The -union of a small family of -subobjects is the unique -subobject containing the and so that the induced map is in . The union is calculated by applying the factorization to the canonical map .
If the map is in , we say is an -union. The set is a preorder under the relation if as -subobjects of . Regarding the as a diagram of shape , the family is an -union if and only if the map colim is in . We say is a filtered union of -subobjects if it is a union of -subobjects and if the category is filtered.
A representable functor preserves the -union of if the functions are jointly surjective, i.e., if each factors through some .
Let be a regular cardinal, and let be a cocomplete category with a proper factorization system .
A small set of objects of is an -generator if in is invertible whenever is bijective for all . Equivalently, is an -generator if for each the family of maps is jointly in , i.e., if the map is in .
A category is called locally -bounded with respect to a proper factorization system if
it has an -generator each of whose objects is -bounded
it has arbitrary cointersections (even large ones) of maps in — that is, it is E-cocomplete.
In a locally -presentable category, every -presentable object is -bounded. Hence a -presentable category is -bounded.
This appears as Lemma 2.3.1 of Freyd-Kelly
Locally bounded categories are necessarily complete
This appears as Corollary 2.2 of Kelly-Lack.
The essential point is an -variant of the special adjoint functor theorem: if is cocomplete, has a proper factorization system , admits arbitrary -cointersections, and has an -generator, then every cocontinuous functor has a right adjoint.
The following examples are discussed in Section 6.1 of Kelly’s Basic concepts of enriched category theory.
Compactly generated spaces, and likewise based compactly generated spaces, with the surjections and the subspace inclusions. The point is an -generator.
Quasi-topological spaces. Note that this category is not -well-copowered.
Banach spaces with the epimorphisms, equivalently the dense maps, and the extremal monomorphisms, equivalently the inclusions of closed subspaces with the induced norm. The base field is an -generator.
The term locally ranked is sometimes used to refer to a locally bounded category which in addition is co-wellpowered. For example, this terminology is used in Adámek et. al..
The contents of this page are taken from:
Max Kelly, Basic concepts of enriched category theory.