1
$\begingroup$

Look at the following definition of the shefification (the source is the Stacks Project):


enter image description here


I don't understand what is the projection $\prod_{u\in U}\mathcal F_u\longrightarrow\prod_{v\in V}\mathcal F_v$. In particular, if $u\in U\setminus V$ what is the image of $s_u$ under this map?

Thanks in advance.

$\endgroup$
1
  • $\begingroup$ Oh, yes! It was very simple. I apologize for the stupid question. $\endgroup$
    – Dubious
    Feb 13, 2014 at 11:29

1 Answer 1

4
$\begingroup$

The actual question turns out to not be particular to sheaves, so I will adjust my notation accordingly. The elements of a product $\prod_{a\in\mathcal A}X_a$ are tuples (well, choice functions) $(x_a)_{a\in\mathcal A}$. If $\mathcal B\subset\mathcal A$, then we can assign $$(x_a)_{a\in\mathcal A}\mapsto (x_b)_{b\in\mathcal B}.$$ This map $\prod_{a\in\mathcal A}X_a\to\prod_{b\in\mathcal B}X_b$ is what the author was referring to.

Also, note that you can use the universal property of products in categories to determine the corresponding projections in non-concrete categories.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.