Treat uncertainty not as random noise on one big global picture, but as local beliefs that must agree where they overlap; coherence across overlaps is enforced by descent (gluing), and Bayesian updating becomes “make the locals compatible, then glue.”

• Base space: Choose your cover (e.g., rooms in a house × time windows; or sensor fields of view).
• Stalks (local data): For each open set, store a belief object (distribution, credal set, Dirichlet parameters, subjective-logic opinion, etc.).
• Restrictions: Maps that push beliefs to smaller overlaps (marginalize variables, project coordinates, or calibrate units).
• Compatibility loss: Measure mismatch on overlaps (e.g., KL(P|Q)+KL(Q|P), Wasserstein, or an interval/credal distance).
• Inference = sheaf gluing: Minimize total incompatibility subject to each node respecting its likelihood/prior. This is the sheaf Laplacian or message-passing analogue; the optimizer returns a glued belief.
• Bayesian updates: New observation = update the local node, then re-solve the coherence problem (fast incremental solvers are possible).
• Diagnostics: Large overlap residuals pinpoint bad sensors, drift, or miscalibration; cohomology classes (if you go that far) flag "global contradictions" that no local tweak can fix.

where P̃ᵢ is the sensor-updated local posterior. The solution gives glued Pᵢ*; take any global section if needed.

• Transport: Move beliefs between contexts via restriction/pushforward; cost = your transport metric (Wasserstein/KL).
• Epistemic: Beliefs may be second-order (uncertainty about parameters); encode as richer stalks (credal sets/intervals, or hyperpriors).
• Objective: Literally optimal transport across overlaps + data terms; descent enforces that the transported beliefs agree.

# Represent the cover and restrictions (graph or poset)
cover = build_cover(rooms, time_windows)

# Pick a belief carrier
# NumPy/SciPy for Gaussians; Dirichlet for discrete
beliefs = {node: GaussianBelief(mu, cov) for node in cover}

# Build overlap list with restriction matrices
overlaps = compute_overlaps(cover)
restrictions = {(i,j): linear_restriction_map(i, j)
                for (i,j) in overlaps}

# Solver: ADMM over overlaps
def sheaf_glue(beliefs, overlaps, restrictions, lambda_reg):
    while not converged:
        # Push to overlaps
        for (i, j) in overlaps:
            Ri, Rj = restrictions[(i,j)]
            mismatch = compute_wasserstein(Ri @ beliefs[i],
                                           Rj @ beliefs[j])
        # Take proximal/gradient steps
        beliefs = update_beliefs(beliefs, mismatch, lambda_reg)
    return beliefs