# Why an integral symbol for the category of elements of a presheaf?

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 $\mathbf C$ ?

• 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. Feb 7, 2014 at 18:13
• It's a kind of colimit, and colimits are a kind of "generalised sum". Feb 8, 2014 at 1:36