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nLab > nLab General Discussions: Van Kampen colimits

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    • CommentRowNumber1.
    • CommentAuthorMichał Przybyłek
    • CommentTimeJun 28th 2015
    • (edited Jun 29th 2015)
    Hi guys,

    I do not want to spam MathOverflow again, so a quick question here.

    Theorem 2 on the page about Van Kampen colimits says: "A colimit in C is van Kampen if and only if it is preserved by the inclusion C ---> Span(C) into the bicategory of spans in C." and its proof is referenced to a paper. But, is not this theorem actually almost a tautology? Let me write J : C ---> Span(C) for the above inclusion. Then hom(J(-), 1) : C^op ---> Cat is the slice functor C/(-) from Definition 1. Now the preservation of colimits by J is equivalent to the preservation of colimits by hom(J(-), 1) by the usual property of presheaves.

    Well... my question really is --- what have I misunderstood now?

    ***
    OK, I can answer it by myself. Generally, it follows that:

    C/(-)^\hom(-, X) ~ C/(-)^\hom(-, Y) in Fib(C) iff X ~ Y in Span(C)

    but C/X ~ C/Y in Cat does not imply X ~ Y in Span(C). For example Set^2/(0,1) ~ Set ~ Set^2/(1,0), yet (0,1) /~ (1,0) in Span(Set). I think I made the same error in the past :-)


    -- Michał R. Przybyłek

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