The Stacks project

7.28 Localization and morphisms

The following lemma is important in order to understand relation between localization and morphisms of sites and topoi.

Lemma 7.28.1. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ and set $U = u(V)$. Then the functor $u' : \mathcal{D}/V \to \mathcal{C}/U$, $V'/V \mapsto u(V')/U$ determines a morphism of sites $f' : \mathcal{C}/U \to \mathcal{D}/V$. The morphism $f'$ fits into a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). } \]

Using the identifications $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/h_ V^\# $ of Lemma 7.25.4 the functor $(f')^{-1}$ is described by the rule

\[ (f')^{-1}(\mathcal{H} \xrightarrow {\varphi } h_ V^\# ) = (f^{-1}\mathcal{H} \xrightarrow {f^{-1}\varphi } h_ U^\# ). \]

Finally, we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$.

Proof. It is clear that $u'$ is continuous, and hence we get functors $f'_* = (u')^ s = (u')^ p$ (see Sections 7.5 and 7.13) and an adjoint $(f')^{-1} = (u')_ s = ((u')_ p\ )^\# $. The assertion $f'_*j_ U^{-1} = j_ V^{-1}f_*$ follows as

\[ (j_ V^{-1}f_*\mathcal{F})(V'/V) = f_*\mathcal{F}(V') = \mathcal{F}(u(V')) = (j_ U^{-1}\mathcal{F})(u(V')/U) = (f'_*j_ U^{-1}\mathcal{F})(V'/V) \]

which holds even for presheaves. What isn't clear a priori is that $(f')^{-1}$ is exact, that the diagram commutes, and that the description of $(f')^{-1}$ holds.

Let $\mathcal{H}$ be a sheaf on $\mathcal{D}/V$. Let us compute $j_{U!}(f')^{-1}\mathcal{H}$. We have

\begin{eqnarray*} j_{U!}(f')^{-1}\mathcal{H} & = ((j_ U)_ p(u'_ p\mathcal{H})^\# )^\# \\ & = ((j_ U)_ pu'_ p\mathcal{H})^\# \\ & = (u_ p(j_ V)_ p\mathcal{H})^\# \\ & = f^{-1}j_{V!}\mathcal{H} \end{eqnarray*}

The first equality by unwinding the definitions. The second equality by Lemma 7.13.4. The third equality because $u \circ j_ V = j_ U \circ u'$. The fourth equality by Lemma 7.13.4 again. All of the equalities above are isomorphisms of functors, and hence we may interpret this as saying that the following diagram of categories and functors is commutative

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_{U!}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_{V!}} \ar[u]^{(f')^{-1}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/h_ V^\# \ar[r] \ar[u]^{f^{-1}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[u]^{f^{-1}} } \]

The middle arrow makes sense as $f^{-1}h_ V^\# = (h_{u(V)})^\# = h_ U^\# $, see Lemma 7.13.5. In particular this proves the description of $(f')^{-1}$ given in the statement of the lemma. Since by Lemma 7.25.4 the left horizontal arrows are equivalences and since $f^{-1}$ is exact by assumption we conclude that $(f')^{-1} = u'_ s$ is exact. Namely, because it is a left adjoint it is already right exact (Categories, Lemma 4.24.5). Hence we only need to show that it transforms a final object into a final object and commutes with fibre products (Categories, Lemma 4.23.2). Both are clear for the induced functor $f^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/h_ V^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $. This proves that $f'$ is a morphism of sites.

We still have to verify that $(f')^{-1}j_ V^{-1} = j_ U^{-1}f^{-1}$. To see this use the formula above and the description in Lemma 7.25.7. Namely, combined these give, for any sheaf $\mathcal{G}$ on $\mathcal{D}$, that

\[ j_{U!}(f')^{-1}j_ V^{-1}\mathcal{G} = f^{-1}j_{V!}j_ V^{-1}\mathcal{G} = f^{-1}(\mathcal{G} \times h_ V^\# ) = f^{-1}\mathcal{G} \times h_ U^\# = j_{U!}j_ U^{-1}f^{-1}\mathcal{G}. \]

Since the functor $j_{U!}$ induces an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ we conclude. $\square$

The following lemma is a special case of the more general Lemma 7.28.1 above.

Lemma 7.28.2. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a functor. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Set $U = u(V)$. Assume that

  1. $\mathcal{C}$ and $\mathcal{D}$ have all finite limits,

  2. $u$ is continuous, and

  3. $u$ commutes with finite limits.

