Milestone 2: Sheaf Reveal
Sheaf Enriched Autonomous Multi-Agent Networks (SEAMAN)
January 28, 2026
The COMPASS Problem
Context: Multi-agent systems (UAVs/UUVs) in communication-impaired environments.
COMPASS Problem:
- Conflicts arise in decentralized decision making
- Disagreement about information and goals
- Standard network control methods such as multi-agent consensus fail for logical or task-based data.
The SEAMAN solution:
- Use cellular sheaves to model local-to-global consistency.
- Use enriched categories to unify logical constraints, assumptions/guarantees, and numerical costs.
- Consistent assigments to sheaves called global sections represent feasible operational plans.
The SEAMAN Solution
- Milestone 2: Translate task agreement problem into a cellular sheaf.
- Milestone 3: Use sheaf Laplacian to compute global sections of the cellular sheaf.
- Milestone 4: Proof-of-concept demo in the Georgia Tech Robotarium.
Abstractions for Mission Requirements
Complex multi-agent missions operate on multiple layers of abstraction.
- High-Level specificaiton: “Eventually visit Region A and avoid Region B.”
- Low-level controllers: “Find \(u(t)\) such that \(\dot{x} = f(x,u)\) minimizing \(\int_{t_i}^{t_f} \ell\left(u(t)\right) dt\).”
Related work on Layered Control Architectures1 2
We focus on the interface between agents
- Heterogeneous: COM, sensors, battery/computational resources, etc.
- Example: Agents with different maps rendevous in an overlapping region.
Key Point
(Dis)agreement between agents is modulated by the level of abstraction.
\(\mathcal{Q}\)-Adjunctions
Adjunctions describe “optimal dictionaries” between different layers of abstraction. They are crucial for constructing the Sheaf Laplacian.
In the literature, a \(\mathcal{B}\)-adjunction is called Galois connection: \(\quad\quad\underline{F}(c) \preceq_{\mathsf{C}'} c' \iff c \preceq_\mathsf{C} \overline{F}(c')\).
Contract Embeddings
We use Galois connections to link local agent specifications with shared specifications via contract embeddings:
We think of…
- \(\underline{F}(c) \in \mathcal{C}_{\Sigma'}\) as being an “abstract embedding” of \(c \in \mathcal{C}_{\Sigma}\)
- \(\overline{F}(c') \in \mathcal{C}_{\Sigma}\) as being an “concrete embedding” of \(c' \in \mathcal{C}_{\Sigma'}\)
Induced Contract Embeddings
We can generate adjunctions automatically from maps \(\Sigma \xrightarrow{f} \Sigma'\) (or from \(\mathcal{A}\mathcal{P} \to \mathcal{A}\mathcal{P}'\)).
Lemma
Let \(f: \Sigma \to \Sigma'\) be a map between variable sets. Suppose \(c = (A,G) \in \mathcal{C}_\Sigma\) and \(c' = (A',G') \in \mathcal{C}_{\Sigma'}\). Then, there exist \(\mathcal{B}\)-functors
defined by
- Pullback (\(f^\ast\)): \(\quad\quad f^*(A', G') = \big(f^{-1}(A'), f^{-1}(G')\big)\)
- Pushforward (\(f_!\)): \(\quad\quad f_!(A, G) = \big(\{\sigma \in \Sigma' : f^{-1}(\sigma) \subseteq A\}, f(G)\big)\)
- Right Adjoint (\(f_\ast\)): \(\quad\quad f_*(A, G) = \big(f(A), \{\sigma \in \Sigma' : f^{-1}(\sigma) \subseteq G\}\big)\)
Interpreting the Functors
- \(f_!(c)\) is the strongest abstract contract implementable by \(c\)
- \(f_\ast(c)\) is the weakest abstract contract requiring \(c\).
Cellular Sheaves of \(\mathcal{Q}\)-Categories
A sheaf attaches a “space of possibilities” to every part of the network.
Let \(G=(V,E)\) be the communication graph. The incidence category (poset) \(\mathcal{G}\) is generated by
- Objects: \(V \sqcup E\)
- Morphisms: An arrow \(v \to e\) iff vertex \(v\) is incident to edge \(e\) written \(v \unlhd e\)
Contract Sheaves
We define a Contract Sheaf as a cellular sheaf valued in \(\mathcal{B}\mathsf{Cat}\).
Stalks: For each agent \(v\), \(F(v) := \mathcal{C}_{\Sigma_v}\) is the quantale of contracts over the agent’s local variables \(\Sigma_v\).
Edge Stalks: For each communication link \(e=\{u,v\}\), \(F(e) := \mathcal{C}_{\Sigma_{uv}}\) is the category of contracts over shared interface variables \(\Sigma_{e}\) between Agent \(u\) and Agent \(v\).
Restriction Maps: Contract embeddings \(\underline{F}_{u \unlhd e}: \mathcal{C}_{\Sigma_u} \to \mathcal{C}_{\Sigma_e}\) which are guaranteed to have right adjoints \(\overline{F}_{u \unlhd e}: \mathcal{C}_{\Sigma_e} \to \mathcal{C}_{\Sigma_u}\).
Global Sections: A tuple of local contracts \((c_v)_{v \in V}\) is a global section if for all edges \(e = \{u,v\}\) \[F_{u \unlhd e}(c_u) = F_{v \unlhd e}(c_v)\]
Tip
With weights \(W: V \times V \to \mathcal{B}\), we can also require refinement in either direction: \(F_{u \unlhd e}(c_u) \preceq F_{v \unlhd e}(c_v)\) or \(F_{u \unlhd e}(c_u) \succeq F_{v \unlhd e}(c_v)\).
Contract Sheaves from Sets
Where \(F: \mathcal{G} \to \mathsf{Set}\) is a cellular sheaf of sets:
- \(F(u) := \Sigma_u\)
- \(F(e) := \Sigma_e\)
- \(F_{u \unlhd e}: \Sigma_u \to \Sigma_e\)
Where \(F: \mathcal{G} \to \mathsf{Set}\) is a cellular sheaf of sets:
- \(F(u) := \Sigma_u\)
- \(F(e) := \Sigma_e\)
- \(F_{u \unlhd e}: \Sigma_u \to \Sigma_e\)
Mine Countermeasure Mission (MCM)
Heterogeneous UUVs must clear mines in a contest littoral zone.
Global View (Interface): The area is divided into sectors (\(\mathcal{A}\mathcal{P}_{e}\)).
Local View (Agents): Each agent uses onboard SLAM, segmenting the area into mapped regions (\(\mathcal{A}\mathcal{P}_{u}\)).
Agent maps differ from each other and the global map due to sensor drift and resolution.
They must agree on a coverage plan (guarantees) and safe zones (assumptions).
The Conflict
Agent \(u\) assumes “Region 3 is safe,” but Agent \(v\) has identified a mine in its local “Region 5.” Both regions physically map to “Sector 3.”
Mine Countermeasure Mission (MCM)
How do we resolve disagreements into sections? Standard consensus algorithms fail here.
Next Step: Introduce a sheaf Laplacian & distributive algorithms to resolve inconsistencies in Milestone 3.
Project Status
Subject Matter Experts (SMEs)
Dr. Matthew Hale
Georgia Tech
Dr. Pierluigi Nuzzo
Berkeley
Dr. Paige North
Utrecht
Dr. Gioele Zardini
MIT
Any Questions?
