# Integrals in analysis and category theory

Are integrals in analysis special cases of coends in category theory? They are both seen as weighted sums, denoted by $\displaystyle\int$ and share the same formal properties (for example Fubini's Theorem). Also notice that there is a very beautiful reformulation of the Yoneda Lemma $$F(a) = \int^{p \in I} \mathrm{Hom}(a,p) \otimes F(p)$$ for functors $F : I^{op} \to \mathcal{C}$ into a cocomplete category $\mathcal{C}$ which looks like the equation $$\mu(A) = \int_{p \in I} \chi_A(p) ~ d \mu(p)$$ for a measure $\mu$ on a measurable space $I$.

Notice that this question is not as silly as it seems: Limits in analysis can be seen as limits in category theory, see MO/9951 or Wikipedia (Topological limits).

If the measurable space $I$ is discrete, the question is about (infinite) series - are they special cases of coends?

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Every colimit is a coend, namely $\varinjlim F = \int^{i : \mathcal{I}} F i$. –  Zhen Lin May 8 at 7:20
I'm pretty sure that what you wrote is the co-Yoneda lemma. The Yoneda lemma would be $F(a) = \int_{p \in I} [\hom(p,a), F(p)]$ (with covariant $F : I \to \mathcal{C}$ this time). –  Najib Idrissi May 8 at 7:20
Did you take a look at tac.mta.ca/tac/volumes/23/12/23-12.pdf ? It does not answer your question about coends, but it might prove useful if you are interested in categorical integration... –  Marco Vergura May 8 at 7:21
@Najib: These are formally equivalent statements (look at $F^{op}$ etc.). –  Martin Brandenburg May 8 at 7:25
Of course, the name is pretty telling :). I simply wanted to point out that it has a different name and interpretation (in terms of presheaves and colimits of representable functors) under this form. –  Najib Idrissi May 8 at 7:29

As an example of the last corollary [see the paper], we could take for $X$ the space of nonnegative step functions of a probability space $S$ and for $Y$ the space of all nonnegative functions with the natural sup metric, and consider as $\varphi$ the elementary integral; thus in general we might call $y\in Y$ $\varphi$-integrable if [a certain right Kan extension, the inner measure] = [a certain left Kan extension, the outer measure].