The Stacks project

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.

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