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

We define Dist(X,Y):=V−Cat(Xop⊗Y,V).

Recall the definition of [ends][1]. Taking the end is an operation of signature

Dist(X,X)→V.

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

V−Cat(X?⊗X?⊗X?,V)→V

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

MOTIVATION: I like to think of the composition ⊗ of (2-ary)distributors and the right adjoints to D⊗− and −⊗E as "[horn-filling][2]" (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∈Dist3(X,Y,A)
V∈Dist3(Y,Z,A)
W∈Dist3(Z,X,A)

there should be
?(U,V,W)∈Dist3(X,Y,Z)

and related adjoints.

Of course Dist3 remains to be defined.


  [1]: http://ncatlab.org/nlab/show/end%E2%80%8E
  [2]: http://ncatlab.org/nlab/show/horn