[Paper Review] Enriched factorization systems

Rory B. B. Lucyshyn-Wright|arXiv (Cornell University)|Jan 1, 2014

Homotopy and Cohomology in Algebraic Topology15 references6 citations

TL;DR

This paper establishes the foundational theory of enriched factorization systems in $β$-categories enriched over a symmetric monoidal category $β$, generalizing classical factorization systems via a $β$-enriched orthogonality condition based on hom-objects. The key contribution is proving the existence of enriched $(̀\text{Epi},\text{Strong Mono})$ and $(̀\text{Strong Epi},\text{Mono})$-factorization systems in broad classes of enriched categories, including locally presentable, topological, and model-like categories.

ABSTRACT

In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects. We set forth the basic theory of such enriched factorization systems. In particular, we establish stability properties for enriched prefactorization systems, we examine the relation of enriched to ordinary factorization systems, and we provide general results for obtaining enriched factorizations by means of wide (co)intersections. As a special case, we prove results on the existence of enriched factorization systems involving enriched strong monomorphisms or strong epimorphisms.

Motivation & Objective

• To formalize and develop the foundational theory of enriched factorization systems, extending classical factorization systems to categories enriched over a symmetric monoidal category $β$.
• To clarify the relationship between enriched factorization systems and ordinary factorization systems in the underlying ordinary category.
• To provide general existence results for enriched factorization systems using wide (co)intersections and orthogonality conditions in hom-objects.
• To establish conditions under which enriched strong monomorphisms and strong epimorphisms form part of a factorization system.
• To demonstrate the existence of canonical enriched factorization systems in broad classes of categories, such as $β$-algebras and models of enriched sketches.

Proposed method

• Define $β$-enriched orthogonality via pullback conditions on hom-objects in $β$, generalizing the classical diagonal lifting property.
• Introduce enriched prefactorization systems as classes of morphisms closed under certain lifting and closure conditions in the enriched setting.
• Use wide (co)intersections of morphisms to construct factorization systems, particularly focusing on $β$-monos and $β$-epis.
• Apply stability results for enriched prefactorization systems under limits and colimits in $β$-categories.
• Leverage properties of well-poweredness, cotensoring, and limits/colimits in $β$-categories to derive existence theorems.
• Dualize results to obtain factorization systems involving strong epimorphisms and monomorphisms, using enriched orthogonality and closure under (co)limits.

Experimental results

Research questions

• RQ1 Under what conditions does an enriched category admit a factorization system involving enriched strong monomorphisms or strong epimorphisms?
• RQ2 How do enriched factorization systems relate to their underlying ordinary factorization systems in the underlying category?
• RQ3 What general categorical constructions (e.g., wide intersections) can be used to build enriched factorization systems?
• RQ4 In which classes of enriched categories—such as $β$-algebras or models of enriched sketches—do enriched $(̀\text{Epi},\text{Strong Mono})$ and $(̀\text{Strong Epi},\text{Mono})$-factorization systems exist?
• RQ5 What structural properties (e.g., well-poweredness, existence of limits/colimits) are sufficient for the existence of enriched factorization systems?

Key findings

• The pair $(\operatorname{\textnormal{{Epi}}}_{\mathscr{V}}\mathscr{B}, \operatorname{\textnormal{{StrMono}}}_{\mathscr{V}}\mathscr{B})$ forms a $β$-factorization system on $β$-categories $β$ that are cotensored, well-powered with respect to $β$-strong-monos, and admit small $β$-limits and $β$-cokernel-pairs.
• The pair $(\operatorname{\textnormal{{StrEpi}}}_{\mathscr{V}}\mathscr{B}, \operatorname{\textnormal{{Mono}}}_{\mathscr{V}}\mathscr{B})$ is a $β$-factorization system on $β$-categories $β$ that are tensored, co-well-powered with respect to $β$-strong-epis, and admit small $β$-colimits and $β$-kernel-pairs.
• In $β$-algebras over a closed, locally presentable, or well-powered $β$-category, the existence of enriched factorization systems is guaranteed under standard completeness and well-poweredness conditions.
• For categories of models of $β$-enriched weighted-limit sketches, the existence of both canonical enriched factorization systems is established via $β$-reflectivity and well-poweredness of the ambient category.
• The theory unifies and generalizes classical factorization systems in topoi and algebraic categories, showing that enriched strong monomorphisms and epimorphisms naturally form part of a $β$-factorization system.
• The dual of the existence result for $(\operatorname{\textnormal{{Epi}}}_{\mathscr{V}}\mathscr{B}, \operatorname{\textnormal{{StrMono}}}_{\mathscr{V}}\mathscr{B})$ yields the existence of $(\operatorname{\textnormal{{StrEpi}}}_{\mathscr{V}}\mathscr{B}, \operatorname{\textnormal{{Mono}}}_{\mathscr{V}}\mathscr{B})$ under dual conditions, confirming symmetry in the theory.