  An algebraic quantum field theory (AQFT) may be expressed as a functor from a
category of spacetimes to a category of algebras of observables. However, a
generic category $\mathsf{C}$ whose objects admit interpretation as spacetimes
is not necessarily viable as the domain of an AQFT functor; often, additional
constraints on the morphisms of $\mathsf{C}$ must be imposed. We introduce
disjointness relations, a generalisation of the orthogonality relations of
Benini, Schenkel and Woike (arXiv:1709.08657). In any category $\mathsf{C}$
equipped with a disjointness relation, we identify a subcategory
$\mathsf{D}_\mathsf{C}$ which is suitable as the domain of an AQFT. We verify
that when $\mathsf{C}$ is the category of all globally hyperbolic spacetimes of
dimension $d+1$ and all local isometries, equipped with the disjointness
relation of spacelike separation, the specified subcategory
$\mathsf{D}_\mathsf{C}$ is the commonly-used domain $\mathsf{Loc}_{d+1}$ of
relativistic AQFTs. By identifying appropriate chiral disjointness relations,
we construct a category $\chi\mathsf{Loc}$ suitable as domain for chiral
conformal field theories (CFTs) in two dimensions. We compare this to an
established AQFT formulation of chiral CFTs, and show that any chiral CFT
expressed in the established formulation induces one defined on
$\chi\mathsf{Loc}$.
