Presheaf of Value Assignments

• Presheaf of value assignments is a structure assigning local sections to open subsets with compatible restriction maps.
• It quantifies local-to-global consistency through a measurable consistency radius, critical in topological data analysis and forcing semantics.
• The sheafification process reveals global extension properties, underpinning robust applications in Boolean valued models and metric spaces.

A presheaf of value assignments is a categorical and topological structure arising in several areas of mathematics including logic, sheaf theory, and topological data analysis. It encapsulates families of local "choices" or "assignments" over open subsets of a given base (such as a topological space or a Stone space), together with compatible restriction maps, providing the framework necessary to analyze local-to-global properties, data gluing, and consistency of sections. This structure has been central in studying Boolean valued models, the semantics of forcing, and the quantitative analysis of data compatibility in applied topology.

1. Formal Definitions and Constructions

Let $(X,T)$ denote a topological space where $T$ is the lattice of open sets. A presheaf of sets $P$ on $X$ assigns to each open $U \in T$ a set $P(U)$ and to each inclusion $U \subseteq V$ a restriction map $r_{U,V}: P(V) \to P(U)$, subject to the usual functoriality axioms ($r_{U,U} = \mathrm{id}$, $r_{U,W} = r_{U,V} \circ r_{V,W}$ for $U \subseteq V \subseteq W$).

A presheaf of value assignments is canonically produced in several contexts:

• Given a sheaf $S$ of pseudometric spaces, one can define the value-assignment presheaf $\mathrm{Assgn}_S$ by

$$\mathrm{Assgn}_S(U) := \left\{ a: U\text{-assignment} \,\middle|\, c_S(a;U)=0 \right\},$$

that is, the set of assignments over $U$ which do extend to global sections, with restriction as forgetting information outside any given open subset (Robinson, 2018).

• In the setting of Boolean valued models, the presheaf $F_\mathcal{M}$ on a Stone space $X_\mathsf{B}$ (consisting of ultrafilters on a complete Boolean algebra $\mathsf{B}$) assigns to each open $U \subset X_\mathsf{B}$ the set of continuous local sections $s: U \to E_\mathcal{M}$, with $E_\mathcal{M}$ the étalé space of value assignments from the underlying $\mathsf{B}$-valued model $\mathcal{M}$ (Pierobon et al., 2020).

Presheaves serve as the precursor to sheaves, and the distinction often becomes crucial when discussing gluing (the sheaf condition), leading to additional properties and characterizations, notably in logical and metric settings.

2. Value Assignments, Consistency, and Quantitative Measures

In the context of pseudometric-valued sheaves, a value assignment is any selection of (potentially inconsistent) local sections $a \in \prod_{U \in T} S(U)$. The measure of how close such an assignment is to being a global section—i.e., how well the local values agree when restricted to overlaps—is quantified by the consistency radius:

$$c_S(a) = \sup_{U \subseteq V} d_U(r_{U,V}(a(V)),\, a(U)),$$

where $d_U$ is the pseudometric on $S(U)$ and $r_{U,V}$ is the restriction map (Robinson, 2018). This generalizes the notion of compatible local data by providing a real-valued (typically non-negative) measure of self-consistency. The value $c_S(a) = 0$ if and only if $a$ is a genuine global section.

Partial assignments over an open $U$ are similarly analyzed via the (restricted) consistency radius $c_S(a;U)$, with lower values indicating greater local agreement.

3. Presheaf-Sheaf Correspondence, Mixing, and Fullness

The relationship between presheaves of value assignments and their sheafifications is tightly coupled to logical properties. In Boolean valued model theory, the mixing property for a $\mathsf{B}$-valued structure $\mathcal{M}$ states:

$$\forall\{b_i\}_{i<\kappa} \subseteq \mathsf{B}\text{ antichain} \;\;\forall\{\tau_i\}_{i<\kappa} \subseteq M \;\; \exists\tau \in M \; \bigwedge_{i<\kappa} (b_i \leq \llbracket \tau = \tau_i \rrbracket_\mathcal{M}).$$

This property equates to the sheaf condition for the dense topology on $\mathsf{B}^+$: a $\mathsf{B}$-valued model $\mathcal{M}$ has mixing if and only if its associated presheaf $F_\mathcal{M}$ is a sheaf for the dense Grothendieck topology (Pierobon et al., 2020).

