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I would like to add to the page category of elements the following characterization:
The category of elements of a functor F:C→Set is equivalent to the subcategory of Fun([0]⋆C,Set) which coincide with F when restricted to C and send [0] to the terminal object in 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?
No, that can’t be correct. Suppose C is a discrete two-object category, for instance.
I am sorry, but where do you see the flaw in this example? The category of elements of F:{0,1}→Set is the discrete category with objects the elements of F(0)⊔F(1); the category of functors blah blah is the category of spans F(0)←*→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×F1 objects, where the first has F0⊔F1. Ok, nevermind. Thank you for your time.
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