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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. 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" (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.

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Just one quick thought: I like the interpretation/definition of ends as weighted limits (with Hom: C x C^op -> V as weight). Do you think the potential 3-ary version above has a similar simple analog as weighted limit? – Stephan Müller Nov 14 '13 at 14:45
I thought about it but could not come up with a 3-ary replacement for the hom. – Garlef Wegart Nov 14 '13 at 16:31
distributor has another formulation as bimodule, which can be viewed as a category over $[1]$. Following this line, you may define $n$-distributor as simplicial set over $\Delta^n$. – Ma Ming Apr 4 '14 at 16:29

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