Source
Hansen and Ghrist (2019), LIT-1659, §4.2 and Fig. 4.1 of arXiv 1808.01513v2 (pp. 18–21); read in NOTE-1233.
What was actually shown
For graphs, reduction stays pairwise. Given boundary vertices B and interior S, harmonic extension (Prop. 4.1) gives the Schur complement L_G(B, B) − L_G(B, S)L_G(S, S)⁻¹L_G(S, B), which “is the Laplacian of a graph with vertex set B” (Theorem 4.2, cited from Dörfler and Bullo, p. 19): a resistor network reduces to an equivalent network on its terminals.
For sheaves, it need not. The authors ask whether for a sheaf F with boundary B there is a sheaf F_B on a graph with vertex set B whose Laplacian is the reduced one, and answer “In general, Kron reduction is not possible for sheaves” (p. 19). The example (Fig. 4.1, p. 20): a star with three leaves, stalk ℝ² at the centre and ℝ at the leaves and edges, the centre’s maps being the first coordinate, the second coordinate and their sum. Every section is fixed by its value (p, q) at the centre, so the boundary values are (p + q, q, p): a two-dimensional space with basis x₁ = (1, 1, 0)ᵀ, x₂ = (1, 0, 1)ᵀ. A sheaf on a graph with vertex set B has only conditions on pairs of boundary vertices. For x₁ to satisfy the conditions on the edges to v₃, the maps from v₁ and v₂ into them must be zero; for x₂, the maps from v₁ and v₃ on the edges to v₂; and the two together force the remaining maps. Then every restriction map is zero and (1, 0, 0)ᵀ is a section too, so no such sheaf has exactly the two-dimensional space. “The problem is that the internal node allows for constraints between boundary nodes that cannot be expressed by purely pairwise interactions. This fact is a fundamental obstruction to Kron reduction for general sheaves” (p. 20). The reduced constraint is v₁ = v₂ + v₃, a relation among three values.
Where it holds. With vertex stalks of dimension at most 1, sheaf Laplacians are the matrices of factor width at most two, and Theorem 4.3 (pp. 20–21) shows that class is closed under Schur complements, so reduction stays within sheaves on graphs.
What could have come out otherwise: a pairwise sheaf on B with exactly the section space spanned by x₁ and x₂ would refute it; the argument above shows there is none, and it was checked for this reading.
What this does not say
- How common the obstruction is, or for which sheaves beyond the scalar-stalk case reduction still works. The paper gives one example and one sufficient condition.
- What would express the reduced constraint. The paper does not propose hypergraphs, higher cells or any other target for the reduction.
- Anything about dynamics. The statement is about sections and the Schur complement. It does not say what a boundary observer of a relaxing interior sees over time; that is CLAIM-480’s memory case, which this paper does not treat.
- That the interactions are physical. “Pairwise interactions” here means compatibility conditions on edges of a graph. Reading the centre as a physical subsystem whose couplings the boundary inherits (CLAIM-480, CLAIM-471) is the record’s interpretation.