[Math] History of integral notation for coends

Question

I'm searching the wheres and whys about the integral notation for co/ends. Who was the first to adopt it? Can you give me a precise pointer or tell me the whole story about it? Was s/he motivated by the Fubini rule only, or there is another additional reason?

Thanks a lot!

Answer 1

From Ross Street's An Australian conspectus of higher categories:

Kelly developed the theory of enriched categories describing enriched adjunction and introducing the variety of limit he called end. I later pointed out that Yoneda had used this concept in the special case of additive categories using an integral notation which Brian Day and Max Kelly adopted. Following this, Mac Lane discussed ends for ordinary categories.

The Day-Kelly paper referred to is

B. Day and G.M. Kelly, Enriched functor categories, Reports of the Midwest Category Seminar, III, Springer, Berlin, 1969, pp. 178–191.

whose introduction states

What we call ends and coends were introduced in the case $V = \text{Ab}$ by Yoneda; we borrow from him the "integral notation".

The Yoneda paper referred to is

Yoneda, N., On Ext and exact sequences. Jour. Fac. Sci. Univ. Tokyo 8 (1960), 507 - 576.

where we find written on page 546 (thanks to Francesco for the page reference)

We shall write $\int_C H$ for the integration $I$ of $H$, and $\int_C^\ast H$ for the cointegration $J$ of $H$.

where his "integration" is our "coend", and dually.