Sheaves as a Means of Maintaining Consistency in Model-based Systems Engineering

Abstract: We propose that the sheaf condition on a presheaf of design spaces provides a mathematical model for multi-view consistency in the architecture of cyber-physical systems (CPS). In model-based systems engineering, multiple engineering views -- electrical, thermal, mechanical, and software -- must be kept mutually consistent, yet current practice relies on informal procedures without a precise semantic account of global consistency. We construct an architectural site: a topological space whose points are pairwise interfaces between engineering domains and whose open sets represent engineering views. A design presheaf assigns to each view its local design space and to each inclusion the corresponding restriction map. We show that the sheaf condition on this presheaf is equivalent to compatibility on pairwise overlaps, yielding a local criterion for global multi-view consistency. The equivalence and a concrete three-view worked example are machine-verified in Lean 4 using Mathlib. The formalization establishes that the design presheaf is a sheaf, that the sheaf condition is equivalent to pairwise overlap compatibility, and that compatible local design families glue to unique global designs. Global consistency of an arbitrary number of views can be certified by checking only pairwise interface compatibility; compatible local designs determine a unique global design; derived properties computed by limit-preserving functors inherit the same consistency guarantee; and the entire verification chain admits machine-checkable proofs in Lean.

• The paper introduces a sheaf-theoretic framework that reduces global multi-view consistency verification in MBSE to simpler pairwise interface validations, cutting down verification complexity.
• Machine verification in Lean 4 confirms the equivalence between the sheaf condition and pairwise interface consistency, ensuring deterministic and unambiguous design integration.
• Functorial inheritance of consistency guarantees that derived engineering properties retain the sheaf property, offering robust, compositional benefits for CPS design.

Sheaf-Theoretic Semantics for Multi-View Consistency in Model-Based Systems Engineering

Introduction

This work provides a categorical, sheaf-theoretic framework for formalizing multi-view consistency in the architecture of cyber-physical systems (CPS) within the context of Model-Based Systems Engineering (MBSE). It addresses the semantic and methodological limitations of prevailing, largely informal, practices for cross-domain design integration. The proposed approach models the architecture as a topological space whose points correspond to engineering domain interfaces and whose open sets represent engineering views (domains such as electrical, thermal, mechanical, and software). Consistency of local design data across views is precisely characterized by the sheaf condition for a presheaf of design spaces, and the equivalence of the sheaf condition with pairwise interface consistency is both mathematically formalized and machine-verified in Lean 4.

Architectural Sites, Presheaves, and the Sheaf Condition

The core mathematical structure of the paper is the architectural site, a topological space $X$ constructed from the pairwise interfaces between engineering domains. Each engineering view is modeled as an open set in $X$, containing those interfaces in which it participates. Local design spaces, capturing design parameters relevant to each interface, are organized as a presheaf $F$ on the site: to each open set (view), $F$ assigns the product of the parameter types at the included interfaces, with restriction maps given by projections.

The sheaf condition for $F$ guarantees that, for any open cover of a view (i.e., a collection of engineering domains whose union covers the relevant interfaces), a family of local designs that agree on overlaps can be uniquely 'glued' into a global design. This is shown to admit a crucial equivalence: the sheaf condition, typically formulated globally, is in this setting equivalent to the requirement that local designs must agree only on pairwise overlaps (the intersection of every pair of views). Thus, global architectural consistency reduces to local pairwise compatibility, a nontrivial reduction that resolves the semantics of MBSE integration in a categorical setting.

Formalization and Worked Example

A detailed, Lean 4-verified construction is given for a three-view site (electrical, thermal, mechanical), reflecting scenarios typical of CPS architecture. The system consists of three interface points:

• Electro-Mechanical,
• Electro-Thermal,
• Thermo-Mechanical.

The design presheaf assigns to each view the tuple of coupling parameters relevant to the interfaces covered by that view. Restriction maps are coordinate projections. The framework confirms:

• The design presheaf is a sheaf.
• The sheaf property is equivalent to the pairwise consistency condition.
• Every pairwise-compatible family of designs glues uniquely to a global design.

All critical properties—including the general sheaf consistency equivalence, the gluing theorem for compatible families, and functorial inheritance of consistency for derived properties—are machine-checked in Lean 4, eliminating proof ambiguity.

Theoretical and Practical Implications

Reduction of Verification Complexity

A paramount outcome is that global architectural consistency across $n$ views can always be assured by checking at most $n \choose 2$ pairwise interface agreements, dramatically reducing verification complexity relative to exponential alternatives.

Determinism in System Integration

Once each domain team produces a locally compatible design, there is exactly one way to integrate these into a global system design. No ambiguity, nondeterminism, or hidden inconsistency remains; integration is a fully determined, mechanically verifiable process. This shifts design integration from ad hoc negotiation to guaranteed assembly.

Functoriality and Inheritance of Consistency

Any derived quantity or engineering property that is functorially computed from the space of designs (via a limit-preserving functor) inherits the sheaf (consistency) property in a compositional manner. This robustness extends the assurance of local-to-global validity beyond the raw design spaces to their properties, e.g., total mass, aggregate thermal budget, or power allocation.

Machine-Checkable Certification and Tool Integration

The machine verification of all results in Lean 4 provides a strong foundation for integrating these theoretical guarantees into future architecture verification or MBSE tooling. Proofs are trustworthy and can be leveraged as trusted cores in future automated certification workflows.

Connections with Related Work

Compared with Spivak’s operadic approaches to systems composition and Censi’s resource allocation via profunctors, this sheaf-theoretic framework directly targets multi-view, non-hierarchical consistency rather than modular composition or two-domain co-design. Related categorical ontology frameworks and earlier sheaf semantics for information integration are extended here to explicit design-parameter spaces and interface-level constraints, with the additional contribution of machine-verified local-to-global theorems specialized to MBSE.

Directions for Future Work

Extensions of this framework include:

• Adopting non-discrete or richer topologies on the architectural site to account for continuous parameter spaces.
• Generalizing the target categories for the presheaf, thus capturing richer classes of quantitative or partial-order constraints (e.g., metric spaces, posets, probability spaces).
• Lifting the framework from sheaves of sets/types to higher sheaves or stacks, addressing cases where design decisions are defined up to equivalence or symmetry.
• Deepening integration with categorical tools such as adjunctions, limits, and Kan extensions for more expressive systems engineering methodologies.

Conclusion

This formalization precisely characterizes the necessary and sufficient conditions for integrating multiple engineering views in cyber-physical systems, embedding the consistency problem into a sheaf-theoretic model with machine-verifiable guarantees. The framework eliminates ambiguity from practical MBSE workflows, providing both a semantic foundation and a verification pathway. By reducing global consistency verification to pairwise interface checks and propagating consistency functorially, the approach advances both the theoretical rigor and practical reliability of integrated engineering design.