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nLab: equivalent definitions of local geometric morphism?

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    • CommentRowNumber1.
    • CommentAuthorjoe.hannon
    • CommentTimeApr 6th 2014

    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?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeApr 6th 2014

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

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2014
    • (edited Apr 6th 2014)

    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.

    • CommentRowNumber4.
    • CommentAuthorjoe.hannon
    • CommentTimeApr 6th 2014

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

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