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nLab > nLab General Discussions: Definition of a sheaf

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  1.  
    • CommentRowNumber1.
    • CommentAuthorBubbles
    • CommentTimeOct 7th 2014
    • (edited Oct 7th 2014)
    There is a sentence in the article on the definition of a coverage that I don't quite understand.

    http://ncatlab.org/nlab/show/coverage

    Under the section "Sheaves on a site", it is said that for a family of coterminal morphisms [...] a presheaf X is a sheaf if: [...]
    The condition stated afterwards is what puzzles me. What is exactly the condition on the elements x_i? The sentence itself doesn't really make sense to me.

    Also, what is meant by "a family of coterminal morphisms" in this case, is it a condition on the family? My guess is that it is a family which can be used as the covering family which must exists for any other family (if that makes sense).

    I tried to read the article "sheaf", but this puzzles me more. It speaks of "compatible families of elements", but I don't see how the link provided ("matching family") explains that in that particular setting.
    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 8th 2014

    Also, what is meant by “a family of coterminal morphisms” in this case, is it a condition on the family?

    This is just a family of morphisms with the same codomain, in this case U.

    What is exactly the condition on the elements x_i?

    A family of elements x_i∈X(U_i) is compatible if for all g: V→U_i and h: V→U_j (where V is arbitrary) such that f_i g = f_j h we have X(g)(x_i) = X(h)(x_j).

    The entire condition can now be reformulated as follows:

    For any compatible collection of elements x_i \in X(U_i) there exists a unique x∈X(U) such that X(f_i)(x)=x_i for all i.

    • CommentRowNumber3.
    • CommentAuthorBubbles
    • CommentTimeOct 8th 2014
    Thanks!

    So if I understand it correctly, in the definition of a sheaf in the article "sheaf", the sentence "and for every compatible family of elements given by tuples[...]" not impose any more conditions on the family of elements other than exactly what is stated right afterwards, namely "such that for all morphisms[...]" (which is your definition above). It confused me because it seemed like being a "compatible family" was an additional criterion on the family of elements.
    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 8th 2014

    Yes, compatible family simply means what I wrote above, i.e., a family of sections that match on the intersections. The statement in the article is certainly ambiguous and would clearly benefit from splitting out the definition of a compatible family.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeOct 8th 2014

    Handy link: coverage.

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