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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 1st 2010
   • (edited Oct 1st 2010)
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   Author: Harry Gindi
   Format: MarkdownItexSheafification is the left-adjoint of the inclusion functor $\iota: Sh\hookrightarrow Psh$. Is there a way to show that sheafification is lex without explicit construction by looking at the properties of $\iota$ (and therefore $Sh$)? Does left-exactness of a functor imply that it preserves monics for sheaves of sets? Here, we're taking $Sh$ to be the category of sheaves with respect to an arbitrary Grothendieck topology.

   Sheafification is the left-adjoint of the inclusion functor $\iota: Sh\hookrightarrow Psh$.

   Is there a way to show that sheafification is lex without explicit construction by looking at the properties of $\iota$ (and therefore $Sh$)?

   Does left-exactness of a functor imply that it preserves monics for sheaves of sets?

   Here, we’re taking $Sh$ to be the category of sheaves with respect to an arbitrary Grothendieck topology.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2010
   • (edited Oct 1st 2010)
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   Author: Urs
   Format: MarkdownItexOne useful abstract argument is this: under mlld conditions (presentable categories) a reflective subcategory $C \hookrightarrow D$ has a lex reflector if (and only if) the localization that it arises from is at collections of morphisms that are suitably stable under pullback. So it is precisely that clause in the definition of Grothendieck topology or coverage that makes sheafification be lex.

   One useful abstract argument is this:

   under mlld conditions (presentable categories) a reflective subcategory $C \hookrightarrow D$ has a lex reflector if (and only if) the localization that it arises from is at collections of morphisms that are suitably stable under pullback.

   So it is precisely that clause in the definition of Grothendieck topology or coverage that makes sheafification be lex.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 1st 2010
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   Author: Mike Shulman
   Format: MarkdownItexThat's a nice fact, Urs. Is that stated on the lab somewhere? To answer Harry's other question, a map $A\to B$ is monic precisely when $$\array{A & \overset{id}{\to} & A \\ ^{id}\downarrow && \downarrow \\ A &\to& B}$$ is a pullback. Therefore, any functor which preserves pullbacks must preserve monics as well.

   That’s a nice fact, Urs. Is that stated on the lab somewhere? To answer Harry’s other question, a map $A\to B$ is monic precisely when

   $$\array{A & \overset{id}{\to} & A \\ ^{id}\downarrow && \downarrow \\ A &\to& B}$$

   is a pullback. Therefore, any functor which preserves pullbacks must preserve monics as well.

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 2nd 2010
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   Author: Harry Gindi
   Format: MarkdownItex@Urs: Could you give a sketch of the argument used to prove the "main lemma"?

   @Urs: Could you give a sketch of the argument used to prove the “main lemma”?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2010
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   Author: Urs
   Format: MarkdownItex> That's a nice fact, Urs. Is that stated on the lab somewhere? Yes, at [[reflective sub-(infinity,1)-category]] in the section <a href="http://ncatlab.org/nlab/show/reflective+sub-(infinity%2C1)-category#ExactLocalizations">exact localizations</a>.

   That’s a nice fact, Urs. Is that stated on the lab somewhere?

   Yes, at reflective sub-(infinity,1)-category in the section exact localizations.