## 7.41 Exactness properties of pushforward

Let $f$ be a morphism of topoi. The functor $f_*$ in general is only left exact. There are many additional conditions one can impose on this functor to single out particular classes of morphisms of topoi. We collect them here and note some of the logical dependencies. Some parts of the following lemma are purely category theoretical (i.e., they do not depend on having a morphism of topoi, just having a pair of adjoint functors is enough).

Lemma 7.41.1. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Consider the following properties (on sheaves of sets):

1. $f_*$ is faithful,

2. $f_*$ is fully faithful,

3. $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective for all $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$,

4. $f_*$ transforms surjections into surjections,

5. $f_*$ commutes with coequalizers,

6. $f_*$ commutes with pushouts,

7. $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is an isomorphism for all $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$,

8. $f_*$ reflects injections,

9. $f_*$ reflects surjections,

10. $f_*$ reflects bijections, and

11. for any surjection $\mathcal{F} \to f^{-1}\mathcal{G}$ there exists a surjection $\mathcal{G}' \to \mathcal{G}$ such that $f^{-1}\mathcal{G}' \to f^{-1}\mathcal{G}$ factors through $\mathcal{F} \to f^{-1}\mathcal{G}$.

Then we have the following implications

1. (2) $\Rightarrow$ (1),

2. (3) $\Rightarrow$ (1),

3. (7) $\Rightarrow$ (1), (2), (3), (8), (9), (10).

4. (3) $\Leftrightarrow$ (9),

5. (6) $\Rightarrow$ (4) and (5) $\Rightarrow$ (4),

6. (4) $\Leftrightarrow$ (11),

7. (9) $\Rightarrow$ (8), (10), and

8. (2) $\Leftrightarrow$ (7).

Picture

$\xymatrix{ (6) \ar@{=>}[rd] & & & & & (9) \ar@{=>}[r] \ar@{=>}[rd] & (8) \\ & (4) \ar@{<=>}[r] & (11) & (2) \ar@{<=>}[r] & (7) \ar@{=>}[ru] \ar@{=>}[rd] & & (10) \\ (5) \ar@{=>}[ur] & & & & & (3) \ar@{=>}[r] & (1) }$

Proof. Proof of (a): This is immediate from the definitions.

Proof of (b). Suppose that $a, b : \mathcal{F} \to \mathcal{F}'$ are maps of sheaves on $\mathcal{C}$. If $f_*a = f_*b$, then $f^{-1}f_*a = f^{-1}f_*b$. Consider the commutative diagram

$\xymatrix{ \mathcal{F} \ar@<-1ex>[r] \ar@<1ex>[r] & \mathcal{F}' \\ f^{-1}f_*\mathcal{F} \ar@<-1ex>[r] \ar@<1ex>[r] \ar[u] & f^{-1}f_*\mathcal{F}' \ar[u] }$

If the bottom two arrows are equal and the vertical arrows are surjective then the top two arrows are equal. Hence (b) follows.

Proof of (c). Suppose that $a : \mathcal{F} \to \mathcal{F}'$ is a map of sheaves on $\mathcal{C}$. Consider the commutative diagram

$\xymatrix{ \mathcal{F} \ar[r] & \mathcal{F}' \\ f^{-1}f_*\mathcal{F} \ar[r] \ar[u] & f^{-1}f_*\mathcal{F}' \ar[u] }$

If (7) holds, then the vertical arrows are isomorphisms. Hence if $f_*a$ is injective (resp. surjective, resp. bijective) then the bottom arrow is injective (resp. surjective, resp. bijective) and hence the top arrow is injective (resp. surjective, resp. bijective). Thus we see that (7) implies (8), (9), (10). It is clear that (7) implies (3). The implications (7) $\Rightarrow$ (2), (1) follow from (a) and (h) which we will see below.

