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Have been trying to wrestle with a definition of something like a category, but for relations, but not an allegory, exactly either.
The goal is to try to banish the "two-ness" of composition present in category, allegory, or even multicategory --- in that a multicategory still has a notion of (many objects in the) domain of a morphism, and codomain of a morphism, as two distinct things.
My attempt so far is to say
Suppose you have a category C and a functor [-] : Cop → Cat, and a cospan (fi : Ci → C | i ∈ 1..n) in C. Define an amalgamation of a sequence of objects (xi ∈ [Ci] | i ∈ 1..n) to be an object x ∈ [C] together with morphisms (gi : [fi](x) → xi ∈ [Ci] | i ∈ 1..n). There is an evident notion of morphisms between amalgamations of the same sequence of objects. A composite of a sequence of objects is a terminal object in the category of amalgamations of that sequence of objects.
A relational structure on C is a functor [-] : Cop → Cat such that all composites exist, and all [C] preserve them.
The motivating example of this definition goes like this: pick a set of names of types, S, with a function El : S → Set that gives the meaning of each of those names. Let C be S → Set. What's an object L of S → Set look like? It's essentially a multiset over S; it assigns a multiplicity to each element s ∈ S. So let [L] be the category whose objects are relations with arity L, that is, subsets of Π s ∈ S . (L(s) → El(s)), and whose morphisms are inclusions of relations. The morphism part of [-] realizes projection of columns out of relations, and duplication of existing columns, according to when morphisms in S → Set drop and coalesce domain values in the codomain, respectively. Composition defined above ends up being composition of relations.
The goal is to try to banish the "two-ness" of composition present in category, allegory, or even multicategory --- in that a multicategory still has a notion of (many objects in the) domain of a morphism, and codomain of a morphism, as two distinct things.
My attempt so far is to say
Suppose you have a category C and a functor [-] : Cop → Cat, and a cospan (fi : Ci → C | i ∈ 1..n) in C. Define an amalgamation of a sequence of objects (xi ∈ [Ci] | i ∈ 1..n) to be an object x ∈ [C] together with morphisms (gi : [fi](x) → xi ∈ [Ci] | i ∈ 1..n). There is an evident notion of morphisms between amalgamations of the same sequence of objects. A composite of a sequence of objects is a terminal object in the category of amalgamations of that sequence of objects.
A relational structure on C is a functor [-] : Cop → Cat such that all composites exist, and all [C] preserve them.
The motivating example of this definition goes like this: pick a set of names of types, S, with a function El : S → Set that gives the meaning of each of those names. Let C be S → Set. What's an object L of S → Set look like? It's essentially a multiset over S; it assigns a multiplicity to each element s ∈ S. So let [L] be the category whose objects are relations with arity L, that is, subsets of Π s ∈ S . (L(s) → El(s)), and whose morphisms are inclusions of relations. The morphism part of [-] realizes projection of columns out of relations, and duplication of existing columns, according to when morphisms in S → Set drop and coalesce domain values in the codomain, respectively. Composition defined above ends up being composition of relations.