85.3 Simplicial sites and topoi
It seems natural to define a simplicial site as a simplicial object in the (big) category whose objects are sites and whose morphisms are morphisms of sites. See Sites, Definitions 7.6.2 and 7.14.1 with composition of morphisms as in Sites, Lemma 7.14.4. But here are some variants one might want to consider: (a) we could work with cocontinuous functors (see Sites, Sections 7.20 and 7.21) between sites instead, (b) we could work in a suitable $2$-category of sites where one introduces the notion of a $2$-morphism between morphisms of sites, (c) we could work in a $2$-category constructed out of cocontinuous functors. Instead of picking one of these variants as a definition we will simply develop theory as needed.
Certainly a simplicial topos should probably be defined as a pseudo-functor from $\Delta ^{opp}$ into the $2$-category of topoi. See Categories, Definition 4.29.5 and Sites, Section 7.15 and 7.36. We will try to avoid working with such a beast if possible.
Case A. Let $\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites. This means that for every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a morphism of sites $f_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$. This morphism is given by a continuous functor in the opposite direction which we will denote $u_\varphi : \mathcal{C}_ m \to \mathcal{C}_ n$.
Lemma 85.3.1. Let $\mathcal{C}$ be a simplicial object in the category of sites. With notation as above we construct a site $\mathcal{C}_{total}$ as follows.
An object of $\mathcal{C}_{total}$ is an object $U$ of $\mathcal{C}_ n$ for some $n$,
a morphism $(\varphi , f) : U \to V$ of $\mathcal{C}_{total}$ is given by a map $\varphi : [m] \to [n]$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$, $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$ and a morphism $f : U \to u_\varphi (V)$ of $\mathcal{C}_ n$, and
a covering $\{ (\text{id}, f_ i) : U_ i \to U\} $ in $\mathcal{C}_{total}$ is given by an $n$ and a covering $\{ f_ i : U_ i \to U\} $ of $\mathcal{C}_ n$.
Proof.
Composition of $(\varphi , f) : U \to V$ with $(\psi , g) : V \to W$ is given by $(\varphi \circ \psi , u_\varphi (g) \circ f)$. This uses that $u_\varphi \circ u_\psi = u_{\varphi \circ \psi }$.
Let $\{ (\text{id}, f_ i) : U_ i \to U\} $ be a covering as in (3) and let $(\varphi , g) : W \to U$ be a morphism with $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$. We claim that
\[ W \times _{(\varphi , g), U, (\text{id}, f_ i)} U_ i = W \times _{g, u_\varphi (U), u_\varphi (f_ i)} u_\varphi (U_ i) \]
in the category $\mathcal{C}_{total}$. This makes sense as by our definition of morphisms of sites, the required fibre products in $\mathcal{C}_ m$ exist since $u_\varphi $ transforms coverings into coverings. The same reasoning implies the claim (details omitted). Thus we see that the collection of coverings is stable under base change. The other axioms of a site are immediate.
$\square$
Case B. Let $\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. This means that for every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a cocontinuous functor denoted $u_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$. The associated morphism of topoi is denoted $f_\varphi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m)$.
Lemma 85.3.2. Let $\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors $u_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ have property $P$ of Sites, Remark 7.20.5. Then we can construct a site $\mathcal{C}_{total}$ as follows.
An object of $\mathcal{C}_{total}$ is an object $U$ of $\mathcal{C}_ n$ for some $n$,
a morphism $(\varphi , f) : U \to V$ of $\mathcal{C}_{total}$ is given by a map $\varphi : [m] \to [n]$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$, $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$ and a morphism $f : u_\varphi (U) \to V$ of $\mathcal{C}_ m$, and
a covering $\{ (\text{id}, f_ i) : U_ i \to U\} $ in $\mathcal{C}_{total}$ is given by an $n$ and a covering $\{ f_ i : U_ i \to U\} $ of $\mathcal{C}_ n$.
Proof.
Composition of $(\varphi , f) : U \to V$ with $(\psi , g) : V \to W$ is given by $(\varphi \circ \psi , g \circ u_\psi (f))$. This uses that $u_\psi \circ u_\varphi = u_{\varphi \circ \psi }$.
