## Abstract

We specialise a recently introduced notion of generalised dinaturality for functors \(T : (\mathcal {C}^\mathsf {op})^p \times \mathcal {C}^q \rightarrow \mathcal {D}\) to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein (*p*, *q*)-*ends* (resp., (*p*, *q*)-*coends*). While higher arity co/ends are particular instances of ‘totally symmetrised’ (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory. The first of these, *weighted category theory*, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, *besides that of a limit*. This leads to a host of varied and rich notions, such as *weighted Kan extensions*, *weighted adjunctions*, and *weighted ends*. The second, *diagonal category theory*, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to *dinatural* transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds *diagonal Kan extensions*, *diagonal adjunctions*, and *diagonal ends*.

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## Notes

- 1.
In [11], the author says that a functor has ‘type

*n*’; mimicking this nomenclature, we shortly refer to our functors as having ‘type ’. - 2.
See also [7] for a similar presentation.

- 3.
More generally, the assignments \(F,G\mapsto \text {DiNat}^{(p,q)}(F,G)\) define functors

- 4.
For \((p,q)=(1,1)\), this amounts to the well-known statement that the co/end of \(T : {\mathcal {C}}^\mathsf {op}\times {\mathcal {C}}\) is the weighted co/limit of

*T*by the hom functor \(\text {hom}_{\mathcal {C}}(-,-):{\mathcal {C}}^\mathsf {op}\times {\mathcal {C}} \rightarrow \mathsf {Set}\); see [12, Section 3.10]. - 5.
Shishido Baiken is the name of a Japanese swordsman (his existence is attested in the

*Nitenki*written in 1776, but the reliability of the text is currently an object of debate). Baiken was a skilled master of*kusarigama-jutsu*and, according to the legend, lost a duel (and his life) with Miyamoto Musashi. - 6.
Upon inspection, one observes that there is a striking similarity between Eqs. 6, 7 and 4, 5. This is not a coincidence, as it is possible to show that diagonal Kan extensions are precisely \(\text {hom}\)-weighted Kan extensions):

This is both an analogy as well as a generalisation (Example 4.14) of the fact that co/ends are exactly \(\text {hom}\)-weighted co/limits.

- 7.
More formally, let \(S : \mathsf {Cat}\rightarrow \mathsf {Cat}\) be the 2-monad of pseudomonoids; let \({\tilde{S}} : \mathsf {Prof}\rightarrow \mathsf {Prof}\) be the lifting of

*S*to the bicategory of profunctors (i.e. to the Kleisli bicategory of the presheaf construction \(\textsf {PSh}\)); then, given an object \(\mathcal {C}\) of \(\mathsf {Cat}\), there is a bijection between pseudo-*S*-algebra structures on \(\textsf {PSh}(\mathcal {C})\) and pseudo-\({\tilde{S}}\)-algebras on \(\mathcal {C}\), as an object of \(\mathsf {Prof}\). - 8.
A

*kusarigama*() is a Japanese compound weapon made of a sickle (

*kama*) and a blunt weight (*fundo*) attached to the opposite ends of a chain (*kusari*). The weight was used to disarm the opponent by entangling their sword in the chain or as a single weapon; disarmed or damaged the opponent, the sickle was then used to deliver the final, fatal strike. Kusarigama was probably adapted from an old farming tool, first adopted by Koga ninjas as a fast, compact weapon; its use then spread to tactic-oriented esoteric weaponry schools like Shinkage-ryū and Suiō-ryū. See [9] for more information.The reason for this terminological choice is the following: as proved in Construction 5.1, is computed via the (

*p*,*q*)-coend formulaIn this equation, we have a sickle connected to the weight

*F*by the chain \(\mathsf {h}^{-}_{A_{1}}\times \cdots \times \mathsf {h}^{-}_{A_{q}}\times \mathsf {h}^{A_{1}}_{-}\times \cdots \times \mathsf {h}^{A_{p}}_{-}\) of hom-functors. - 9.
Similarly to how a morphism of \(\mathsf {Tw}^{(p,q)}(\mathcal {C})\) turned out to involve

*pq*arrows of \(\mathcal {C}\), unravelling the construction given in this section for arbitrary (*p*,*q*) gives a category \(\mathsf {Tw}^{(p,q)}(\mathcal {C})\) whose morphisms now consist of 4*pq*morphisms of \(\mathcal {C}\). Additionally,*each of these*points now in a different direction (i.e. they cannot anymore be arranged as morphisms in product categories). Together, these two points make \(\mathsf {Tw}^{(p,q)}(\mathcal {C})\) unusable in practice when*p*and*q*are too large. As a compromise, we work out the case \((p,q)=(1,1)\), which is both the simplest case as well as the most useful one.

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The first author was supported by the ESF funded Estonian IT Academy research measure (Project 2014-2020.4.05.19-0001). The first author would like to thank A. Santamaria and his delightful talk at ItaCa [18], without which this paper would probably not exist, the entire Italian community of category theorists that has made ItaCa possible, and *sensei* Inoue Takehiko. The second author is greatly indebted to many people for their immense generosity and kindness, including their parents, friends, as well as Jonathan Beardsley, Lennart Meier, Igor Mencattini, Oziride Manzoli Neto, and Eric Peterson, among so many others who I’ve had the pleasure of knowing. The second author was supported by grant #2020/02861-7, São Paulo Research Foundation (FAPESP).

Communicated by Nicola Gambino.

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Loregian, F., de Oliveira Santos, T. Coends of Higher Arity.
*Appl Categor Struct* (2021). https://doi.org/10.1007/s10485-021-09653-x

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### Keywords

- Coend
- Cowedge
- Dinatural transformation