[Paper Review] A new Galois structure in the category of internal preorders

Alberto Facchini, Carmelo Antonio Finocchiaro+1 more|arXiv (Cornell University)|Sep 19, 2019

Homotopy and Cohomology in Algebraic Topology10 references4 citations

TL;DR

This paper establishes a new pretorsion theory (Eq(C), ParOrd(C)) in the category of internal preorders in an exact category C, showing that the reflector to internal partial orders has stable units, thus inducing an admissible Galois structure. In the case C = Set, this yields a monotone-light factorization system in PreOrd(Set), with topological interpretations via Alexandroff-discrete spaces and partition topologies.

ABSTRACT

Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C), \mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$, where $\mathsf{Eq}(\mathbb C)$ and $\mathsf{ParOrd}(\mathbb C)$) are the full subcategories of internal equivalence relations and of internal partial orders in $\mathbb C$, respectively. We observe that $\mathsf{ParOrd}(\mathbb C)$ is a reflective subcategory of $\mathsf{PreOrd}(\mathbb C)$ such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector $F:\mathsf{PreOrd}(\mathbb C) o \mathsf{ParOrd}(\mathbb C)$ turns out to have stable units in the sense of Cassidy, Hébert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when $\mathbb C$ is the category $\mathsf{Set}$ of sets, we show that this reflection induces a monotone-light factorization system (in the sense of Carboni, Janelidze, Kelly and Paré) in $\mathsf{PreOrd}(\mathsf{Set})$. A topological interpretation of our results in the category of Alexandroff-discrete spaces is also given, via the well-known isomorphism between this latter category and $\mathsf{PreOrd}(\mathsf{Set})$.

Motivation & Objective

• To establish a pretorsion theory (Eq(C), ParOrd(C)) in the category PreOrd(C) of internal preorders in an exact category C.
• To show that the reflector F: PreOrd(C) → ParOrd(C) has stable units, thereby inducing an admissible categorical Galois structure.
• To characterize the resulting monotone-light factorization system in the special case C = Set.
• To provide a topological interpretation via the isomorphism between PreOrd(Set) and the category of Alexandroff-discrete spaces.
• To link the algebraic structure of preorders with topological properties such as T0 separation and partition topologies.

Proposed method

• Define a pretorsion theory (T, F) in PreOrd(C) using the subcategories Eq(C) of internal equivalence relations and ParOrd(C) of internal partial orders.
• Prove that the inclusion ParOrd(C) ↪ PreOrd(C) is reflective, with the reflector F having pullback-stable regular epimorphisms as components of the unit.
• Show that the reflector F has stable units in the sense of Cassidy, H´ebert, and Kelly, which implies an admissible Galois structure.
• Characterize the factorization system (E, M) induced by the Galois structure, where E consists of morphisms inverted by F and M consists of trivial coverings.
• Specialize to C = Set, showing that the reflection induces a monotone-light factorization system (E′, M*) in PreOrd(Set), with E′ consisting of pullback-stable regular epimorphisms with trivial fibers.
• Use the known isomorphism between PreOrd(Set) and the category Alex of Alexandroff-discrete spaces to translate algebraic results into topological terms, especially relating preorder relations to topologies and T0/separation properties.

Experimental results

Research questions

• RQ1 How can a pretorsion theory be defined in the category PreOrd(C) when the category lacks a zero object?
• RQ2 Under what conditions does the reflector from internal preorders to internal partial orders have stable units?
• RQ3 What is the structure of the monotone-light factorization system induced by the reflection in PreOrd(Set)?
• RQ4 How do the algebraic properties of preorders relate to topological properties in Alexandroff-discrete spaces?
• RQ5 When does the preorder induced by a topology become an equivalence relation, and what does this imply for the topology?

Key findings

• The pair (Eq(C), ParOrd(C)) forms a pretorsion theory in PreOrd(C), with ParOrd(C) as a reflective subcategory.
• The reflector F: PreOrd(C) → ParOrd(C) has stable units, which ensures that the induced Galois structure is admissible.
• In the case C = Set, the reflection induces a monotone-light factorization system (E′, M*) in PreOrd(Set), where E′ consists of pullback-stable regular epimorphisms with trivial fiber topologies.
• A morphism f: A → B in PreOrd(Set) lies in M* if and only if each fiber of f is a T0 space under the subspace topology.
• For Alexandroff-discrete spaces, the preorder induced by the topology is a partial order if and only if the space is T0, and is an equivalence relation if and only if the topology is a partition topology.
• The category Alex of Alexandroff-discrete spaces admits a pretorsion theory (PartAlex, T0Alex), where PartAlex consists of spaces with partition topologies and T0Alex those that are T0, with a reflection to T0Alex given by the quotient map identifying points with the same closure.