There exists a commutative diagram of morphisms of sites

\[ \xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d]_{f'} & \mathcal{C} \ar[d]^ f \\ \mathcal{D}/V \ar[r]^{j_ V} & \mathcal{D} } \]

where the right vertical arrow corresponds to $u$, the left vertical arrow corresponds to the functor $u' : \mathcal{D}/V \to \mathcal{C}/U$, $V'/V \mapsto u(V')/u(V)$ and the horizontal arrows correspond to the functors $\mathcal{C} \to \mathcal{C}/U$, $X \mapsto X \times U$ and $\mathcal{D} \to \mathcal{D}/V$, $Y \mapsto Y \times V$ as in Lemma 7.27.2. Moreover, the associated diagram of morphisms of topoi is equal to the diagram of Lemma 7.28.1. In particular we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$.

Proof. Note that $u$ satisfies the assumptions of Proposition 7.14.7 and hence induces a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ by that proposition. It is clear that $u$ induces a functor $u'$ as indicated. It is clear that this functor also satisfies the assumptions of Proposition 7.14.7. Hence we get a morphism of sites $f' : \mathcal{C}/U \to \mathcal{D}/V$. The diagram commutes by our definition of composition of morphisms of sites (see Definition 7.14.5) and because

\[ u(Y \times V) = u(Y) \times u(V) = u(Y) \times U \]

which shows that the diagram of categories and functors opposite to the diagram of the lemma commutes. $\square$

At this point we can localize a site, we know how to relocalize, and we can localize a morphism of sites at an object of the site downstairs. If we combine these then we get the following kind of diagram.

Lemma 7.28.3. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$, $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f_ c} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). } \]

We have $f_ c = f' \circ j_{U/u(V)}$ where $f' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V)) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V)$ is as in Lemma 7.28.1 and $j_{U/u(V)} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V))$ is as in Lemma 7.25.8. Using the identifications $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/h_ V^\# $ of Lemma 7.25.4 the functor $(f_ c)^{-1}$ is described by the rule

\[ (f_ c)^{-1}(\mathcal{H} \xrightarrow {\varphi } h_ V^\# ) = (f^{-1}\mathcal{H} \times _{f^{-1}\varphi , h_{u(V)}^\# , c} h_ U^\# \rightarrow h_ U^\# ). \]

Finally, given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that

\[ \xymatrix{ U' \ar[r]_-{c'} \ar[d]_ a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) } \]

commutes, then the diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U') \ar[r]_{j_{U'/U}} \ar[d]_{f_{c'}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[d]^{f_ c} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V') \ar[r]^{j_{V'/V}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V). } \]

commutes.

Proof. This lemma proves itself, and is more a collection of things we know at this stage of the development of theory. For example the commutativity of the first square follows from the commutativity of Diagram (7.25.8.1) and the commutativity of the diagram in Lemma 7.28.1. The description of $f_ c^{-1}$ follows on combining Lemma 7.25.9 with Lemma 7.28.1. The commutativity of the last square then follows from the equality

\[ f^{-1}\mathcal{H} \times _{h_{u(V)}^\# , c} h_ U^\# \times _{h_ U^\# } h_{U'}^\# = f^{-1}(\mathcal{H} \times _{h_ V^\# } h_{V'}^\# ) \times _{h_{u(V'), c'}^\# } h_{U'}^\# \]

which is formal using that $f^{-1}h_ V^\# = h_{u(V)}^\# $ and $f^{-1}h_{V'}^\# = h_{u(V')}^\# $, see Lemma 7.13.5. $\square$

In the following lemma we find another kind of functoriality of localization, in case the morphism of topoi comes from a cocontinuous functor. This is a kind of diagram which is different from the diagram in Lemma 7.28.1, and in particular, in general the equality $f'_*j_ U^{-1} = j_ V^{-1}f_*$ seen in Lemma 7.28.1 does not hold in the situation of the following lemma.

Lemma 7.28.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $U$ be an object of $\mathcal{C}$, and set $V = u(U)$. We have a commutative diagram

\[ \xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/V \ar[r]^-{j_ V} & \mathcal{D} } \]

where the left vertical arrow is $u' : \mathcal{C}/U \to \mathcal{D}/V$, $U'/U \mapsto V'/V$. Then $u'$ is cocontinuous also and we get a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^-{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]

where $f$ (resp. $f'$) corresponds to $u$ (resp. $u'$).

