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Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$).

Assume that every $T_3$ cover has a refinement of the form $\stackrel{f_1}{\to} \stackrel{f_2}{\to}$ where $f_2$ is a $T_2$ cover and $f_1$ is a $T_1$ cover (this is true for example, if $T_1$ is the Nisnevich topology, and $T_2$ is the topology consisting of proper cdh covers; in this case $T_3$ is the cdh topology).

If $F$ is a $T_1$-sheaf, is its $T_2$-sheafification still a $T_1$-sheaf (and therefore a $T_3$-sheaf)?

I can show this is true if $F$ is a $T_2$ separated $T_1$ sheaf, but I want to remove this condition.

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How about this for a counter-example: We take the category

$$ A \stackrel{f_2}{\to} B \rightrightarrows C \stackrel{f_1}{\to} D $$

where there is a unique morphism from $A$ to $C$ and from $B$ to $D$. The only $T_1$ cover is $f_1$ and the only $T_2$ cover is $f_2$.

This category and these topologies satisfy my conditions.

Consider the $T_1$ sheaf $F$:

$F(A) = $ a singleton set $\underline{0} = ${$0$}.

$F(B) = $ a two element set $\underline{1} = ${$0, 1$}.

$F(C) = $ a singleton set $\underline{0} = ${$0$}.

$F(D) = \varnothing$ the empty set.

$$ \underline{0} \leftarrow \underline{1} \leftleftarrows \underline{0} \leftarrow \varnothing $$

The two morphisms $F(B) \leftleftarrows F(C)$ are the two distinct morphims $\underline{1} \leftleftarrows \underline{0}$. This is a $T_1$ sheaf.

Then the $T_2$ separated presheaf (which happens to already by a $T_2$ sheaf) associated to this $T_1$ sheaf is

$$ \underline{0} \leftarrow \underline{0} \leftleftarrows \underline{0} \leftarrow \varnothing $$

which is no longer a $T_1$ sheaf.

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