so is the map of (d). $\square$

4.3. Rings and modules in $\operatorname{Top}^{\mathrm o}(X)$. Let $X$ be good. A ring $A$ of $\operatorname{Top}^{\mathrm o}(X)$ is a family of rings $A_i$ with ring maps $A_i\to f^{-1}A_j$. For $f\colon i\to j$, let $f^*\colon\operatorname{Mod}(A_j)\to\operatorname{Mod}(A_i)$ be $f^{-1}$ followed by restriction of scalars along $A_i\to f^{-1}A_j$. As in Proposition 4.2 (a), a left $A$-module is a family of $A_i$-modules $M_i$ with $A_i$-linear maps $M_i\to f^*M_j$, compatible with composites.

Proposition 4.4. (a) The functor $f^*\colon\operatorname{Mod}(A_j)\to\operatorname{Mod}(A_i)$ has a left adjoint $f^A_!$, with $f^A_!(A_i^{(E)})=A_j^{(f_!E)}$ for the free module on an object $E$ of $X_i$. So an $A$-module is the same as a family $M_i$ with $A_j$-linear maps $f^A_!M_i\to M_j$ compatible with composites. (b) The restriction $i^{\mathrm o*}\colon\operatorname{Mod}(A)\to\operatorname{Mod}(A_i)$ has the left adjoint $i^{\mathrm o}_!$, with $i^{\mathrm o}_!(L)_j=\bigoplus_{f\colon i\to j}f^A_!L$.

Proof. (a) For $E\in X_i$ and an $A_j$-module $M$, $\begin{aligned}&\operatorname{Hom}_{A_j}(A_j^{(f_!E)},M)\\&\quad=\operatorname{Hom}_{X_j}(f_!E,M)\\&\quad=\operatorname{Hom}_{X_i}(E,f^{-1}M)\\&\quad=\operatorname{Hom}_{A_i}(A_i^{(E)},f^*M).\end{aligned}$ So the left adjoint is defined on free modules and on the maps between them, by Yoneda. Every $A_i$-module $L$ has a presentation $L_1\to L_0\to L\to0$ by free modules. Put $f^A_!L=\operatorname{coker}(f^A_!L_1\to f^A_!L_0)$. Then $\begin{aligned}&\operatorname{Hom}(f^A_!L,M)\\&\quad=\ker\bigl(\operatorname{Hom}(L_0,f^*M)\\&\qquad\qquad\to\operatorname{Hom}(L_1,f^*M)\bigr)\\&\quad=\operatorname{Hom}(L,f^*M).\end{aligned}$ So $f^A_!L$ represents the functor $M\mapsto\operatorname{Hom}(L,f^*M)$, and it is independent of the presentation up to unique isomorphism. (b) As in Proposition 3.1 (2) of the lesson on fibred topoi, with $f^A_!$ in place of $f_!$. $\square$

When $\operatorname{Top}^{\mathrm o}(X)$ is a topos, $i^{\mathrm o*}$ is the inverse image of a morphism of ringed topoi, and $i^{\mathrm o*}(L\otimes_AM)=L_i\otimes_{A_i}M_i$ as in Proposition 4.2 (c). For example, for the constant simplicial object at the final object of a topos $T$, $\operatorname{Top}^{\mathrm o}$ is the category of simplicial objects of $T$. A ring is then a simplicial ring of $T$, and its modules are the simplicial modules.

5. Diagrams of diagrams

5.1. Diagrams and their types. Let $T$ be a category. Let $\operatorname{Diagr}_1(T)$ be the category whose objects are the pairs $(I,X)$ of a small category $I$ and a functor $X\colon I\to T$, a diagram of type $I$. A morphism $(I,X)\to(J,Y)$ is a pair $(u,v)$ of a functor $v\colon I\to J$ and a natural transformation $u\colon X\to Y\circ v$. The composite of $(u,v)$ with $(u',v')\colon(J,Y)\to(K,Z)$ is $(u'v\circ u,v'v)$, where $(u'v)_i=u'_{v(i)}$. Put $\operatorname{Diagr}_0(T)=T$ and $\operatorname{Diagr}_n(T)=\operatorname{Diagr}_1(\operatorname{Diagr}_{n-1}(T))$ for $n\geq1$. The objects of $\operatorname{Diagr}_n(T)$ are the $n$-diagrams of $T$.