Fullness is a stricter property, requiring that

$$\llbracket \exists x\,\varphi(x,\bar a)\rrbracket_\mathcal{M} = \bigvee_{i<m} \llbracket \varphi(\tau_i, \bar a) \rrbracket_\mathcal{M}$$

for some finite choices $\tau_1, \ldots, \tau_m \in M$ and all formulas $\varphi$, i.e., existential quantification is witnessed by finitely many elements. Fullness corresponds to the completeness of global sections: every global section of the étalé space $E_\mathcal{M} \to X_\mathsf{B}$ is induced by some element $\tau\in M$. Mixing is strictly stronger than fullness (Pierobon et al., 2020).

4. Functoriality, Continuity, and Robustness

The constructions of presheaves of value assignments are functorial and continuous under suitable morphisms. For presheaves arising from pseudometric sheaves, the consistency filtration is a functor

$$\mathrm{CF}: \mathrm{ShvFPA}_L \to \mathrm{CoarseFilt}$$

where $\mathrm{ShvFPA}_L$ is the category of sheaves of pseudometric spaces with Lipschitz stalk-maps over finite topological spaces, and $\mathrm{CoarseFilt}$ the category of coarsening filtrations (levelwise refining filtrations of open covers) (Robinson, 2018). This functoriality ensures structure-preservation under homeomorphic and Lipschitz-continuous data transformations.

The consistency radius map $(S,a) \mapsto c_S(a)$ is continuous when restricted to finite base spaces; small perturbations in the assignment or the sheaf induce proportionally small changes in the consistency radius. Via the induced interleaving distance on filtrations, the consistency filtration inherits robustness: small perturbations in inputs (including noise or model errors) result in small interleavings (perturbations) of the resulting covers and its derived invariants, such as those computed by persistent cohomology. This continuity ensures stability in downstream analyses.

5. Structural Equivalences and Main Theorems

The correspondence between Boolean valued models and presheaves/sheaves of value assignments is formalized by an adjoint pair of contravariant functors:

• $L$: From $\mathsf{B}$-valued models to separated presheaves on $\mathsf{B}^+$
• $R$: From separated presheaves to $\mathsf{B}$-valued models

This adjunction restricts to equivalence between extensional models and those presheaves whose local sections extend globally. Mixing corresponds to the sheaf condition for the dense topology, while fullness corresponds to the surjectivity of global sections (Pierobon et al., 2020). After sheafification, the generalized forcing (Łoś) theorem is obtained: for the mixing-extension $M^+$ of a model $\mathcal{M}$,

$$M^+/G \models \varphi([\tau]_G) \iff \llbracket \varphi(\tau) \rrbracket_{M^+} \in G$$

for every formula $\varphi$ and ultrafilter $G$.

In metric settings, the presheaf of value assignments repackages notions of partial consistency and extension into a categorical context amenable to functorial and persistent analysis (Robinson, 2018).

6. Applications and Significance

Presheaves of value assignments are fundamental in both pure and applied frameworks. In topological data analysis, the consistency filtration associated to value assignments recovers classical filtrations such as the Čech filtration for point clouds, with persistent cohomology as a robust invariant (Robinson, 2018). In distributed sensing, value assignments quantify the consistency of local sensor readings, and the induced filtrations encode internal agreement at varying tolerances.

In logic and set-theoretic forcing, modeling truth assignments as local sections of a presheaf (and their sheafification) clarifies the role of truth values, gluing, and the translation between forcing language and topological sheaf language. The mixing property captures the gluing axiom, and fullness corresponds to finite-witness quantification.

The interplay between categorical, logical, and quantitative formulations underscores the unified role of presheaves of value assignments across theory and applications, revealing common structural features in categorical logic, model theory, and applied topological data analysis (Robinson, 2018, Pierobon et al., 2020).