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.



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