Let $\mathrm{Cat}$ be the category of small categories. For $n\geq1$, the type is the functor $\operatorname{Typ}\colon\operatorname{Diagr}_n(T)\to\operatorname{Diagr}_{n-1}(\mathrm{Cat})$ given for $n=1$ by $\operatorname{Typ}(I,X)=I$ and $\operatorname{Typ}(u,v)=v$, and for $n\geq2$ by $\operatorname{Typ}(K,X)=(K,\operatorname{Typ}\circ X)$ and $\operatorname{Typ}(u,v)=(\operatorname{Typ}\circ u,v)$, where $(\operatorname{Typ}\circ u)_k=\operatorname{Typ}(u_k)$ and $\operatorname{Typ}$ on $\operatorname{Diagr}_{n-1}(T)$ is a functor by induction. For $I\in\operatorname{Diagr}_{n-1}(\mathrm{Cat})$, the category $\operatorname{Diagr}_I(T)$ of $I$-diagrams has the $n$-diagrams of type $I$ as objects and the morphisms of type $\mathrm{id}_I$, the $I$-morphisms, as morphisms.

5.2. The total category of a type. For $n\geq1$, define a functor $\int\colon\operatorname{Diagr}_{n-1}(\mathrm{Cat})\to\mathrm{Cat}$ by induction. For $n=1$, $\int I=I$. For $n\geq2$, let $I=(K,(I_k))$, with transition morphisms $I(f)\colon I_k\to I_{k'}$ for $f\colon k\to k'$. The objects of $\int I$ are the pairs $(k,a)$ with $a$ an object of $\int I_k$. A morphism $(k,a)\to(k',a')$ is a pair $(f,\alpha)$ of an arrow $f\colon k\to k'$ and a morphism $\alpha\colon(\int I(f))(a)\to a'$ of $\int I_{k'}$, and $\begin{aligned}&(f',\alpha')\circ(f,\alpha)\\&\quad=\bigl(f'f,\ \alpha'\circ(\textstyle\int I(f'))(\alpha)\bigr).\end{aligned}$ A morphism $t=(w,v)\colon I\to J$ induces the functor $\int t$ with $(\int t)(k,a)=(v(k),(\int w_k)(a))$ and $(\int t)(f,\alpha)=(v(f),(\int w_{k'})(\alpha))$. The second formula gives a morphism of $\int J$, because $\int w_{k'}\circ\int I(f)=\int J(v(f))\circ\int w_k$, by the naturality of $w$ and the functoriality of $\int$ on $\operatorname{Diagr}_{n-2}(\mathrm{Cat})$. For $n=2$, $\int I$ is the Grothendieck construction of the functor $k\mapsto I_k$. For every $k$, the objects $(k,a)$ and the morphisms $(\mathrm{id}_k,\alpha)$ form the fibre over $k$, a subcategory identified with $\int I_k$.

Lemma 5.3. Let $n\geq1$. To every $n$-diagram $X$ of $T$ of type $I$ there is attached a functor $X^\natural\colon\int I\to T$, and to every morphism $m\colon X\to Y$ of type $t\colon I\to J$ a natural transformation $m^\natural\colon X^\natural\to Y^\natural\circ\int t$, such that $(m'm)^\natural=(m'^\natural\int t)\circ m^\natural$ and $\mathrm{id}^\natural=\mathrm{id}$. The map $X\mapsto X^\natural$ is a bijection from the $n$-diagrams of type $I$ to the functors $\int I\to T$, and $m\mapsto m^\natural$ is a bijection from the morphisms $X\to Y$ of type $t$ to the natural transformations $X^\natural\to Y^\natural\circ\int t$. In particular, $X\mapsto X^\natural$ is an isomorphism from $\operatorname{Diagr}_I(T)$ to the category of functors $\int I\to T$.

