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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 2nd 2011
   • (edited Mar 2nd 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexSuppose $T$ is a (grothendieck) topos, and suppose $C\subseteq T$ is a full subcategory. If the inclusion $C\hookrightarrow T$ has a lex left adjoint, is $C$ necessarily a topos? What if the embedding is not necessarily full?

   Suppose is a (grothendieck) topos, and suppose is a full subcategory. If the inclusion has a lex left adjoint, is necessarily a topos?

   What if the embedding is not necessarily full?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 2nd 2011
   • (edited Mar 2nd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAs you know, Grothendieck toposes are the exact reflective subcategories of presheaf toposes. So if $T$ is a Grothendieck topos, there exists a small category $B$ and a left exact reflective inclusion $T \hookrightarrow PSh(B)$. Since left exact reflective inclusions are closed under composition, it follows that also $$ C \hookrightarrow T \hookrightarrow PSh(B) $$ is a left exact reflective inclusion into a presheaf topos.

   As you know, Grothendieck toposes are the exact reflective subcategories of presheaf toposes. So if is a Grothendieck topos, there exists a small category and a left exact reflective inclusion . Since left exact reflective inclusions are closed under composition, it follows that also

   is a left exact reflective inclusion into a presheaf topos.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 4th 2011
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   Author: Harry Gindi
   Format: MarkdownItexThanks!

   Thanks!