[Paper Review] Lawvere completion and separation via closure

Dirk Hofmann, Walter Tholen|ArXiv.org|Dec 31, 2007

Structural Analysis and Optimization6 references4 citations

TL;DR

This paper introduces a closure-theoretic framework for Lawvere completeness and separation in $τ$-categories, generalizing $ν$-category theory via a topological theory $τ = (\mathbb{T}, \mathsf{V}, \xi)$. It establishes that the L-completion of a $τ$-category is its L-closure in the Yoneda image within the category of $τ$-modules, yielding an epi-reflective subcategory of L-complete $τ$-categories.

ABSTRACT

For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.

Motivation & Objective

• To generalize Lawvere's notion of completeness and separation from $ν$-categories to $τ$-categories using closure operators.
• To resolve the asymmetry in the Yoneda embedding for $τ$-categories by defining $X^{\mathrm{op}}$ as $TX$ with a $τ$-structure.
• To establish a categorical framework for completeness and separation that unifies metric, topological, and continuity spaces.
• To show that L-completion arises as the L-closure of the Yoneda image in the category of $τ$-modules.
• To prove that the full subcategory of L-complete $τ$-categories is epi-reflective in the category of L-separated $τ$-categories.

Proposed method

• Define L-density for $ν$-functors as the property that $f \cdot m = g \cdot m$ implies $f \cong g$, generalizing epimorphisms.
• Introduce the L-closure of a subobject $M \to X$ as the largest subobject for which $M \to \overline{M}$ is L-dense.
• Construct the Yoneda functor $\mathpzc{y}: X \to \hat{X} = \mathsf{V}^{X^{\mathrm{op}}}$ with $X^{\mathrm{op}} = TX$ equipped with a $τ$-algebra structure.
• Use the internal hom $\multimap$ in the quantale $\mathsf{V}$ and the monad $\mathbb{T}$ to define the $τ$-module structure on $\hat{X}$.
• Define the L-completion $\tilde{X}$ as the L-closure of $\operatorname{\mathpzc{y}}(X)$ in $\hat{X}$, ensuring full faithfulness and L-density of $\operatorname{\mathpzc{y}}$.
• Establish that $\psi \in \hat{X}$ is a right adjoint $τ$-module iff $\psi \in \overline{\operatorname{\mathpzc{y}}(X)}$, linking adjointness to closure.

Experimental results

Research questions

• RQ1 How can Lawvere completeness and separation be generalized from $ν$-categories to $τ$-categories using closure operators?
• RQ2 What is the correct generalization of the Yoneda embedding in the context of $τ$-categories, given the asymmetry of the $τ$-structure?
• RQ3 How does the L-closure construction yield a completion process that is both complete and separated?
• RQ4 What conditions ensure that the L-completion of a $τ$-category is an epi-reflective subcategory of the category of L-separated $τ$-categories?
• RQ5 In what sense is the L-completion of a $τ$-category equivalent to the closure of its Yoneda image in the module category?

Key findings

• The L-completion of a $τ$-category $X$ is given by $\tilde{X} = \overline{\operatorname{\mathpzc{y}}(X)}$, the L-closure of the Yoneda image in $\hat{X}$, making $\tilde{X}$ L-complete.
• The Yoneda embedding $\operatorname{\mathpzc{y}}: X \to \tilde{X}$ is fully faithful and L-dense, ensuring that $\tilde{X}$ is the smallest L-complete extension of $X$.
• A $τ$-module $\psi \in \hat{X}$ is a right adjoint if and only if $\psi \in \overline{\operatorname{\mathpzc{y}}(X)}$, linking adjointness to closure.
• The full subcategory $\mathscr{T}\text{-}\mathsf{Cat}_{\mathrm{cpl}}$ of L-complete $τ$-categories is epi-reflective in $\mathscr{T}\text{-}\mathsf{Cat}_{\mathrm{sep}}$, with reflection given by $\operatorname{\mathpzc{y}}: X \to \tilde{X}$.
• The construction generalizes known cases: for $\mathbb{T}$ the ultrafilter monad and $\mathsf{V} = \mathsf{2}$, it recovers Barr’s relational description of topological spaces.
• For $\mathbb{T}$ the ultrafilter monad and $\mathsf{V}$ the Lawvere half-line, the framework recovers Lowen’s approach spaces, validating its generality.