Proof. For $n=1$, put $X^\natural=X$ and $(u,v)^\natural=u$. Let $n\geq2$ and $X=(K,X)$, so that $X_k$ has type $I_k$ and $X(f)\colon X_k\to X_{k'}$ has type $I(f)$. Put $X^\natural(k,a)=X_k^\natural(a)$ and $X^\natural(f,\alpha)=X_{k'}^\natural(\alpha)\circ X(f)^\natural_a.$ For a composite $(f',\alpha')\circ(f,\alpha)$, the rule $X(f'f)^\natural=(X(f')^\natural\int I(f))\circ X(f)^\natural$ and the naturality of $X(f')^\natural$ at $\alpha$ show that $X^\natural$ is a functor. For $m=(u,v)$, put $m^\natural_{(k,a)}=(u_k)^\natural_a$. Its naturality at $(f,\alpha)$ follows from $u_{k'}\circ X(f)=Y(v(f))\circ u_k$, the composition rule one level down and the naturality of $u_{k'}^\natural$; the composition rule for $m$ follows from that for the $u_k$.

Conversely, let $F\colon\int I\to T$ be a functor. It restricts on the fibres to functors $F_k\colon\int I_k\to T$. The morphisms $(f,\mathrm{id})\colon(k,a)\to(k',(\int I(f))(a))$ give natural transformations $F_k\to F_{k'}\circ\int I(f)$, because $\begin{aligned}&(f,\mathrm{id})\circ(\mathrm{id}_k,\alpha)\\&\quad=(\mathrm{id}_{k'},(\textstyle\int I(f))(\alpha))\circ(f,\mathrm{id}).\end{aligned}$ By induction, the $F_k$ are the $X_k^\natural$ of unique $(n-1)$-diagrams $X_k$ of type $I_k$, and these transformations are the $X(f)^\natural$ of unique morphisms $X(f)$ of type $I(f)$. The relations $(f',\mathrm{id})\circ(f,\mathrm{id})=(f'f,\mathrm{id})$ and $(\mathrm{id}_k,\mathrm{id})=\mathrm{id}$ give $X(f'f)=X(f')X(f)$ and $X(\mathrm{id}_k)=\mathrm{id}$, so $X$ is an $n$-diagram of type $I$. Since $(f,\alpha)=(\mathrm{id}_{k'},\alpha)\circ(f,\mathrm{id})$, we have $F=X^\natural$, and $F$ determines $X$. The same argument, on the fibres and at the morphisms $(f,\mathrm{id})$, applies to natural transformations $X^\natural\to Y^\natural\circ\int t$. $\square$

5.4. Topoi of diagrams. Let $T$ be a topos. For an $n$-diagram $X$ of type $I$, $n\geq1$, let $D(X)=T^{\int I}/X^\natural$ be the category of functors $E\colon\int I\to T$ with a natural transformation $E\to X^\natural$; for $n=0$, put $D(X)=T/X$. It is a topos, as a localization of the topos $T^{\int I}$ (the remark after Proposition 5.1 of the lesson on fibred topoi). For a morphism $m\colon X\to Y$ of type $t$, let $\begin{gathered}m^*\colon D(Y)\to D(X),\\ E\mapsto(E\circ\textstyle\int t)\times_{Y^\natural\circ\int t}X^\natural,\end{gathered}$ the fibre product being taken along $m^\natural$. For $n=0$ the formula reads $E\mapsto E\times_YX$.

