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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 9th 2010

    In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor PSh(S)Sh(S,J)PSh(S) \to Sh(S,J). To quote my definition again

    Definition: Let (C,J) be a site (J a pretopology). A map f:abf:a \to b is a J-local isomorphism if there are covering families (v ib)(v_i \to b) and (u ja)(u_j \to a) such that for each u ju_j the restriction f|u jf|u_j is an isomorphism onto some v iv_i.

    I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: [Top op,Set][Top^{op},Set] is very big), so if they are equivalent, I’d like to put this somewhere.

    Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.

    Any thoughts?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 11th 2010

    This is to alert Urs the last question of #1 is directed at him :)