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Fan Zheng
Fan Zheng
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I am just adding a tiny bit that is missed from the previous 2 answers, that is, the resp. part of Question 1:

If $F^l$ are sheaves on $X$, is $\oplus_l F^l$, which exists in the category of presheaves on $X$, automatically a sheaf on $X$?

The answer is NO. For a counterexample see this answerthis answer.

I am just adding a tiny bit that is missed from the previous 2 answers, that is, the resp. part of Question 1:

If $F^l$ are sheaves on $X$, is $\oplus_l F^l$, which exists in the category of presheaves on $X$, automatically a sheaf on $X$?

The answer is NO. For a counterexample see this answer.

I am just adding a tiny bit that is missed from the previous 2 answers, that is, the resp. part of Question 1:

If $F^l$ are sheaves on $X$, is $\oplus_l F^l$, which exists in the category of presheaves on $X$, automatically a sheaf on $X$?

The answer is NO. For a counterexample see this answer.

Source Link
Fan Zheng
Fan Zheng
  • 5.2k
  • 20
  • 37

I am just adding a tiny bit that is missed from the previous 2 answers, that is, the resp. part of Question 1:

If $F^l$ are sheaves on $X$, is $\oplus_l F^l$, which exists in the category of presheaves on $X$, automatically a sheaf on $X$?

The answer is NO. For a counterexample see this answer.