Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.
Gomi,Magnitude homology of geodesic space, arXiv preprint arXiv:1902.07044 (2019).↑3, 11, 23, 29, 33
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abstract
This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then, on a geodesic metric space, we interpret the description by means of geodesics. The third magnitude homology of a geodesic metric space also admits a description in terms of a simplicial complex. Under an assumption on a metric space, the simplicial description allows us to introduce an invariant of third magnitude homology classes as an intersection number. Finally, we provide a complete description of all the magnitude homology groups of a geodesic metric space which fulfils a certain non-branching assumption.
years
2026 4representative citing papers
Orlik–Solomon sheaf homology on a geometric lattice concentrates in top degree and decomposes as a sum of local OS algebras tensored with top homology of complementary geometric semilattices.
A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.
Presents a geometric framework for measuring AI agent identity via √JSD spaces and magnitude homology, identifies two conditioning mechanisms, and attributes apparent drift to padding artifacts rather than context length.
citing papers explorer
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Homotopy theories via the magnitude-path spectral sequence
Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.
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Orlik--Solomon sheaf homology of geometric lattices
Orlik–Solomon sheaf homology on a geometric lattice concentrates in top degree and decomposes as a sum of local OS algebras tensored with top homology of complementary geometric semilattices.
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Magnitude of metric measure spaces and integrals over geodesics
A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.
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Measuring What Persists: Conditioning Mechanisms and a Geometric Framework for AI Agent Identity
Presents a geometric framework for measuring AI agent identity via √JSD spaces and magnitude homology, identifies two conditioning mechanisms, and attributes apparent drift to padding artifacts rather than context length.