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    • CommentRowNumber1.
    • CommentAuthorFosco
    • CommentTimeNov 6th 2013

    I would like to add to the page category of elements the following characterization:

    The category of elements of a functor F:CSetF\colon C\to \Set is equivalent to the subcategory of Fun([0]C,Set)\Fun ([0]\star C, \Set ) which coincide with FF when restricted to CC and send [0] to the terminal object in Set\Set. This seems quite natural to me and I wasn’t able to disprove it.

    Can you provide me either a counterexample or a reference for this result I can add to the page?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeNov 6th 2013

    No, that can’t be correct. Suppose CC is a discrete two-object category, for instance.

    • CommentRowNumber3.
    • CommentAuthorFosco
    • CommentTimeNov 6th 2013
    • (edited Nov 6th 2013)

    I am sorry, but where do you see the flaw in this example? The category of elements of F:{0,1}SetF\colon \{0,1\}\to \Set is the discrete category with objects the elements of F(0)F(1)F(0)\sqcup F(1); the category of functors blah blah is the category of spans F(0)*F(1)F(0) \leftarrow * \to F(1). To be honest I argued in general, but in this and in other toy examples it seems to work.

    Edit: Now I see by myself; both categories are discrete, but the second has F0×F1F0\times F1 objects, where the first has F0F1F0\sqcup F1. Ok, nevermind. Thank you for your time.