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I think the generalization I want is something like:

A "topological multicategory" is:

a set of objects X

for each x ∈ X, a set H

for each x ∈ X, a homset operator [—, x] : 2

for each x ∈ X, there's an identity arrow id

If f : W → x and for each w ∈ W we have a g

If Z ⊆ W and f : Z → x, then also f : W → x

Presumably composition has to be associative and respect identities. I think also if V ⊆ W and f : V → x, and for all w ∈ W we have g

f ⚬ (g

That is, if f

A functor from X to Y is a map F from objects of X to objects of Y, and a map of homsets

[F

for any Z ⊆ Y and x ∈ X. Presumably it respects composition and identities in some appropriate way.

A "topological multicategory" is:

a set of objects X

for each x ∈ X, a set H

_{x}of morphisms whose codomain is x.for each x ∈ X, a homset operator [—, x] : 2

^{X}→ 2^{Hx}. Given a*set*Z of domain objects, it yields [Z, x], the set of morphisms that are allowed to go from Z to x. We write f : Z → x for f ∈ [Z, x].for each x ∈ X, there's an identity arrow id

_{x}: {x} → xIf f : W → x and for each w ∈ W we have a g

_{w}: Z → w, then we can form the composite f ⚬ (g_{w})_{w ∈ W}: Z → x.If Z ⊆ W and f : Z → x, then also f : W → x

Presumably composition has to be associative and respect identities. I think also if V ⊆ W and f : V → x, and for all w ∈ W we have g

_{w}: Z → w, then we should have thatf ⚬ (g

_{w})_{w ∈ W}= f ⚬ (g_{v})_{v ∈ V}That is, if f

*does*know how to have a more tidied up reduced domain, then composites of arrows along the irrelevant input ports shouldn't matter.A functor from X to Y is a map F from objects of X to objects of Y, and a map of homsets

[F

^{*}Z, x] → [Z, Fx]for any Z ⊆ Y and x ∈ X. Presumably it respects composition and identities in some appropriate way.