Coends of higher arity (Q2667936)

The concept of end appeared, for additive categories, in the paper [\textit{N. Yoneda}, J. Fac. Sci., Univ. Tokyo, Sect. I 8, 507--576 (1960; Zbl 0163.26902)] where an integral notation was used since coends resemble the integrals of analysis. The terms end and coend were coined in the development of enriched category theory where it is even more necessary than in ordinary category theory; see [\textit{B. J. Day} and \textit{G. M. Kelly}, Lect. Notes Math. 106, 178--191 (1969; Zbl 0214.03202)] The end of a functor \(F : \mathcal{C}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{X}\) is an object \(\int_AF(A,A) \in \mathcal{X}\) equipped with a terminally universal dinatural transformation (wedge) \(\int_AF(A,A)\to F\). The authors introduce higher arity ends and coends, remarking that they serve an important technical role in the study of both weighted and diagonal category theory. For non-negative integers \(p\) and \(q\), they write \(\mathcal{C}^{(p,q)}\) for the category \((\mathcal{C}^{\mathrm{op}})^p\times \mathcal{C}^q\). There is a \((p,q)\)-diagonal functor \(\Delta_{p,q} : \mathcal{C}\to \mathcal{C}^{(p,q)}\). Proposition 3.1 item (PE3) on page 192 expresses higher arity ends in terms of ordinary ends: for \(D : \mathcal{C}^{(p,q)} \to \mathcal{D}\), \[_{(p,q)}\int_{A\in \mathcal{C}}D(A,A) \cong \int_{A\in \mathcal{C}}D(\Delta_pA,\Delta_qA) \ .\] Reviewer's remark: Unfortunately, I believe Lemma 3.3 on page 196 is false. The claim in the first sentence of the ``proof'', that the \((p,q)=(2,1)\) case is indicative of the general case, is false. (I have contacted one of the authors who agrees there is a problem.) When either \(p=1\) or \(q=1\), the lemma holds by applying Yoneda's Lemma in a way that cannot be done otherwise. This causes a problem for Proposition 3.2 and their proof of Fubini. However, their Fubini Theorem follows from the usual Fubini Theorem for ends. Section 4 deals with examples of concepts that can be expressed in terms of \((p,q)\)-ends or coends. In particular, their weighted co/ends are \((2,2)\)-co/ends and they also have Kan extensions weighted by an endomodule. For a monoidal category \(\mathcal{C}\), the paper introduces Day \((n,n)\)-convolution as an \(n\)-fold tensor product on the category of \(\mathcal{C}\)-presheaves using an \((n,n)\)-coend formula. This is related to, but does not give, the convolution of [\textit{B. Day}, Lect. Notes Math. 137, 1--38 (1970; Zbl 0203.31402)] as a special case, rather, the sequence of tensors determines an operad. Section 5 is entitled ``Kusarigamas and Twisted Arrow Categories'' and aims to introduce a kind of replacement for the Yoneda Lemma to provide higher arity coends with a fairly rich calculus. [I noticed some typographical errors. At the end of the Proof of Theorem 3.1, in \((A\pitchfork B)\pitchfork C\cong (A\times B) \pitchfork C\), the left hand side is wrongly bracketed; similarly at the beginning of that Proof. At the bottom of page 192 there is some rogue material.]