Proof of (d). Assume (3). Suppose that $a : \mathcal{F} \to \mathcal{F}'$ is a map of sheaves on $\mathcal{C}$ such that $f_*a$ is surjective. As $f^{-1}$ is exact this implies that $f^{-1}f_*a : f^{-1}f_*\mathcal{F} \to f^{-1}f_*\mathcal{F}'$ is surjective. Combined with (3) this implies that $a$ is surjective. This means that (9) holds. Assume (9). Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. We have to show that the map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective. It suffices to show that $f_*f^{-1}f_*\mathcal{F} \to f_*\mathcal{F}$ is surjective. And this is true because there is a canonical map $f_*\mathcal{F} \to f_*f^{-1}f_*\mathcal{F}$ which is a one-sided inverse.

Proof of (e). We use Categories, Lemma 4.13.3 without further mention. If $\mathcal{F} \to \mathcal{F}'$ is surjective then $\mathcal{F}' \amalg _\mathcal {F} \mathcal{F}' \to \mathcal{F}'$ is an isomorphism. Hence (6) implies that

$f_*\mathcal{F}' \amalg _{f_*\mathcal{F}} f_*\mathcal{F}' = f_*(\mathcal{F}' \amalg _\mathcal {F} \mathcal{F}') \longrightarrow f_*\mathcal{F}'$

is an isomorphism also. And this in turn implies that $f_*\mathcal{F} \to f_*\mathcal{F}'$ is surjective. Hence we see that (6) implies (4). If $\mathcal{F} \to \mathcal{F}'$ is surjective then $\mathcal{F}'$ is the coequalizer of the two projections $\mathcal{F} \times _{\mathcal{F}'} \mathcal{F} \to \mathcal{F}$ by Lemma 7.11.3. Hence if (5) holds, then $f_*\mathcal{F}'$ is the coequalizer of the two projections

$f_*(\mathcal{F} \times _{\mathcal{F}'} \mathcal{F}) = f_*\mathcal{F} \times _{f_*\mathcal{F}'} f_*\mathcal{F} \longrightarrow f_*\mathcal{F}$

which clearly means that $f_*\mathcal{F} \to f_*\mathcal{F}'$ is surjective. Hence (5) implies (4) as well.

Proof of (f). Assume (4). Let $\mathcal{F} \to f^{-1}\mathcal{G}$ be a surjective map of sheaves on $\mathcal{C}$. By (4) we see that $f_*\mathcal{F} \to f_*f^{-1}\mathcal{G}$ is surjective. Let $\mathcal{G}'$ be the fibre product

$\xymatrix{ f_*\mathcal{F} \ar[r] & f_*f^{-1}\mathcal{G} \\ \mathcal{G}' \ar[u] \ar[r] & \mathcal{G} \ar[u] }$

so that $\mathcal{G}' \to \mathcal{G}$ is surjective also. Consider the commutative diagram

$\xymatrix{ \mathcal{F} \ar[r] & f^{-1}\mathcal{G} \\ f^{-1}f_*\mathcal{F} \ar[r] \ar[u] & f^{-1}f_*f^{-1}\mathcal{G} \ar[u] \\ f^{-1}\mathcal{G}' \ar[u] \ar[r] & f^{-1}\mathcal{G} \ar[u] }$

and we see the required result. Conversely, assume (11). Let $a : \mathcal{F} \to \mathcal{F}'$ be surjective map of sheaves on $\mathcal{C}$. Consider the fibre product diagram

$\xymatrix{ \mathcal{F} \ar[r] & \mathcal{F}' \\ \mathcal{F}'' \ar[u] \ar[r] & f^{-1}f_*\mathcal{F}' \ar[u] }$

Because the lower horizontal arrow is surjective and by (11) we can find a surjection $\gamma : \mathcal{G}' \to f_*\mathcal{F}'$ such that $f^{-1}\gamma$ factors through $\mathcal{F}'' \to f^{-1}f_*\mathcal{F}'$:

$\xymatrix{ & \mathcal{F} \ar[r] & \mathcal{F}' \\ f^{-1}\mathcal{G}' \ar[r] & \mathcal{F}'' \ar[u] \ar[r] & f^{-1}f_*\mathcal{F}' \ar[u] }$

Pushing this down using $f_*$ we get a commutative diagram

$\xymatrix{ & f_*\mathcal{F} \ar[r] & f_*\mathcal{F}' \\ f_*f^{-1}\mathcal{G}' \ar[r] & f_*\mathcal{F}'' \ar[u] \ar[r] & f_*f^{-1}f_*\mathcal{F}' \ar[u] \\ \mathcal{G}' \ar[u] \ar[rr] & & f_*\mathcal{F}' \ar[u] }$

which proves that (4) holds.

