nLab locally bounded category

Idea

Local boundedness of a category is a generalization of the notion of local presentability that includes the category of topological spaces.

Context

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 $X \to A_j$.

Definition

Let $\lambda$ be a regular cardinal, and let $C$ be a cocomplete category with a proper factorization system $(E,M)$.

Definition (locally bounded)

An object $X$ in $C$ is $\lambda$-bounded if $C(X,-)$ preserves $\lambda$-filtered unions of $M$-subobjects.

Definition ($(E,M)$-generator)

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 $E$.

Definition (locally bounded category)

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.

Features

Proposition (relation to local presentability)

In a locally $\lambda$-presentable category, every $\lambda$-presentable object is $\lambda$-bounded. Hence a $\lambda$-presentable category is $\lambda$-bounded.

Proof

This appears as Lemma 2.3.1 of Freyd-Kelly

Proposition (completeness)

Locally bounded categories are necessarily complete

Proof

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.

Examples

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.

Related notions

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..

References

The contents of this page are taken from:

• Max Kelly, Steve Lack, $V$-cat is locally presentable or locally bounded if $V$ is so TAC (2001)

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.

• Jírí Adámek?, Horst Herrlich, Jírí Rosickỳ?, Walter Tholen, On a generalized small-object argument for the injective subcategory problem. Cah. Topol. Géom. Différ. Catég 43 (2002) 83–106.