Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.


We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$


Recall the definition of [ends](http://ncatlab.org/nlab/show/end%E2%80%8E). Taking the end is an operation of signature


$$\mathrm{Dist}(X,X)→\mathbb V.$$


QUESTION: Is there an analogue for functors into V of higher arity. More explicitly: Is there a canonical operation of signature


$$\mathbb V−\mathrm{Cat}(X^?⊗X^?⊗X^?,V)→\mathbb V$$


where the ? are to be replaced by either op or nothing.


MOTIVATION: I like to think of the composition $\otimes$ of (2-ary)distributors and the right adjoints to $D\otimes−$ and $−\otimes E$ as "[horn-filling](http://ncatlab.org/nlab/show/horn)" (in the sense of viewing categories as simplicial sets).


I hope to find a similar situation for "3-ary distributors" - Whenever there is a (oriented?) tetrahedron of 3-ary functors with one side missing we should be able to find the missing side.


So given


$$U\in\mathrm{Dist}^3(X,Y,A)$$
$$V\in\mathrm{Dist}^3(Y,Z,A)$$
$$W\in\mathrm{Dist}^3(Z,X,A)$$


there should be
$$?(U,V,W)\in\mathrm{Dist}^3(X,Y,Z)$$


and related adjoints.


Of course $\mathrm{Dist}^3$ remains to be defined.



Ends and coends – analogues for higher arity – Horn Filling
This is exactly the subject of the paper [Coends of higher arity](https://arxiv.org/abs/2011.13881) by Loregian and de Oliveira Santos.