Proof of (g). Assume (9). We use Categories, Lemma 4.13.3 without further mention. Let $a : \mathcal{F} \to \mathcal{F}'$ be a map of sheaves on $\mathcal{C}$ such that $f_*a$ is injective. This means that $f_*\mathcal{F} \to f_*\mathcal{F} \times _{f_*\mathcal{F}'} f_*\mathcal{F} = f_*(\mathcal{F} \times _{\mathcal{F}'} \mathcal{F})$ is an isomorphism. Thus by (9) we see that $\mathcal{F} \to \mathcal{F} \times _{\mathcal{F}'} \mathcal{F}$ is surjective, i.e., an isomorphism. Thus $a$ is injective, i.e., (8) holds. Since (10) is trivially equivalent to (8) $+$ (9) we are done with (g).

Proof of (h). This is Categories, Lemma 4.24.4. $\square$

Here is a condition on a morphism of sites which guarantees that the functor $f_*$ transforms surjective maps into surjective maps.

Lemma 7.41.2. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for any object $U$ of $\mathcal{C}$ and any covering $\{ V_ j \to u(U)\}$ in $\mathcal{D}$ there exists a covering $\{ U_ i \to U\}$ in $\mathcal{C}$ such that the map of sheaves

$\coprod h_{u(U_ i)}^\# \to h_{u(U)}^\#$

factors through the map of sheaves

$\coprod h_{V_ j}^\# \to h_{u(U)}^\# .$

Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves.

Proof. Let $a : \mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves on $\mathcal{D}$. Let $U$ be an object of $\mathcal{C}$ and let $s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U))$. By assumption there exists a covering $\{ V_ j \to u(U)\}$ and sections $s_ j \in \mathcal{F}(V_ j)$ with $a(s_ j) = s|_{V_ j}$. Now we may think of the sections $s$, $s_ j$ and $a$ as giving a commutative diagram of maps of sheaves

$\xymatrix{ \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ h_{u(U)}^\# \ar[r]^ s & \mathcal{G} }$

By assumption there exists a covering $\{ U_ i \to U\}$ such that we can enlarge the commutative diagram above as follows

$\xymatrix{ & \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ \coprod h_{u(U_ i)}^\# \ar[r] \ar[ur] & h_{u(U)}^\# \ar[r]^ s & \mathcal{G} }$

Because $\mathcal{F}$ is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections $s_ i \in \mathcal{F}(u(U_ i))$, i.e., sections $s_ i \in f_*\mathcal{F}(U_ i)$. The commutativity of the diagram implies that $a(s_ i)$ is equal to the restriction of $s$ to $U_ i$. In other words we have shown that $f_*a$ is a surjective map of sheaves. $\square$

Example 7.41.3. Assume $f : \mathcal{D} \to \mathcal{C}$ satisfies the assumptions of Lemma 7.41.2. Then it is in general not the case that $f_*$ commutes with coequalizers or pushouts. Namely, suppose that $f$ is the morphism of sites associated to the morphism of topological spaces $X = \{ 1, 2\} \to Y = \{ *\}$ (see Example 7.14.2), where $Y$ is a singleton space, and $X = \{ 1, 2\}$ is a discrete space with two points. A sheaf $\mathcal{F}$ on $X$ is given by a pair $(A_1, A_2)$ of sets. Then $f_*\mathcal{F}$ corresponds to the set $A_1 \times A_2$. Hence if $a = (a_1, a_2), b = (b_1, b_2) : (A_1, A_2) \to (B_1, B_2)$ are maps of sheaves on $X$, then the coequalizer of $a, b$ is $(C_1, C_2)$ where $C_ i$ is the coequalizer of $a_ i, b_ i$, and the coequalizer of $f_*a, f_*b$ is the coequalizer of

