subassignments

Subassignments are a concept used mainly in algebraic geometry and logic to describe a subobject of a functor. Given a category C and a functor F: C → Sets, a subassignment G of F consists of, for every object X in C, a subset G(X) ⊆ F(X), and for every morphism f: X → Y in C, F(f)(G(X)) ⊆ G(Y). Equivalently, G is a subfunctor of F with inclusion i: G → F. This general framework captures how a subset can vary functorially with respect to the test objects in C.

Examples illustrate the idea. If X is a k-scheme and F is the functor of points F(T) = Hom_k(T, X), then a closed subscheme Z ⊆ X yields a subassignment G with G(T) = Hom_k(T, Z). More generally, definable or constructible subsets of jet or arc spaces can be described as subassignments of the corresponding functor of points. In model-theoretic contexts, subassignments encode parameterized families of elements satisfying a given property.

Properties and interpretation. Subassignments are stable under base change in the sense that pullbacks along morphisms induce corresponding subassignments. They can often be glued from local data and form a natural lattice under inclusion. They provide a flexible description of subsets that vary with test objects, without necessarily being representable by a single geometric object.

Relation to other notions. Every closed subscheme yields a subassignment of the functor of points; under suitable representability conditions, a subassignment can correspond to a subscheme. Subassignments thus generalize subschemes and align with the functor-of-points perspective.

Use and references. Subassignments appear in motivic integration literature and in discussions of functor-of-points approaches to algebraic geometry and logic.