A very elementary question on the definition of sheaf on a site

Question

I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.

In the page 26 of the book, 'a family of effective epimorphisms' is introduced.

'A family $\{ U_{i} \rightarrow V \}$ is a family of effective epimorphisms if the diagram

$Hom(V,Z) \rightarrow \prod_{i} Hom(U_{i}, Z) \rightrightarrows \prod_{i,j}(U_{i}\times_{U}U_{j},Z)$ is exact for all the objects $Z$.

The question is 'in the condition, do we restrict the pairs $i,j$ to be distinct?'

I gave a thought on this question, but I'm not sure whether the two versions give us the equivalent results or not.

Thank you very much in advance.

Answer 1

That exactness conditions can be rephrased more explicitely as:

$$ Hom(V,Z) = \left\lbrace (v_i) \in \prod_i Hom(U_i,Z) \ \middle| \ \forall i,j,v_i \circ \pi_1 = v_j \circ \pi_2 \right\rbrace $$

where $\pi_1,\pi_2$ denotes the two projections $U_i \times_V U_j \rightrightarrows U_i,U_j$.

When you write it like this, the condition in the case $i=j$ is clearly vacuous when all the map $U_i \to V$ are monomorphisms. Indeed in this case $U_i \times_V U_j$ is justs the intersection of $U_i$ and $U_j$, so that $\pi_1=\pi_2$ when $i = j$. This case is very frequent, and you can very often restrict to it by considering the "image" of the $U_i$ in $V$.

But in some situation (for e.g. if you want to keep your objects $U_i$ be to in some specified site that do not admit image factorization like the étale site) it might not be the case. and in general you need the case $i=j$. Consider the case where the you only have a single map $U \to V$. Then the condition becomes

$$ Hom(V,Z) = \left\lbrace f \in Hom(U,Z) \ \middle| \ f\circ \pi_1 = f \circ \pi_2 \right\rbrace $$

where $\pi_1$ and $\pi_2$ are the two projections $U \times_V U \rightrightarrows U$.

You can think of $U \times_V U \rightrightarrows U$ as a map $U \times_V U \to U \times U$ which is a monomorphisms and corresponds to the "equivalence relation such that $V$ should be the quotient of $U$ by this relations". Or rephrased this as $V$ being the coequalizer ('in the category of sheaves') $U \times_V U \rightrightarrows U \to V$, i.e. $V = U /R$ where $R$ is the equivalence relation $U \times_V U$.

And a function from $V \to Z$ can be described as a function $U \to Z$ which is compatible to the equivalence relation $R$ such that $U/R \simeq V$.

Also note that in the general case (with several map) you can think of the general condition as being in two part: you have the condition for $i=j$ that assert that each maps $U_i \to Z$ factors through "the image $V_i$ of $U_i$ in $V$" (if this make sense) , and the condition for $i \neq j$ that implement the usual compatibility condition.

Answer 2

Yes, you have to include the case $i=j$. Just look at what happens in the case of a single $U_1$, in order for this to boil down to the concept of a single effective epimorphism (introduced in the preceding paragraph in the book) you need to include $i=j=1$.