TITLE: Why an integral symbol for the category of elements of a presheaf?
QUESTION [18 upvotes]: 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 $\mathbf C$ ?

REPLY [5 votes]: 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!