# History of integral notation for coends

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!

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.

• Thank you Alex! Any hope to find a scanned copy of Yoneda's paper? – Fosco Dec 1 '14 at 16:18
• Do you know how we came to write the end variable at the bottom of the integral sign and the coend variable at the top? – Tom Leinster Dec 1 '14 at 17:26
• @TomLeinster: I followed the discussion on the $n$-cafe' and I remember the discussion on this point. What about the association coends = colimits, and something "co-" deserves to be a superscript, because of tensor calculus? – Fosco Dec 1 '14 at 19:12
• So Yoneda has the variable at the bottom for both, with our "end" being the dual notion (denoted by an $\ast$). Max used the terms end and coend before adopting the integral notation, which allows for this change in variance. The bottom/top convention is used in the Day-Kelly paper, I imagine following the common covariant=subscript, contravariant=superscript convention that Fosco mentions, (compare Grothendieck's $f_! \dashv f^\ast \dashv f_\ast \dashv f^!$). – Alexander Campbell Dec 5 '14 at 8:02
• Another interesting fact about this Yoneda paper is that it contains the notion of two-sided fibration, under the name "regular span". – Alexander Campbell Dec 5 '14 at 8:15

According to the book Galois Theory, Hopf Algebras, and Semiabelian Categories by George Janelidze, Bodo Pareigis and Walter Tholen, the integral notation for (co)ends, generally attributed to MacLane, is originally due to Yoneda. See in particular the interesting historical footnote 1, page 201.

The precise reference given is page 546 of

Yoneda, Nobuo: On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507–576.