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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 $F\colon C\to \Set$ is equivalent to the subcategory of $\Fun ([0]\star C, \Set )$ 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?

• CommentRowNumber2.
• CommentAuthorZhen Lin
• CommentTimeNov 6th 2013

No, that can’t be correct. Suppose $C$ 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 $F\colon \{0,1\}\to \Set$ is the discrete category with objects the elements of $F(0)\sqcup F(1)$; the category of functors blah blah is the category of spans $F(0) \leftarrow * \to F(1)$. 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 $F0\times F1$ objects, where the first has $F0\sqcup F1$. Ok, nevermind. Thank you for your time.