Proof. The commutativity of the first diagram is clear. It implies the commutativity of the second diagram provided we show that $u'$ is cocontinuous.

Let $U'/U$ be an object of $\mathcal{C}/U$. Let $\{ V_ j/V \to u(U')/V\} _{j \in J}$ be a covering of $u(U')/V$ in $\mathcal{D}/V$. Since $u$ is cocontinuous there exists a covering $\{ U_ i' \to U'\} _{i \in I}$ such that the family $\{ u(U_ i') \to u(U')\} $ refines the covering $\{ V_ j \to u(U')\} $ in $\mathcal{D}$. In other words, there exists a map of index sets $\alpha : I \to J$ and morphisms $\phi _ i : u(U_ i') \to V_{\alpha (i)}$ over $U'$. Think of $U_ i'$ as an object over $U$ via the composition $U'_ i \to U' \to U$. Then $\{ U'_ i/U \to U'/U\} $ is a covering of $\mathcal{C}/U$ such that $\{ u(U_ i')/V \to u(U')/V\} $ refines $\{ V_ j/V \to u(U')/V\} $ (use the same $\alpha $ and the same maps $\phi _ i$). Hence $u' : \mathcal{C}/U \to \mathcal{D}/V$ is cocontinuous. $\square$

Lemma 7.28.5. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $V$ be an object of $\mathcal{D}$. Let ${}^ u_ V\mathcal{I}$ be the category introduced in Section 7.19. We have a commutative diagram

\[ \vcenter { \xymatrix{ \, _ V^ u\mathcal{I} \ar[r]_ j \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/V \ar[r]^-{j_ V} & \mathcal{D} } } \quad \text{where}\quad \begin{matrix} j : (U, \psi ) \mapsto U \\ u' : (U, \psi ) \mapsto (\psi : u(U) \to V) \end{matrix} \]

Declare a family of morphisms $\{ (U_ i, \psi _ i) \to (U, \psi )\} $ of ${}^ u_ V\mathcal{I}$ to be a covering if and only if $\{ U_ i \to U\} $ is a covering in $\mathcal{C}$. Then

  1. ${}^ u_ V\mathcal{I}$ is a site,

  2. $j$ is continuous and cocontinuous,

  3. $u'$ is cocontinuous,

  4. we get a commutative diagram of topoi

    \[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ({}^ u_ V\mathcal{I}) \ar[r]_ j \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^-{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]

    where $f$ (resp. $f'$) corresponds to $u$ (resp. $u'$), and

  5. we have $f'_*j^{-1} = j_ V^{-1}f_*$.

Proof. Parts (1), (2), (3), and (4) are straightforward consequences of the definitions and the fact that the functor $j$ commutes with fibre products. We omit the details. To see (5) recall that $f_*$ is given by ${}_ su = {}_ pu$. Hence the value of $j_ V^{-1}f_*\mathcal{F}$ on $V'/V$ is the value of ${}_ pu\mathcal{F}$ on $V'$ which is the limit of the values of $\mathcal{F}$ on the category ${}^ u_{V'}\mathcal{I}$. Clearly, there is an equivalence of categories

\[ {}^ u_{V'}\mathcal{I} \to {}^{u'}_{V'/V}\mathcal{I} \]

Since the value of $f'_*j^{-1}\mathcal{F}$ on $V'/V$ is given by the limit of the values of $j^{-1}\mathcal{F}$ on the category ${}^{u'}_{V'/V}\mathcal{I}$ and since the values of $j^{-1}\mathcal{F}$ on objects of ${}^ u_ V\mathcal{I}$ are just the values of $\mathcal{F}$ (by Lemma 7.21.5 as $j$ is continuous and cocontinuous) we see that (5) is true. $\square$

The following two results are of a slightly different nature.

Lemma 7.28.6. Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors

\[ \xymatrix{ \mathcal{C}' \ar[r]_{v'} \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}' \ar[r]^ v & \mathcal{D} } \]

Assume

  1. $u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of topoi $f$, $f'$, $g$, and $g'$ by Lemma 7.21.1,

  2. $v \circ u' = u \circ v'$,

  3. $v$ and $v'$ are continuous as well as cocontinuous, and

  4. for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal.

Then $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.