Let $\{ (\text{id}, f_ i) : U_ i \to U\} $ be a covering as in (3) and let $(\varphi , g) : W \to U$ be a morphism with $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$. We claim that
\[ W \times _{(\varphi , g), U, (\text{id}, f_ i)} U_ i = W \times _{g, U, f_ i} U_ i \]
in the category $\mathcal{C}_{total}$ where the right hand side is the object of $\mathcal{C}_ m$ defined in Sites, Remark 7.20.5 which exists by property $P$. Compatibility of this type of fibre product with compositions of functors implies the claim (details omitted). Since the family $\{ W \times _{g, U, f_ i} U_ i \to W\} $ is a covering of $\mathcal{C}_ m$ by property $P$ we see that the collection of coverings is stable under base change. The other axioms of a site are immediate.
$\square$
Situation 85.3.3. Here we have one of the following two cases:
$\mathcal{C}$ is a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites. For every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a morphism of sites $f_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ given by a continuous functor $u_\varphi : \mathcal{C}_ m \to \mathcal{C}_ n$.
$\mathcal{C}$ is a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors having property $P$ of Sites, Remark 7.20.5. For every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a cocontinuous functor $u_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ which induces a morphism of topoi $f_\varphi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m)$.
As usual we will denote $f_\varphi ^{-1}$ and $f_{\varphi , *}$ the pullback and pushforward. We let $\mathcal{C}_{total}$ denote the site defined in Lemma 85.3.1 (case A) or Lemma 85.3.2 (case B).
Let $\mathcal{C}$ be as in Situation 85.3.3. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}_{total}$. It is clear from the definition of coverings, that the restriction of $\mathcal{F}$ to the objects of $\mathcal{C}_ n$ defines a sheaf $\mathcal{F}_ n$ on the site $\mathcal{C}_ n$. For every $\varphi : [m] \to [n]$ the restriction maps of $\mathcal{F}$ along the morphisms $(\varphi , f) : U \to V$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$ and $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ m)$ define an element $\mathcal{F}(\varphi )$ of
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m)}(\mathcal{F}_ m, f_{\varphi , *}\mathcal{F}_ n) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)}(f_\varphi ^{-1}\mathcal{F}_ m, \mathcal{F}_ n) \]
Moreover, given $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ the diagrams
\[ \vcenter { \xymatrix{ \mathcal{F}_ l \ar[rr]_{\mathcal{F}(\varphi \circ \psi )} \ar[rd]_{\mathcal{F}(\psi )} & & f_{\varphi \circ \psi , *} \mathcal{F}_ n \\ & f_{\psi , *}\mathcal{F}_ m \ar[ur]_{f_{\psi , *}\mathcal{F}(\varphi )} } } \quad \text{and}\quad \vcenter { \xymatrix{ f_{\varphi \circ \psi }^{-1}\mathcal{F}_ l \ar[rr]_{\mathcal{F}(\varphi \circ \psi )} \ar[rd]_{f_\varphi ^{-1}\mathcal{F}(\psi )} & & \mathcal{F}_ n \\ & f_\varphi ^{-1}\mathcal{F}_ m \ar[ur]_{\mathcal{F}(\varphi )} } } \]
commute. Clearly, the converse statement is true as well: if we have a system $(\{ \mathcal{F}_ n\} _{n \geq 0}, \{ \mathcal{F}(\varphi )\} _{\varphi \in \text{Arrows}(\Delta )})$ satisfying the commutativity constraints above, then we obtain a sheaf on $\mathcal{C}_{total}$.
Lemma 85.3.4. In Situation 85.3.3 there is an equivalence of categories between
$\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$, and
the category of systems $(\mathcal{F}_ n, \mathcal{F}(\varphi ))$ described above.
In particular, the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ only depends on the topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)$ and the morphisms of topoi $f_\varphi $.
Proof.
See discussion above.
$\square$
Lemma 85.3.5. In Situation 85.3.3 the functor $\mathcal{C}_ n \to \mathcal{C}_{total}$, $U \mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ satisfies
$g_ n^{-1}$ associates to the sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ the sheaf $\mathcal{F}_ n$ on $\mathcal{C}_ n$,
$g_ n^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)$ has a left adjoint $g^{Sh}_{n!}$,
for $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)$ the restriction of $g_{n!}^{Sh}\mathcal{G}$ to $\mathcal{C}_ m$ is $\coprod \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G}$,
$g_{n!}^{Sh}$ commutes with finite connected limits,
$g_ n^{-1} : \textit{Ab}(\mathcal{C}_{total}) \to \textit{Ab}(\mathcal{C}_ n)$ has a left adjoint $g_{n!}$,
for $\mathcal{G}$ in $\textit{Ab}(\mathcal{C}_ n)$ the restriction of $g_{n!}\mathcal{G}$ to $\mathcal{C}_ m$ is $\bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G}$, and
$g_{n!}$ is exact.
