Sheaf Consistency Mbse

---
name: sheaf-consistency-mbse
description: >
  Sheaf-theoretic framework for multi-view consistency in model-based systems engineering (MBSE).
  Uses presheaves on architectural sites to certify global design consistency via pairwise interface
  checks. Provides a formal mathematical criterion for when local engineering views (electrical, thermal,
  mechanical, software) determine a coherent global design.
  Use when: (1) analyzing multi-domain CPS architecture consistency, (2) formalizing MBSE verification,
  (3) building design space composition tools, (4) category theory applied to systems engineering,
  (5) formal verification of system architectures in Lean 4 or Mathlib.
  Activation: sheaf consistency, MBSE verification, multi-view architecture, design presheaf,
  architectural site, category theory systems engineering, formal architecture verification,
  CPS consistency, cross-domain design, SysML formal semantics.
---

# Sheaf Consistency for MBSE

Framework from arXiv:2605.08609 — "Sheaves as a Means of Maintaining Consistency in Model-based Systems Engineering" (Josh Gibson, 2026-05-09).

## Problem

In model-based systems engineering (MBSE), CPS architectures span multiple engineering views (electrical, thermal, mechanical, software). Each view has domain-specific design parameters, but they must remain mutually consistent. Current practice uses informal reviews, traceability matrices, and simulation — none providing a mathematical definition of what multi-view consistency means or a proof that local checks are sufficient for global consistency.

## Core Methodology

### Architectural Site

Construct a topological space where:
- **Points** = pairwise interfaces between engineering domains: `X = {(i,j) | V_i and V_j interact}`
- **Open sets** = engineering views: `U_k = {(i,j) ∈ X | k ∈ {i,j}}`
- The topology is generated by these opens

### Design Presheaf

A contravariant functor F: Op(X)^op → Type that:
- Assigns to each view U its **local design space** F(U) = product of coupling parameters
- Assigns **restriction maps** (projections) for view inclusions
- Encodes: restricting from larger to smaller view via intermediate is same as direct restriction

### Sheaf Consistency Theorem

**F is a sheaf ⟺ F satisfies pairwise intersection condition**

This means: global multi-view consistency is completely determined by compatibility on pairwise overlaps.

### Key Consequences

1. **Pairwise sufficiency**: Checking compatibility on pairwise interfaces certifies global consistency of arbitrary number of views
2. **Unique global design**: Compatible local designs determine a unique global design
3. **Inherited consistency**: Properties computed by limit-preserving functors inherit consistency
4. **Formal verification**: Entire chain from theorem to instance admits machine-checkable proof in Lean 4/Mathlib

## Implementation Pattern

```python
# Pseudocode: checking multi-view consistency via sheaf condition

def check_multiview_consistency(views, interfaces, design_params):
    """
    views: list of engineering domains (e.g., ["electrical", "thermal", "mechanical"])
    interfaces: dict of (view_a, view_b) -> list of shared parameters
    design_params: dict of view -> dict of parameter -> value
    """
    # 1. Build architectural site
    site = build_architectural_site(views, interfaces)
    
    # 2. Construct design presheaf
    presheaf = DesignPresheaf(site, design_params)
    
    # 3. Check pairwise compatibility (sufficient for global consistency)
    for (v1, v2), shared in interfaces.items():
        params_1 = design_params[v1]
        params_2 = design_params[v2]
        for p in shared:
            if params_1[p] != params_2[p]:
                return False, f"Inconsistency at {v1}-{v2}: parameter {p}"
    
    return True, "All views globally consistent"

def build_design_category(views, design_spaces, interface_projections):
    """
    Build SystemDiagram with CompatiblePair as pullback objects.
    Supports refinement adjunctions with unit/counit preservation.
    """
    diagram = SystemDiagram(views, design_spaces)
    for v1, v2 in view_pairs(views):
        pullback = CompatiblePair(
            design_spaces[v1], design_spaces[v2],
            interface_projections[(v1, v2)]
        )
        diagram.add_compatible_pair(pullback)
    return diagram
```

## Lean 4 Formalization

The paper provides machine-checkable proofs in Lean 4 using Mathlib:

```lean
-- SheafConsistency.lean
theorem sheaf_consistency {C : Type (u + 1)} [Category.{u} C]
    {X : ArchitecturalSite.{u}} (F : DesignPresheaf X C) :
    F.IsSheaf ↔ F.IsSheafPairwiseIntersections :=
  F.isSheaf_iff_isSheafPairwiseIntersections

-- DesignCategory.lean
structure SystemDiagram where
  designSpaces : Type → Type
  interfaceProjections : Type → Type → Type
  
structure CompatiblePair (D1 D2 : Type) (π₁ π₂ : Type) where
  pullback : Type
  equiv : pullback ≃ {p : D1 × D2 // π₁ p.1 = π₂ p.2}
```

## Applications

### CPS Architecture Verification

For autonomous vehicles, aircraft, satellites:
1. Identify all engineering views (electrical, thermal, mechanical, software, etc.)
2. Define interface parameters between each pair of views
3. Check pairwise compatibility using restriction maps
4. By Sheaf Consistency Theorem, pairwise checks guarantee global consistency

### Design Space Composition

When composing subsystem designs:
1. Each subsystem is a local section of the design presheaf
2. Composition is gluing via the sheaf condition
3. If pairwise interfaces are compatible, composition is guaranteed to exist and be unique

### Cross-Domain Dependency Tracking

- Dependencies between views modeled as functors between design space categories
- Hierarchical decomposition uses limits and colimits
- Refinement (abstract → detailed) captured by adjunctions

## Related Concepts

- **Mathlib**: `TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections`
- **Category theory**: Presheaves, sheaves, limits, adjunctions
- **MBSE tools**: SysML, AADL, Capella (this framework provides formal semantics for their consistency mechanisms)
- **Formal verification**: Lean 4 proof assistant