Lemma 5.5. The functor $m^*$ preserves all small limits and colimits, and it has a left adjoint $m_!$ and a right adjoint $m_*$. So $(m^*,m_*)$ is a morphism of topoi $D(X)\to D(Y)$. The canonical isomorphisms between iterated fibre products give isomorphisms $m^*m'^*\cong(m'm)^*$ and $\mathrm{id}^*\cong\mathrm{id}$ that satisfy the conditions of Definition 1.1 of the lesson on fibred topoi. In particular, an $n$-diagram $X\colon K\to\operatorname{Diagr}_{n-1}(T)$ defines a good fibred topos $k\mapsto D(X_k)$ over $K$, with the transition morphisms of the $X(f)$.

Proof. The functor $m^*$ is the composite of the functor $\pi\colon E\mapsto E\circ\int t$, from $T^{\int J}/Y^\natural$ to $T^{\int I}/(Y^\natural\circ\int t)$, and of the base change $\beta$ along $m^\natural$. Since $T$ has small limits and colimits and $\int I$ is small, the left and right Kan extensions along $\int t$ exist. The functor $\pi$ has the left adjoint $G\mapsto\operatorname{Lan}_{\int t}G$, with the map to $Y^\natural$ induced by $G\to Y^\natural\circ\int t$ and the counit, and the right adjoint $G\mapsto\operatorname{Ran}_{\int t}G\times_{\operatorname{Ran}_{\int t}(Y^\natural\circ\int t)}Y^\natural$, through the unit. The functor $\beta$ has the left adjoint given by composition with $m^\natural$, and a right adjoint, because $\beta$ is the inverse image of the localization morphism of the topos $T^{\int I}/(Y^\natural\circ\int t)$ at its object $m^\natural$ [Stacks, Tag 04GZ]. Composing gives $m_!$ and $m_*$, so $m^*$ preserves all small limits and colimits. The isomorphism $m^*m'^*\cong(m'm)^*$ comes from $(m'm)^\natural=(m'^\natural\int t)\circ m^\natural$ and $\int(t't)=\int t'\circ\int t$: both sides are fibre products of $E\circ\int t'\circ\int t$ with $X^\natural$ over $Z^\natural\circ\int t'\circ\int t$. The conditions on triple composites hold because the isomorphisms between iterated fibre products are the unique ones compatible with the projections. $\square$

5.6. The topoi of an $n$-diagram. Let $T$ be a topos. Define topoi $\operatorname{Top}(X)$ and $\operatorname{Top}^{\mathrm o}(X)$ of $n$-diagrams $X$, with morphisms of topoi for the morphisms of $n$-diagrams and isomorphisms for their composites as in Lemma 5.5, by induction on $n$. For $n=0$, both are $T/X$, with the localization morphisms. For $n\geq1$ and $X\colon K\to\operatorname{Diagr}_{n-1}(T)$, the levels $\operatorname{Top}(X_k)$ and the morphisms of topoi of the $X(f)$ form a fibred topos over $K$, and $\operatorname{Top}(X)$ is its total topos $\operatorname{Top}$. A morphism $(u,v)\colon X\to Y$ gives a morphism of fibred topoi, formed by $v$ and the morphisms of topoi of the $u_k$, with the isomorphisms $\theta_f$ that come from $u_{k'}\circ X(f)=Y(v(f))\circ u_k$ and the isomorphisms for composites one level down. Proposition 2.1 gives the morphism of topoi $\operatorname{Top}(X)\to\operatorname{Top}(Y)$, and the formulas of Proposition 1.2 give the isomorphisms for composites. The same construction with $\operatorname{Top}^{\mathrm o}$ and Proposition 3.1 defines $\operatorname{Top}^{\mathrm o}(X)$, provided that the fibred topos $k\mapsto\operatorname{Top}^{\mathrm o}(X_k)$ is good and its total category $\operatorname{Top}^{\mathrm o}$ is a topos. This is part of the following theorem, which describes $\operatorname{Top}^{\mathrm o}(X)$.

Theorem 5.7. Let $T$ be a topos, $n\geq1$, and $X$ an $n$-diagram of $T$ of type $I$. Regard $X^\natural\colon\int I\to T$ as a diagram in $T$, with the fibred topos $c\mapsto T/X^\natural(c)$ over $\int I$ of Examples 1.2 (c) of the lesson on fibred topoi.

