Let $\mathbf C$ be a category and $P \colon \mathbf C^{\rm op} \to \mathbf{Set}$ a presheaf. One can associate to $P$ the category of elements of $P$ (also called Grothendieck construction over $P$), denoted $\int_{\mathbf C} P$, whose objects are $(C,p)$ for $C$ object of $\mathbf C$ and $p \in P( C )$, and whose morphism $(C',p') \to (C,p)$ are those $u \colon C' \to C$ such that $P(u)( p ) = p'$.

What is the origin of the symbol $\int$ for such a construction ? Does it actually link to any kind of integration with some good choice of $\mathbf C$ ?

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    $\begingroup$ I think it is more natural to replace $\mathsf{Set}$ by $\mathsf{Cat}$ here. Then $P(u)(p)=p'$ is replaced by a morphism $P(u)(p) \to p'$, for instance. Then $P$ can be thought of a "category-valued function" (pre-stack) and $\int P$ is the "mean" of this function, which is a category. Perhaps one can fill this with life by looking at some specific examples. $\endgroup$ Feb 7, 2014 at 18:13
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    $\begingroup$ It's a kind of colimit, and colimits are a kind of "generalised sum". $\endgroup$
    – Zhen Lin
    Feb 8, 2014 at 1:36

1 Answer 1


This is not a real explanation, but I'm kinda proud of having told André Joyal (which wasn't surprised at all, but amused of seeing it) that there is a funny coincidence for this notation, namely that the category of elements of a presheaf can be written as a coend. If you go to the nlab page "category of elements" you'll find more informations!

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    $\begingroup$ This doesn't seem coincidental to me. "Elements" are, in some particular sense, defined to be initial among maps into something. Euclid was doing category theory when he said that points have "no part". $\endgroup$ Feb 10, 2014 at 20:20

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