$a_1 \times a_2, b_1 \times b_2 : A_1 \times A_2 \longrightarrow B_1 \times B_2$

which is in general different from $C_1 \times C_2$. Namely, if $A_2 = \emptyset$ then $A_1 \times A_2 = \emptyset$, and hence the coequalizer of the displayed arrows is $B_1 \times B_2$, but in general $C_1 \not= B_1$. A similar example works for pushouts.

The following lemma gives a criterion for when a morphism of sites has a functor $f_*$ which reflects injections and surjections. Note that this also implies that $f_*$ is faithful and that the map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is always surjective.

Lemma 7.41.4. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for every object $V$ of $\mathcal{D}$ there exist objects $U_ i$ of $\mathcal{C}$ and morphisms $u(U_ i) \to V$ such that $\{ u(U_ i) \to V\}$ is a covering of $\mathcal{D}$. In this case the functor $f_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ reflects injections and surjections.

Proof. Let $\alpha : \mathcal{F} \to \mathcal{G}$ be maps of sheaves on $\mathcal{D}$. By assumption for every object $V$ of $\mathcal{D}$ we get $\mathcal{F}(V) \subset \prod \mathcal{F}(u(U_ i)) = \prod f_*\mathcal{F}(U_ i)$ by the sheaf condition for some $U_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and similarly for $\mathcal{G}$. Hence it is clear that if $f_*\alpha$ is injective, then $\alpha$ is injective. In other words $f_*$ reflects injections.

Suppose that $f_*\alpha$ is surjective. Then for $V, U_ i, u(U_ i) \to V$ as above and a section $s \in \mathcal{G}(V)$, there exist coverings $\{ U_{ij} \to U_ i\}$ such that $s|_{u(U_{ij})}$ is in the image of $\mathcal{F}(u(U_{ij}))$. Since $\{ u(U_{ij}) \to V\}$ is a covering (as $u$ is continuous and by the axioms of a site) we conclude that $s$ is locally in the image. Thus $\alpha$ is surjective. In other words $f_*$ reflects surjections. $\square$

Example 7.41.5. We construct a morphism $f : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Lemma 7.41.4. Namely, let $\varphi : G \to H$ be a morphism of finite groups. Consider the sites $\mathcal{D} = \mathcal{T}_ G$ and $\mathcal{C} = \mathcal{T}_ H$ of countable $G$-sets and $H$-sets and coverings countable families of jointly surjective maps (Example 7.6.5). Let $u : \mathcal{T}_ H \to \mathcal{T}_ G$ be the functor described in Section 7.16 and $f : \mathcal{T}_ G \to \mathcal{T}_ H$ the corresponding morphism of sites. If $\varphi$ is injective, then every countable $G$-set is, as a $G$-set, the quotient of a countable $H$-set (this fails if $\varphi$ isn't injective). Thus $f$ satisfies the hypothesis of Lemma 7.41.4. If the sheaf $\mathcal{F}$ on $\mathcal{T}_ G$ corresponds to the $G$-set $S$, then the canonical map

$f^{-1}f_*\mathcal{F} \longrightarrow \mathcal{F}$

corresponds to the map

$\text{Map}_ G(H, S) \longrightarrow S,\quad a \longmapsto a(1_ H)$

If $\varphi$ is injective but not surjective, then this map is surjective (as it should according to Lemma 7.41.4) but not injective in general (for example take $G = \{ 1\}$, $H = \{ 1, \sigma \}$, and $S = \{ 1, 2\}$). Moreover, the functor $f_*$ does not commute with coequalizers or pushouts (for $G = \{ 1\}$ and $H = \{ 1, \sigma \}$).

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