Covers and Restriction

There is only a perspective seeing, only a perspective 'knowing.'

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. Chosen restriction maps satisfying the identity and composition laws 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 specify universal relationships between translations, 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 now receives a more specific representation: 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 $U_w$. Applicability requires the current context to lie within its certified scope, with the stated time and property conditions satisfied. Refinement preserves a witness only along maps for which that preservation has been established; a new constraint or changed predicate does not inherit a certificate merely by being more specific. 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:

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 view also needs a policy for reasoning about absence¹. 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. The differing status follows from the declared treatment of absence; the two views may use the same proof logic.

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)

The word “sustainable” here abbreviates different standards. Those standards must be represented before the values can be compared. Identical wording does not establish a common proposition.

Refinement and Restriction

Contexts are related by refinement. A context U refines a context V (written U ⪯ V, or equivalently U → V) under the following chosen comparison discipline. Adding predicates or constraints alone will not construct the required maps, and closed-world and open-world policies do not supply a general ordering of logical strength.

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 logic with the declared additional constraints.

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.

Restriction maps must be well-behaved:

Covers and Overlaps

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

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; separate material evidence
B (Fast Fashion)	material = silk_blend	OWA; supplier assertion

Assume the two records concern the same item, date and material classification, and that “silk” excludes blends in this classification. Their restricted material values then disagree. Their absence policies neither establish nor defeat the material evidence.

Two reconciliation paths exist:

1. 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.

2. A coarser comparison: Map silk and silk blend to textile in an explicitly defined classification. Agreement about textile leaves the disagreement about composition unresolved. Silk and silk blend need not be comparable by specificity. An adjunction can organize an abstraction when its order and maps satisfy the adjunction law; this small coarsening alone has not proved one.

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

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

Overlaps When Logics Differ

Different logics can require a comparison interface: a shared signature or explicit translation, a logic in which the relevant claims can be compared, and maps from each record into that interface. This is not automatically the categorical overlap $U\times_X V$ in the refinement site defined above. There, projections from the overlap to U and V require it to refine both; its signature must include both signatures under the stated inclusion convention.

A weaker common language answers a different question. If U uses closed-world inference and V uses open-world inference, an adapter may replace an unsupported negative conclusion with “undetermined.” That loses a conclusion; it is not the derivability-preserving refinement map defined above. Incorporating such an adapter into a presheaf requires separately specified records, maps and composition laws. Naming it an overlap supplies none of those conditions.

Logic reconciliation in A15 uses these comparison interfaces where needed. The site construction below concerns the category and topology actually specified.

The Site Structure

With identities and composition satisfying the category laws, the chosen contexts and refinements form a category. A coverage satisfying the required axioms then supplies a site. These are requirements on the construction, not automatic properties of every collection of schemas.

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
• Where it exists, 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.

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

If the project requires those modules to use one common dependency instance, their version requirements 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 removes the shared-instance requirement for that dependency. Other interfaces between the modules may still require compatibility checks.
• Explicit witnessing: Introduce a version shim that mediates between v1.0 and v2.0. The shim needs an explicit behavioral contract. It is not an adjunction merely because it translates versions.

A dependency graph can help construct a category of contexts; version resolution can then be studied through the chosen compatibility relations. Ordinary build systems already perform this work. The proposed representation makes its assumptions available for comparison with other forms of reconciliation.

Truth-in-View

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

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.)

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: can be analyzed with this structure once the contexts, maps and coverage are supplied. Local derivability and shared constraints then become inspectable obligations rather than assurances borrowed from the vocabulary of a site.

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.