$\operatorname{Top}^{\mathrm o}(X)$ is defined, and it is equivalent to $\operatorname{Top}^{\mathrm o}(X^\natural)$, that is, to the topos $D(X)=T^{\int I}/X^\natural$, hence, by Lemma 5.3, to the category $\operatorname{Diagr}_I(T)/X$ of $I$-diagrams over $X$.
$\operatorname{Top}(X)$ is equivalent to $\operatorname{Top}(X^\natural)$.
The equivalences carry the restriction to a level $X_k$ of $X$ to the restriction to the fibre of $\int I$ over $k$, where the fibred topos is that of $X_k^\natural$. They carry the inverse images of a morphism $m$ of $n$-diagrams of type $t$ to those of the morphism of diagrams $(m^\natural,\int t)\colon X^\natural\to Y^\natural$, up to isomorphisms compatible with composites.

Proof. Induction on $n$. For $n=1$, $X^\natural=X$, and Proposition 5.1 of the lesson on fibred topoi identifies $\operatorname{Top}^{\mathrm o}(X)$ with $D(X)=T^I/X$. For a morphism $m=(u,v)$ of $1$-diagrams, the inverse image of Proposition 1.2 is $G\mapsto(G_{v(i)}\times_{Y(v(i))}X(i))_i$, which is $m^*G$.

Let $n\geq2$ and $X\colon K\to\operatorname{Diagr}_{n-1}(T)$, with $I_k$ the type of $X_k$. By induction, the equivalences for the $X_k$, together with the isomorphisms that relate them to the inverse images of the $X(f)$, form $K$-morphisms from the fibred topoi $k\mapsto\operatorname{Top}(X_k)$ and $k\mapsto\operatorname{Top}^{\mathrm o}(X_k)$ to the fibred topoi $k\mapsto\operatorname{Top}(X_k^\natural)$ and $k\mapsto\operatorname{Top}^{\mathrm o}(X_k^\natural)$, whose transition morphisms are those of the morphisms of diagrams $(X(f)^\natural,\int I(f))$. The levels of these $K$-morphisms are equivalences. The last fibred topos is $k\mapsto D(X_k)$, which is good by Lemma 5.5; hence so is $k\mapsto\operatorname{Top}^{\mathrm o}(X_k)$. A $K$-morphism whose levels are equivalences induces equivalences of both total categories, by the solution of Exercise 6.2 and the same argument with the transition maps reversed.

An object of the total topos $\operatorname{Top}$ of $k\mapsto\operatorname{Top}(X_k^\natural)$ is a family of objects $F_k$ of $\operatorname{Top}(X_k^\natural)$, that is, of objects $F_{(k,a)}$ of $T/X^\natural(k,a)$ with maps $\alpha^*F_{(k,a')}\to F_{(k,a)}$ for the morphisms $(\mathrm{id}_k,\alpha)$ of the fibre, together with maps $X(f)^*F_{k'}\to F_k$, compatible with composites. By Proposition 1.2, the components of the latter are maps $(X(f)^\natural_a)^*F_{(k',(\int I(f))(a))}\to F_{(k,a)}$. Since $X^\natural(f,\alpha)=X_{k'}^\natural(\alpha)\circ X(f)^\natural_a$ and $(f,\alpha)=(\mathrm{id}_{k'},\alpha)\circ(f,\mathrm{id})$, these data are the same as maps $X^\natural(f,\alpha)^*F_{(k',a')}\to F_{(k,a)}$ for all morphisms $(f,\alpha)$ of $\int I$, compatible with composites, as in the proof of Lemma 5.3: an object of $\operatorname{Top}(X^\natural)$. Morphisms correspond in the same way. The same argument, with the maps reversed, identifies the total category $\operatorname{Top}^{\mathrm o}$ of $k\mapsto\operatorname{Top}^{\mathrm o}(X_k^\natural)$ with $\operatorname{Top}^{\mathrm o}(X^\natural)$, which is $D(X)$ by Proposition 5.1 of the lesson on fibred topoi. So $\operatorname{Top}^{\mathrm o}(X)$ is a topos. On the fibre over $k$, these identifications restrict to the identities of $\operatorname{Top}(X_k^\natural)$ and $\operatorname{Top}^{\mathrm o}(X_k^\natural)$.

