Covers and Restriction: How global truth decomposes into local pieces

A12 · A12bMachine-checkedThe formal structure of context sites and coverage.

Showing also what relation the merchandize of one country or of one city bears to that of others.
— Francesco Balducci Pegolotti, Pratica della Mercatura (c. 1340)

This chapter formalizes context as a mathematical structure rather than an ambient notion. It defines Anchor A12 (covers of contexts with overlap reconciliation) and Anchor A12b (the Grothendieck site structure on categories of contexts), establishing the stage on which all subsequent coherence conditions operate. Restriction maps between contexts are shown to form a presheaf, and conflict witnesses are introduced as first-class objects that record disagreement with provenance. The motivating examples and touchstones treated here correspond to the narrative discussions of contextual fragmentation in Vol I, Chapters 5 ("The Empire of Tables") and 6 ("Evidence without Custody").

The Problem with "Context"

The word "context" has become a hedge. Systems claim to be "context-aware" when they use GPS coordinates. Analysts say "in this context" to avoid specifying which assumptions they are making. Large language models have "context windows" measured in tokens, a precise quantity that tells you nothing about what the tokens mean or how they relate. The word has become a placeholder for "the stuff around the thing," which is not a definition.

Part II gave us a calculus of sameness: invariants that survive transformations, isomorphisms that witness structural identity, adjunctions that measure the cost of imperfect translation, and witnesses that carry kind, scope, and transport rules. But that calculus assumed we knew what "scope" meant. Every witness in A10 carried a scope predicate S, and we left S unanalyzed. Now we pay the debt.

A scope is a context. The scope predicate S(ctx, t) from A10 is now implemented: ctx ∈ U means "ctx belongs to context U," and time restrictions live in the constraint set $I_U$ . Each witness is indexed by a base context $U_w$ ; it is valid at runtime precisely when ctx ⪯ $U_w$ (the current context refines the witness's base) and the time constraints in $I_{U_w}$ are satisfied. The Part II machinery (invariants, witnesses, transport) gains a home.

A context is not ambient but a restriction structure: a view with its own vocabulary, its own constraints, its own logic, and explicit maps that translate claims from one view to another. Contexts overlap, and where they overlap, their claims must agree or be explicitly reconciled. That structure is not metaphor but the stage on which coherence conditions live.

Context as Restriction

A context U is a view together with the rules of inference that are valid in that view. Formally, a context is a triple:

A context U consists of:

A signature Σ (the predicates and sorts that exist in this view)
A constraint set I (the invariants that must hold)
A logic L (the inference rules: closed-world vs open-world, classical vs intuitionistic)

The local theory T(U) is the set of sentences derivable in U under L. Think of T(U) as what U knows, not as the underlying data.

The signature tells you what distinctions you can draw. A formal-wear context might include predicates like is_suit, is_evening_gown, is_tuxedo. A casual-wear context might lack is_tuxedo entirely; the distinction does not exist there.

The constraints tell you what must hold. In formal wear, perhaps is_tuxedo → is_suit (every tuxedo is a suit). In casual wear, that constraint is irrelevant because tuxedos are not in the vocabulary.

The logic tells you how to reason about absence(Reiter 1978). In a closed-world inventory context, if a product is not listed, it is not available. In an open-world trend-forecasting context, absence of a trend does not mean the trend will not emerge. Same data, different epistemic status, because different logics.

This is where A4 (Epistemic Status) becomes operational. A4 said: true/false/undetermined is relative to a view and its logic. A12 operationalizes that relativity. The sentence "This dress is sustainable" might be:

Undetermined in a marketing context (open-world, promotional claims)
True in a compliance context (closed-world over certified suppliers)
False in an activist context (stricter standard than compliance)

Same sentence, different truth values, because different contexts. This is not relativism; it is precision.

Contexts are related by refinement. A context U refines a context V (written U ⪯ V, or equivalently U → V) when U is more specific: U has at least as many predicates, at least as many constraints, and at least as strong a logic.

U refines V (U ⪯ V) iff:

Σ_U ⊇ Σ_V (U can draw all the distinctions V can, and possibly more)
I_U ⊇ I_V (U satisfies all the constraints V requires, and possibly more)
There exists a restriction map ρ : T(V) → T(U) that preserves derivability on the shared vocabulary

The third condition handles logic differences: when L_U and L_V differ, the restriction map translates into a common fragment or the weaker logic. The map's existence is the requirement, not a global ordering on logics.