Proof. The categories ${}^{u'}_{V'}\mathcal{I}$ and ${}^{\ \ \ u}_{v(V')}\mathcal{I}$ are defined in Section 7.19. The functor in condition (4) sends the object $\psi : u'(U') \to V'$ of ${}^{u'}_{V'}\mathcal{I}$ to the object $v(\psi ) : uv'(U') = vu'(U') \to v(V')$ of ${}^{\ \ \ u}_{v(V')}\mathcal{I}$. Recall that $g^{-1}$ is given by $v^ p$ (Lemma 7.21.5) and $f_*$ is given by ${}_ su = {}_ pu$. Hence the value of $g^{-1}f_*\mathcal{F}$ on $V'$ is the value of ${}_ pu\mathcal{F}$ on $v(V')$ which is the limit

\[ \mathop{\mathrm{lim}}\nolimits _{u(U) \to v(V') \in \mathop{\mathrm{Ob}}\nolimits ({}^{\ \ \ u}_{v(V')}\mathcal{I}^{opp})} \mathcal{F}(U) \]

By the same reasoning, the value of $f'_*(g')^{-1}\mathcal{F}$ on $V'$ is given by the limit

\[ \mathop{\mathrm{lim}}\nolimits _{u'(U') \to V' \in \mathop{\mathrm{Ob}}\nolimits ({}^{u'}_{V'}\mathcal{I}^{opp})} \mathcal{F}(v'(U')) \]

Thus assumption (4) and Categories, Lemma 4.17.4 show that these agree and the first equality of the lemma is proved. The second equality follows from the first by uniqueness of adjoints. $\square$

Lemma 7.28.7. Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors

\[ \xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^ v \ar[u]^{u'} & \mathcal{D} \ar[u]_ u } \]

With notation as in Sections 7.14 and 7.21 assume

  1. $u$ and $u'$ are continuous giving rise to morphisms of sites $f$ and $f'$,

  2. $v$ and $v'$ are cocontinuous giving rise to morphisms of topoi $g$ and $g'$,

  3. $u \circ v = v' \circ u'$, and

  4. $v$ and $v'$ are continuous as well as cocontinuous.

Then1 $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.

Proof. Namely, we have

\[ f'_*(g')^{-1}\mathcal{F} = (u')^ p((v')^ p\mathcal{F})^\# = (u')^ p(v')^ p\mathcal{F} \]

The first equality by definition and the second by Lemma 7.21.5. We have

\[ g^{-1}f_*\mathcal{F} = (v^ pu^ p\mathcal{F})^\# = ((u')^ p(v')^ p\mathcal{F})^\# = (u')^ p(v')^ p\mathcal{F} \]

The first equality by definition, the second because $u \circ v = v' \circ u'$, the third because we already saw that $(u')^ p(v')^ p\mathcal{F}$ is a sheaf. This proves $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and the equality $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ follows by uniqueness of left adjoints. $\square$

[1] In this generality we don't know $f \circ g'$ is equal to $g \circ f'$ as morphisms of topoi (there is a canonical $2$-arrow from the first to the second which may not be an isomorphism).

Comments (3)

Comment #8458 by ZL on

I would like to ask how strict the equalities "" in the section are?

To elaborate my question, first suppose that we have morphisms of topoi , and . We say if and . However, if we only know , then by adjunction we know that there exists a canonical isomorphism . Hence we get a canoical -isomorphism .

In this regard, for example in lemma 7.28.4, we have . Hence we have . However, we only have canonical isomorphisms , for and both involving taking inductive limits. Hence what seems to me is that we have a canonical isomorphism .

In the same manner, in lemma 7.28.1, we verified the equlity to conclude that . But again by adjunction of functors, we a priori only knows that canonically.

Is there a remedy for this problem?

Comment #9075 by on

Dear ZL, thank you for the question and apologies for the late reply. Really the category of topoi (and/or of sites) is a 2-category (just like the category of categories is a 2-category). So for some of the diagrams in this section, we shouldn't say that they commute but really that they commute up to a 2-arrow. In the Stacks project we are a bit laks about this and we often use the equal sign "" between functors to indicate that there exists a canonical identification between them, i.e., that there is a canonical invertible 2 arrow from one to the other. Here the word "canonical" indicates that we're too lazy to pin down the isomorphism more precisely.

This works rather well up to a point. If more precision is needed (say beyond 2 catgories into 3 or higher categories), then the best thing is probably to work with -categories which are off limits in the Stacks project.

Comment #9316 by ZL on

Thanks for answering my question and share your philosophy on how to deal with the -morphisms! The canonical -arrows sometimes really bug me a lot.

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