For a morphism $m=(u,v)\colon X\to Y$ of type $t$, the inverse images of Proposition 1.2 are computed on the levels by those of the $u_k$, which correspond by induction to those of the morphisms of diagrams $(u_k^\natural,\int t_k)$, where $t_k$ is the type of $u_k$. The functor $\int t$ maps the fibre over $k$ to the fibre over $v(k)$ by $\int t_k$, and $m^\natural$ restricts there to $u_k^\natural$. So the inverse images of $(m^\natural,\int t)$ restrict on the fibres to those of the $(u_k^\natural,\int t_k)$, and the transition maps agree. $\square$

Examples 5.8. (a) Let $K$ and $I$ be small categories, and let $X$ be the $2$-diagram of $T$ with $X_k$ the constant $I$-diagram at the final object $e$ for every $k$, and $X(f)=(\mathrm{id},\mathrm{id})$. Its type is $(K,k\mapsto I)$ with identity transition functors, and $\int$ of it is $K\times I$ (Exercise 6.4). So $\operatorname{Top}^{\mathrm o}(X)$ is the category of functors $K\times I\to T$. For $K=I=\Delta^{\mathrm{op}}$, it is the category of bisimplicial objects of $T$.

(b) In the situation of (a), $\operatorname{Top}(X_k)$ is the category of functors $I^{\mathrm{op}}\to T$, by Example 5.2 (c) of the lesson on fibred topoi, and the transition morphisms are identities. By the same example, $\operatorname{Top}(X)$ is the category of functors $K^{\mathrm{op}}\to T^{I^{\mathrm{op}}}$, that is, of functors $(K\times I)^{\mathrm{op}}\to T$. For $K=I=\Delta^{\mathrm{op}}$, it is the category of bicosimplicial objects of $T$.

6. Exercises

Exercise 6.1. Let $X$ be the fibred topos over $\{0\to1\}$ defined by a morphism of topoi $X_0\to X_1$. Show that $\Gamma(\operatorname{Top}(X),F)=\Gamma(X_1,F_1)$.

Exercise 6.2. Let $m$ be an $I$-morphism such that every $u_i$ is an equivalence of topoi. Show that $m^*\colon\operatorname{Top}(Y)\to\operatorname{Top}(X)$ is an equivalence.

Exercise 6.3. Let $X$ be the constant fibred topos over $\{0\to1\}$ with value the category of sets, and $A$ the constant ring $\mathbb Z$. Let $L=(L_0,L_1)=(0,\mathbb Z)$ and $M=(\mathbb Z,\mathbb Z)$ with identity transition map. Show that the map of Proposition 4.2 (d) for $i=1$ is not an isomorphism.

Exercise 6.4. Let $K$ and $I$ be small categories, and let $\mathbf I$ be the type $(K,k\mapsto I)$ with identity transition functors. Show that $\int\mathbf I$ is isomorphic to $K\times I$. Deduce from Lemma 5.3 that a $K$-diagram of $I$-diagrams of a category $T$ whose transition morphisms have identity types is the same as a functor $K\times I\to T$.

7. Complete solutions

6.1. By Corollary 2.2 (a), $\Gamma(\operatorname{Top}(X),F)=\varprojlim_{i\in I}\Gamma(X_i,F_i)$, the limit of the contravariant functor with the map $\Gamma(X_1,F_1)\to\Gamma(X_0,F_0)$ induced by $F_f$. The object $1$ is final in $I=\{0\to1\}$, and the limit of a contravariant functor is its value at a final object. So the limit is $\Gamma(X_1,F_1)$.