The "formal evening wear" context refines the "evening wear" context: it has the same predicates plus is_black_tie, the same constraints plus is_black_tie → is_formal, and the same or stronger logic.

When U refines V, there is a restriction map ρ : T(V) → T(U) that translates claims on the shared fragment. If "this is an evening gown" is derivable in V, then its translation is derivable in U under the refinement's chosen comparison discipline. Restriction goes "upward" in the refinement order: from coarser contexts to finer ones.

Remark

Contexts refine downward (toward more specific); truth restricts upward (contravariantly). This is the key orientation. A morphism U → V in the category of contexts means U is finer; the restriction map T(V) → T(U) goes the opposite direction.

Restriction maps must be well-behaved:

For restriction maps ρ along refinement morphisms:

Identity: $\rho_{\text{id}} = \text{id}$ (restricting along identity does nothing)
Composition: $\rho_{g \circ f} = \rho_f \circ \rho_g$ (restricting twice equals restricting along the composition)

These laws make T into a presheaf: a contravariant functor from contexts to theories(Lane 1992, ch. I). This is the law that prevents pipelines from inventing contradictions when they refine a view.

Covers and Overlaps

A12

A cover is a family of contexts that jointly describe a larger context.

A cover of context U is a family $\{U_i \to U\}$ of refinement morphisms such that the $U_i$ are declared jointly sufficient for U.

"Jointly sufficient" means: any question answerable in U can be answered by combining answers from the Uᵢ, provided those answers agree on overlaps.

A cover is not "all the subcontexts we can think of." It is the set of views you commit to as jointly sufficient for deciding a class of questions. What counts as a cover is a design choice.

Consider three merchants who contribute to a fashion catalog:

A (Luxury): Verified materials, high prices, closed-world inventory
B (Fast Fashion): Trendy, affordable, open-world claims (promotional material may not be verified)
C (Vintage): Authentic items, provenance-tracked, closed-world

They share some products. The catalog context U is covered by $\{A, B, C\}$ . But the cover is only useful if we can reconcile claims on the overlaps.

The overlap A ∧ B (the pullback in general; the meet/greatest-lower-bound in the poset simplification) is where agreement must happen. Suppose a dress appears in both:

Context	Claim	Logic
A (Luxury)	material = silk	CWA (verified)
B (Fast Fashion)	material = silk_blend	OWA (promotional)

The restriction of A's claim to the overlap says "silk (certified)." The restriction of B's claim says "silk_blend (asserted)." These do not agree.

Two reconciliation paths exist:

Witnessed conflict: Fork into two interpretations, each scoped. Downstream queries must declare which fork they consume. The conflict is not hidden; it is recorded as a first-class object with provenance.
Adjunction-shaped approximation: Map both claims into a common lattice (e.g., material types ordered by specificity: silk ⊑ silk_blend ⊑ textile ⊑ unknown) and transport via the coarser predicate. The luxury claim becomes "at least silk"; the fast-fashion claim becomes "at least silk_blend." These are compatible in the lattice, though information is lost.

Both paths require explicit structure. Neither pretends the disagreement does not exist.

A conflict witness for predicate P on entity x in overlap U ∧ V records:

The left claim (with provenance from U)
The right claim (with provenance from V)
The overlap context U ∧ V
Admissible transports: fork policy, precedence rule, or approximation map

This keeps the A10 theme: sameness is an artifact, not a bare predicate. So is disagreement.

Overlaps When Logics Differ

The overlap U ∧ V is not always straightforward. If U uses closed-world logic and V uses open-world logic, what logic does U ∧ V use?

The overlap is constructed, not found. It carries:

A shared signature (the predicates both U and V have, or a chosen translation)
A comparison logic (often the weaker one, or a common fragment both logics extend)
Explicit witness maps from each side into the overlap

If U is closed-world and V is open-world, the overlap might use open-world logic (the weaker assumption). Then the restriction from U to U ∧ V forgets the closed-world guarantee: a claim that was "false by omission" in U becomes "undetermined" in U ∧ V. That is information loss, and it must be explicit.

This is where logic functors (A15) will eventually live. For now, the point is that overlap is not set intersection but a negotiated interface.

The Site Structure

A12b

Contexts with refinement morphisms form a category. When we add a specification of which families count as covers, we get a site.