Proof.
Case A. If $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_ n$ then the image $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_{total}$ by definition (Lemma 85.3.1). For a morphism $V \to U$ of $\mathcal{C}_ n$, the fibre product $V \times _ U U_ i$ in $\mathcal{C}_ n$ is also the fibre product in $\mathcal{C}_{total}$ (by the claim in the proof of Lemma 85.3.1). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of $U$ in $\mathcal{C}_ n$ and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is certainly the case that our functor is cocontinuous.
Case B. If $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_ n$ then the image $\{ U_ i \to U\} _{i \in I}$ is a covering in $\mathcal{C}_{total}$ by definition (Lemma 85.3.2). For a morphism $V \to U$ of $\mathcal{C}_ n$, the fibre product $V \times _ U U_ i$ in $\mathcal{C}_ n$ is also the fibre product in $\mathcal{C}_{total}$ (by the claim in the proof of Lemma 85.3.2). Therefore our functor is continuous. On the other hand, our functor defines a bijection between coverings of $U$ in $\mathcal{C}_ n$ and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is certainly the case that our functor is cocontinuous.
At this point part (1) and the existence of $g^{Sh}_{n!}$ and $g_{n!}$ in cases A and B follows from Sites, Lemmas 7.21.1 and 7.21.5 and Modules on Sites, Lemma 18.16.2.
Proof of (3). Let $\mathcal{G}$ be a sheaf on $\mathcal{C}_ n$. Consider the sheaf $\mathcal{H}$ on $\mathcal{C}_{total}$ whose degree $m$ part is the sheaf
\[ \mathcal{H}_ m = \coprod \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G} \]
given in part (3) of the statement of the lemma. Given a map $\psi : [m] \to [m']$ the map $\mathcal{H}(\psi ) : f_\psi ^{-1}\mathcal{H}_ m \to \mathcal{H}_{m'}$ is given on components by the identifications
\[ f_\psi ^{-1} f_\varphi ^{-1} \mathcal{G} \to f_{\psi \circ \varphi }^{-1}\mathcal{G} \]
Observe that given a map $\alpha : \mathcal{H} \to \mathcal{F}$ of sheaves on $\mathcal{C}_{total}$ we obtain a map $\mathcal{G} \to \mathcal{F}_ n$ corresponding to the restriction of $\alpha _ n$ to the component $\mathcal{G}$ in $\mathcal{H}_ n$. Conversely, given a map $\beta : \mathcal{G} \to \mathcal{F}_ n$ of sheaves on $\mathcal{C}_ n$ we can define $\alpha : \mathcal{H} \to \mathcal{F}$ by letting $\alpha _ m$ be the map which on components
\[ f_\varphi ^{-1}\mathcal{G} \to \mathcal{F}_ m \]
uses the maps adjoint to $\mathcal{F}(\varphi ) \circ f_\varphi ^{-1}\beta $. We omit the arguments showing these two constructions give mutually inverse maps
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)}(\mathcal{G}, \mathcal{F}_ n) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) \]
Thus $\mathcal{H} = g^{Sh}_{n!}\mathcal{G}$ as desired.
Proof of (4). If $\mathcal{G}$ is an abelian sheaf on $\mathcal{C}_ n$, then we proceed in exactly the same ammner as above, except that we define $\mathcal{H}$ is the abelian sheaf on $\mathcal{C}_{total}$ whose degree $m$ part is the sheaf
\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}\mathcal{G} \]
with transition maps defined exactly as above. The bijection
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C}_ n)}(\mathcal{G}, \mathcal{F}_ n) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F}) \]
is proved exactly as above. Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired.
The exactness properties of $g^{Sh}_{n!}$ and $g_{n!}$ follow from formulas given for these functors.
$\square$
slogan
Lemma 85.3.6. In Situation 85.3.3. If $\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n$ is injective in $\textit{Ab}(\mathcal{C}_ n)$. If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n^\bullet $ is K-injective in $\textit{Ab}(\mathcal{C}_ n)$.
Proof.
The first statement follows from Homology, Lemma 12.29.1 and Lemma 85.3.5. The second statement from Derived Categories, Lemma 13.31.9 and Lemma 85.3.5.
$\square$
Comments (2)
Comment #7493 by Bogdan on
Comment #7497 by Johan on