6.2. Choose quasi-inverses $w_i^*$ of the $u_i^*$, with isomorphisms $u_i^*w_i^*\cong\mathrm{id}$ and $w_i^*u_i^*\cong\mathrm{id}$. Composing $w_i^*\theta_fw_j^*\colon w_i^*f^*u_j^*w_j^*\to w_i^*u_i^*f^*w_j^*$ with these isomorphisms gives isomorphisms $\theta'_f\colon w_i^*f^*\to f^*w_j^*$, compatible with composites because the $\theta_f$ are. Define $w^*\colon\operatorname{Top}(X)\to\operatorname{Top}(Y)$ by $w^*(E)_i=w_i^*E_i$ and $w^*(E)_f=w_i^*(E_f)\circ\theta'^{-1}_f$, the composite $f^*w_j^*E_j\to w_i^*f^*E_j\to w_i^*E_i$. The isomorphisms $u_i^*w_i^*E_i\cong E_i$ and $w_i^*u_i^*F_i\cong F_i$ are compatible with the transition maps, by the definition of $\theta'$ and the naturality of the chosen isomorphisms. So they give isomorphisms $m^*w^*\cong\mathrm{id}$ and $w^*m^*\cong\mathrm{id}$, and $m^*$ is an equivalence.

6.3. Here $f^*=\mathrm{id}$, $A_0=A_1=\mathbb Z$, and $L_f\colon\mathbb Z\to0$. By Proposition 4.2 (d) and its proof, the level-1 value of $\mathcal Hom_A(L,M)$ is $\operatorname{Hom}_A(1_!(\mathbb Z)\otimes_AL,M)$, where $1_!(\mathbb Z)=(\mathbb Z,\mathbb Z)$ with identity transition map. Then $1_!(\mathbb Z)\otimes_AL=L$. A morphism $L\to M$ is a pair $(u_0\colon0\to\mathbb Z,u_1\colon\mathbb Z\to\mathbb Z)$ with $u_0\circ L_f=M_f\circ u_1$. The left side is $0$, so $u_1=0$. Hence the level-1 value is $0$, while $\operatorname{Hom}(L_1,M_1)=\mathbb Z$.

6.4. An object of $\int\mathbf I$ is a pair $(k,a)$ with $a\in I$. Since $\int\mathbf I(f)$ is the identity of $I$, a morphism $(k,a)\to(k',a')$ is a pair $(f,\alpha)$ with $f\colon k\to k'$ and $\alpha\colon a\to a'$, and the composition rule of 5.2 becomes $(f',\alpha')\circ(f,\alpha)=(f'f,\alpha'\alpha)$. So $\int\mathbf I=K\times I$. By Lemma 5.3, $X\mapsto X^\natural$ is an isomorphism from $\operatorname{Diagr}_{\mathbf I}(T)$ to the category of functors $K\times I\to T$. An object of $\operatorname{Diagr}_{\mathbf I}(T)$ is a functor $k\mapsto X_k$ to $\operatorname{Diagr}_1(T)$ with $X_k$ of type $I$ and $X(f)$ of type $\mathrm{id}_I$, that is, a $K$-diagram of $I$-diagrams whose transition morphisms are natural transformations $X_k\to X_{k'}$.

Sources

Morphisms of fibred topoi and the inverse image on total topoi are in SGA 4, Exposé VI, introduced by Grothendieck and Verdier. The formulas for direct images on $\operatorname{Top}^{\mathrm o}$, modules on $\operatorname{Top}^{\mathrm o}$ and the enveloping functors $f^A_!$ are in Illusie's work on the cotangent complex. Diagrams of diagrams, their types and the description of the second total topos of an $n$-diagram as the topos of diagrams of its type over it are in Illusie's work on the cotangent complex. The total category of a type is the construction of Grothendieck in SGA 1, Exposé VI, iterated; the proof given here reduces the description to diagrams of one level through it. The results on modules used here are in the Stacks project, Chapter Modules on Sites.