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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.15778 |
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| _version_ | 1866917499751104512 |
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| author | Ghrist, Robert |
| author_facet | Ghrist, Robert |
| contents | We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator $Φ=A\circ D$ factored into collective distribution $D$ and aggregation $A$; an institution-edge duality identifies it equivalently with the equalizer of the dual operator $D\circ A$ on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category. The construction is functorial under change of coefficient category: a Clearing Invariance Theorem shows that a finite-limit-preserving functor compatible with constraint subobjects induces a canonical isomorphism on global-section objects, enabling uniform comparison of clearing problems across categories of payment data. Existence, uniqueness, and iterative computation of clearing sections are organized by the structure carried on payment objects: Tarski's theorem yields existence and a complete-lattice structure under complete-lattice global elements; Scott continuity refines this to convergent Kleene iteration; an acyclic underlying graph admits a unique clearing section in finitely many steps with no order or metric hypothesis; and Banach's theorem on global elements yields uniqueness under metric contraction. The Eisenberg--Noe model and lattice liability networks arise as special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15778 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Clearing in Liability Networks via Sheaves on Directed Hypergraphs Ghrist, Robert Mathematical Finance Category Theory Primary 91G45, Secondary 91G40, 06B23, 18A30, 18F20 We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator $Φ=A\circ D$ factored into collective distribution $D$ and aggregation $A$; an institution-edge duality identifies it equivalently with the equalizer of the dual operator $D\circ A$ on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category. The construction is functorial under change of coefficient category: a Clearing Invariance Theorem shows that a finite-limit-preserving functor compatible with constraint subobjects induces a canonical isomorphism on global-section objects, enabling uniform comparison of clearing problems across categories of payment data. Existence, uniqueness, and iterative computation of clearing sections are organized by the structure carried on payment objects: Tarski's theorem yields existence and a complete-lattice structure under complete-lattice global elements; Scott continuity refines this to convergent Kleene iteration; an acyclic underlying graph admits a unique clearing section in finitely many steps with no order or metric hypothesis; and Banach's theorem on global elements yields uniqueness under metric contraction. The Eisenberg--Noe model and lattice liability networks arise as special cases. |
| title | Clearing in Liability Networks via Sheaves on Directed Hypergraphs |
| topic | Mathematical Finance Category Theory Primary 91G45, Secondary 91G40, 06B23, 18A30, 18F20 |
| url | https://arxiv.org/abs/2605.15778 |