nLab
locally bounded category

Contents

Idea

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

Context

Let CC be a small cocomplete category with a proper factorization system, i.e., an orthogonal factorization system (E,M)(E,M) where every map in EE is an epimorphism and every map in MM is a monomorphism.

The MM-union of a small family of MM-subobjects (A jB) jJ(A_j \to B)_{j \in J} is the unique MM-subobject ABA \to B containing the A jA_j and so that the induced map jA jA\sum_j A_j \to A is in EE. The union is calculated by applying the (E,M)(E,M) factorization to the canonical map jA jB\sum_j A_j \to B.

If the map jA jB\sum_j A_j \to B is in EE, we say (A jB) jJ(A_j \to B)_{j \in J} is an MM-union. The set JJ is a preorder under the relation jkj \leq k if A jA kA_j \leq A_k as MM-subobjects of BB. Regarding the A jA_j as a diagram of shape JJ, the family (A jB) jJ(A_j \to B)_{j \in J} is an MM-union if and only if the map colimA jBA_j \to B is in EE. We say (A jB) jJ(A_j \to B)_{j \in J} is a filtered union of MM-subobjects if it is a union of MM-subobjects and if the category JJ is filtered.

A representable functor C(X,)C(X,-) preserves the MM-union of (A jB) jJ(A_j \to B)_{j \in J} if the functions C(X,A j)C(X,A)C(X,A_j) \to C(X,A) are jointly surjective, i.e., if each XAX \to A factors through some XA jX \to A_j.

Definition

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

Definition (locally bounded)

An object XX in CC is λ\lambda-bounded if C(X,)C(X,-) preserves λ\lambda-filtered unions of MM-subobjects.

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

A small set GG of objects of CC is an (E,M)(E,M)-generator if f:ABf \colon A \to B in MM is invertible whenever f *:C(X,A)C(X,B)f_* \colon C(X,A) \to C(X,B) is bijective for all XGX \in G. Equivalently, GG is an (E,M)(E,M)-generator if for each ACA \in C the family of maps XAX \to A is jointly in EE, i.e., if the map XG C(X,A)XA\sum_{X \in G} \sum_{C(X,A)} X \to A is in EE.

Definition (locally bounded category)

A category CC is called locally λ\lambda-bounded with respect to a proper factorization system (E,M)(E,M) if

  • it has an (E,M)(E,M)-generator GG each of whose objects is λ\lambda-bounded

  • it has arbitrary cointersections (even large ones) of maps in EE — that is, it is E-cocomplete.

Features

Proposition (relation to local presentability)

In a locally λ\lambda-presentable category, every λ\lambda-presentable object is λ\lambda-bounded. Hence a λ\lambda-presentable category is λ\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)(E,M)-variant of the special adjoint functor theorem: if CC is cocomplete, has a proper factorization system (E,M)(E,M), admits arbitrary EE-cointersections, and has an (E,M)(E,M)-generator, then every cocontinuous functor CDC \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.

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, VV-cat is locally presentable or locally bounded if VV 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.

Revised on April 22, 2013 01:45:48 by Marc Olschok (212.23.103.132)