Local boundedness of a category is a generalization of the notion of local presentability that includes the category of topological spaces.
Let $C$ be a small cocomplete category with a proper factorization system, i.e., an orthogonal factorization system $(E,M)$ where every map in $E$ is an epimorphism and every map in $M$ is a monomorphism.
The $M$-union of a small family of $M$-subobjects $(A_j \to B)_{j \in J}$ is the unique $M$-subobject $A \to B$ containing the $A_j$ and so that the induced map $\sum_j A_j \to A$ is in $E$. The union is calculated by applying the $(E,M)$ factorization to the canonical map $\sum_j A_j \to B$.
If the map $\sum_j A_j \to B$ is in $E$, we say $(A_j \to B)_{j \in J}$ is an $M$-union. The set $J$ is a preorder under the relation $j \leq k$ if $A_j \leq A_k$ as $M$-subobjects of $B$. Regarding the $A_j$ as a diagram of shape $J$, the family $(A_j \to B)_{j \in J}$ is an $M$-union if and only if the map colim$A_j \to B$ is in $E$. We say $(A_j \to B)_{j \in J}$ is a filtered union of $M$-subobjects if it is a union of $M$-subobjects and if the category $J$ is filtered.
A representable functor $C(X,-)$ preserves the $M$-union of $(A_j \to B)_{j \in J}$ if the functions $C(X,A_j) \to C(X,A)$ are jointly surjective, i.e., if each $X \to A$ factors through some $X \to A_j$.
Let $\lambda$ be a regular cardinal, and let $C$ be a cocomplete category with a proper factorization system $(E,M)$.
An object $X$ in $C$ is $\lambda$-bounded if $C(X,-)$ preserves $\lambda$-filtered unions of $M$-subobjects.
A small set $G$ of objects of $C$ is an $(E,M)$-generator if $f \colon A \to B$ in $M$ is invertible whenever $f_* \colon C(X,A) \to C(X,B)$ is bijective for all $X \in G$. Equivalently, $G$ is an $(E,M)$-generator if for each $A \in C$ the family of maps $X \to A$ is jointly in $E$, i.e., if the map $\sum_{X \in G} \sum_{C(X,A)} X \to A$ is in $E$.
A category $C$ is called locally $\lambda$-bounded with respect to a proper factorization system $(E,M)$ if
it has an $(E,M)$-generator $G$ each of whose objects is $\lambda$-bounded
it has arbitrary cointersections (even large ones) of maps in $E$ — that is, it is E-cocomplete.
In a locally $\lambda$-presentable category, every $\lambda$-presentable object is $\lambda$-bounded. Hence a $\lambda$-presentable category is $\lambda$-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 $(E,M)$-variant of the special adjoint functor theorem: if $C$ is cocomplete, has a proper factorization system $(E,M)$, admits arbitrary $E$-cointersections, and has an $(E,M)$-generator, then every cocontinuous functor $C \to D$ has a right adjoint.
The following examples are discussed in Section 6.1 of Kelly’s Basic concepts of enriched category theory.
Locally presentable categories such as the categories of simplicial sets, categories, abelian groups, sets.
Compactly generated spaces, and likewise based compactly generated spaces, with $E$ the surjections and $M$ the subspace inclusions. The point is an $(E,M)$-generator.
Quasi-topological spaces. Note that this category is not $E$-well-copowered.
Banach spaces with $E$ the epimorphisms, equivalently the dense maps, and $M$ the extremal monomorphisms, equivalently the inclusions of closed subspaces with the induced norm. The base field is an $(E,M)$-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:
See also:
Max Kelly, Basic concepts of enriched category theory.
Peter Freyd, Max Kelly, Categories of continuous functors J. Pure. Appl. Algebra 2 (1972) 169-191.
Last revised on March 31, 2020 at 09:41:08. See the history of this page for a list of all contributions to it.