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Is this question acceptable?: Is the category of elements a coend?

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    I am trying to understand coends. In order to do this, I would like to show that certain things are in fact coends. In particular if the category of elements is a coend, I would like to prove it for myself. Further, (this is secondary since I could find some of theses in books and papers) any other nice examples that I may show for myself would be great. I do not want any proofs of facts, just the facts, I would like to prove this for myself. If this is an acceptable question, should it be a CW. Thank you for any help.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 9th 2012 edited
     

    No. It's a "Grothendieck construction", which uses the integral symbol as well. Specifically, it's the Grothendieck construction applied to a pseudopresheaf of discrete categories (i.e. sets) (which are the same things as presheaves of sets, since there are no coherence data to speak of).

    Also, this is not the correct place to ask this question. Probably a good place to ask something like this is on math.se.

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    I suspect such a question would get closed, mostly because the answer is (I believe) "no": the category of elements is not a coend in any useful way. As Harry surmises, you may have been misled by the fact that some people use the integral sign to denote a category of elements, and the integral sign is also used for ends and coends.

    There's some sense in which a category of elements is an end. It's a lax limit. But that seems some distance from what you're asking. So, if I were you, I wouldn't ask the question you describe.

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    Spice, if you can recognize a tensor product of modules as a coend, and if you can recognize geometric realization of a simplicial set as an coend, then you are probably well on your way to understanding coends. The nLab might help. Possibly you might find Mac Lane's "Milgram bar construction as a tensor product of functors" helpful and illuminating.

    Recently there has been discussion on how effective math.stackexchange.com would be in answering queries at this level. The reports have not been so good.

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    Thank you very much. I think that this thread can be closed, if their is a need.

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