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1. • CommentRowNumber1.
   • CommentAuthorjoe.hannon
   • CommentTimeApr 6th 2014
   • PermaLink
   Author: joe.hannon
   Format: MarkdownItexIn the article on [essential geometric morphisms](http://ncatlab.org/nlab/show/essential+geometric+morphism) it is stated that a local geometric morphism is an essential geometric morphism such that $f_*$ has a right adjoint and $f_!$ is full and faithful. On the other hand, in the article on [local geometric morphisms](http://ncatlab.org/nlab/show/local+geometric+morphism) the definition given is a geometric morphism such that $f_*$ has a right adjoint and $f^!$ is fully faithful. Are these two definitions equivalent? In particular, does $f^*$ need to have a left adjoint $f_!$ in a local geometric morphism? Which functor is required to be fully faithful?

   In the article on essential geometric morphisms it is stated that a local geometric morphism is an essential geometric morphism such that $f_*$ has a right adjoint and $f_!$ is full and faithful. On the other hand, in the article on local geometric morphisms the definition given is a geometric morphism such that $f_*$ has a right adjoint and $f^!$ is fully faithful. Are these two definitions equivalent? In particular, does $f^*$ need to have a left adjoint $f_!$ in a local geometric morphism? Which functor is required to be fully faithful?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeApr 6th 2014
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   Author: Zhen Lin
   Format: MarkdownItexNo, the existence of $f_!$ is not required. If it exists, then there are alternative conditions for $f$ to be local.

   No, the existence of $f_!$ is not required. If it exists, then there are alternative conditions for $f$ to be local.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 6th 2014
   • (edited Apr 6th 2014)
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   Author: Urs
   Format: MarkdownItexThanks for catching that. The paragraph talked about _adding_ to "essentiality" the condition of "connected surjectivity" and "locality", but I can see how that didn't become very clear at all. (Also the notation suddenly changed from $f$ to $p$ etc., quite a mess.) I have edited now.

   Thanks for catching that. The paragraph talked about adding to “essentiality” the condition of “connected surjectivity” and “locality”, but I can see how that didn’t become very clear at all. (Also the notation suddenly changed from $f$ to $p$ etc., quite a mess.) I have edited now.

4. • CommentRowNumber4.
   • CommentAuthorjoe.hannon
   • CommentTimeApr 6th 2014
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   Author: joe.hannon
   Format: MarkdownItexSo that definition is for a local _and_ essential geometric morphism. Thanks Urs.

   So that definition is for a local and essential geometric morphism. Thanks Urs.