A site (C, J) consists of(Verdier 1972--1973)(Lane 1992, ch. III):

A category C of contexts with refinement morphisms
A Grothendieck topology J specifying which families $\{U_i \to U\}$ are covers

Covers satisfy:

Identity: $\{U \to U\}$ covers U
Stability: If $\{U_i \to U\}$ covers U and $V \to U$ is any morphism, then the pullback family $\{U_i \times_U V \to V\}$ covers V
Transitivity: If $\{U_i \to U\}$ covers U and each $U_i$ is covered by $\{U_{ij} \to U_i\}$ , then $\{U_{ij} \to U\}$ covers U

The terminology is heavy, but the structure is light. For most systems applications, you can work with a poset of contexts under refinement. Then:

Morphism U → V exists exactly when U ⪯ V
Overlap U ∧ V is the greatest lower bound (the most general context that refines both)
Cover $\{U_i \to U\}$ means the $U_i$ are declared sufficient for U relative to the topology J

In the poset reading, a cover behaves like an exhausting family: set-union is a useful picture, not the definition. The topology J declares what "sufficient" means for the questions you care about.

The axioms say: covers behave sensibly under zooming and re-covering. If you have a cover and you zoom into a subcontext, the restricted family is still a cover. If you cover each piece of a cover, the composite is a cover.

Remark

In the poset picture, "jointly exhaust" looks like set union. But formally, it is a declared relation: the family is sufficient for the questions we care about, not necessarily for every possible distinction. The topology J is the declaration.

This is where A5 (Coherence Requirement) becomes structural. A5 asked preformally for "overlap agreement." A12b makes that request precise: a claim is coherent over a cover iff its restrictions to all pairwise overlaps agree.

Build Systems as Sites

The fashion catalog is not the only example. Software build systems have the same structure.

A software project has modules: auth, payments, ui, shared-lib. Each module is a context with:

A signature (exported types and functions)
Constraints (invariants, contracts, API guarantees)
A logic (strict typing vs permissive, fail-fast vs lenient error handling)

The project is covered by its modules. But modules share dependencies, and dependencies have versions. Suppose:

Module auth depends on shared-lib v1.0
Module payments depends on shared-lib v2.0

The overlap (auth ∧ payments via shared-lib) has a conflict. The restriction of auth's dependency claim says "v1.0." The restriction of payments' claim says "v2.0." These do not agree.

Build systems resolve this by:

Forcing agreement: Require a single version project-wide. This corresponds to "choose one restriction as authoritative on overlap."
Scoped resolution: Allow different versions in different contexts. auth sees v1.0; payments sees v2.0; they never interact. This makes the overlap empty by design: no gluing required because no shared claims.
Explicit witnessing: Introduce a version shim that mediates between v1.0 and v2.0. The shim is an adjunction-shaped approximation witness, lossy in one direction or both.

The "dependency graph" is a category of contexts. "Version resolution" is finding a cover with compatible restrictions. The problem is not new; the structure is.

Truth-in-View

We can now state precisely what "truth" means in a system with multiple contexts.

A sentence p is true in context U (written p ∈ T(U)) iff p is derivable in the local theory of U under U's logic L_U.

A sentence p is coherent over a cover $\{U_i \to U\}$ iff:

p restricts to a sentence pᵢ ∈ T(Uᵢ) for each i
For all overlaps Uᵢ ∧ Uⱼ, the restrictions of pᵢ and pⱼ agree

Coherence is not "true everywhere." It is "locally true in a compatible way." A global claim is one whose local forms agree wherever the views meet. (We formalize this as the sheaf condition in the next chapter.)

Example

"The cocktail dress is silk" might be:

True in U_luxury (verified, closed-world)
Undetermined in U_fast_fashion (unverified, open-world)
False in U_vintage (provenance shows polyester blend)

This is not a global claim. It is three local claims that do not cohere. The system can record all three, scoped to their contexts. It cannot assert a global "is silk" without reconciliation.

Consequence

A12 and A12b give context a structure. A context is not vague surroundings; it is a triple (signature, constraints, logic) that determines what is sayable and what is derivable. Contexts refine one another; finer contexts see more distinctions. Restriction maps translate claims from coarser to finer, contravariantly.

A cover is a family of contexts declared jointly sufficient for a larger context. The site structure (category + topology) specifies which families count as covers and ensures covers behave sensibly under zooming and re-covering.

The fashion catalog, the build system, the entity-resolution pipeline: all are instances of the same structure. Local contexts with local truths, overlapping in ways that demand reconciliation.

A5 asked for overlap agreement. A12b answers: agreement on overlaps is the coherence requirement, now made structural. A claim is coherent over a cover iff its restrictions to all pairwise overlaps match.

We have a space of views. Now we need a rule for when local views glue into global truths.