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Magnitude Homology Is the Associated Graded of the Length Filtration

Luciano Melodia

arXiv:2608.21479v1math.ATcs.CGcs.LGcs.LO

TL;DR

Magnitude homology records length levels, while its persistent refinement records sublevel-set behavior without directly identifying the relationship between graded and persistent information. The paper filters the length nerve, identifies magnitude homology as the associated graded, and links both through a long exact sequence. It then derives barcode critical-value and stability results and applies the construction to quantitative equational theories.

  • Problem

    Magnitude homology is graded by length and persistent magnitude homology is persistence-based, but their relationship as constructions is not established.

  • Method

    The paper filters normalized nerve chains by length, identifies the associated graded with the magnitude complex, and connects the resulting homologies through a long exact sequence.

  • Results

    The construction locates barcode critical values through nonzero magnitude homology, gives a stability bound of (n + 1)δ in degree n, and yields bounded barcode comparisons for inclusions of theories.

  • Takeaways & Limitations

    Graded magnitude-homology computations list lengths at which barcode endpoints can occur, while theory inclusions compare barcodes and quantify axiomatic strength.

  • Takeaways & Limitations

    The invariant can miss slack in the triangle inequality at a single length, and its stability is (n + 1)δ rather than nonexpansive.

Abstract

from arXiv · show

Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, while a computed perturbation moves a barcode by more than $δ$, so the factor cannot be dropped. We apply this to quantitative equational theories, whose free algebras are metric spaces built from syntax: an inclusion of theories induces a morphism of the presenting monads and a comparison of barcodes with an explicit bound, so the invariant measures axiomatic strength. Four examples are computed, one in every degree.

1 Introduction

The paper identifies magnitude homology and persistent magnitude homology as two views of one length-filtered nerve construction. It connects their algebraic structures, stability behavior, and application to quantitative equational theories.

  • Main construction: Magnitude homology is the associated graded of the length filtration on normalized nerve chains.The persistent theory uses sublevel sets, while magnitude homology uses level sets of length.
  • Main construction: A long exact sequence ties magnitude homology to persistent magnitude homology and shows that barcode endpoints occur at lengths where magnitude homology is nonzero.Thus graded computations identify possible critical values of the persistence module.
  • Stability: Moving every distance by at most δ moves the persistence module by at most (n + 1)δ in degree n.A computed perturbation moves the barcode by more than δ, so the factor cannot be dropped.
  • Applications: An inclusion of quantitative equational theories induces a morphism of presenting monads and a barcode comparison with an explicit bound.The resulting invariant measures axiomatic strength.
  • Paper organization: The paper develops the construction through the length nerve, barcode stability, theory morphisms, and four computed examples.Sections 2–7 establish the framework, while Section 8 computes the examples.

2 The Base of Enrichment

The enrichment base is the ordered monoidal category of extended nonnegative distances, whose enriched categories are extended quasi-pseudometric spaces. Its symmetric separated subcategory is Met, with nonexpansive maps as morphisms.

  • The enrichment base: The enrichment category has objects [0, ∞], order ≥, tensor addition, unit 0, and a truncated subtraction-like operation ⊖.Its tensor is not the cartesian product, and the unit is terminal.
  • Enriched categories: A category enriched in this base is a set equipped with an extended, possibly asymmetric distance satisfying d(x, x) = 0 and the triangle inequality.Distances may be infinite, vanish off the diagonal, or fail to be symmetric.
  • Enriched functors: Enriched functors are exactly nonexpansive maps satisfying d(x, x′) ≥ d(f x, f x′).The enrichment axioms impose no additional data because the base category is thin.
  • The category Met: Met is the full subcategory of enriched categories with symmetric distances and separation, so distance zero implies equality.Its morphisms are the same nonexpansive maps as in the ambient enriched category.
  • Finite-distance restriction: Finite-distance equivalence classes of an enriched category form Vfin-categories, restricting attention to pairs mutually connected by finite distances.This relation is reflexive, symmetric, and transitive by the triangle inequality in both directions.

3 Quantitative Equational Theories and Their Monad

Quantitative equational theories are built from graded syntax, deduction rules, and metric models, yielding free quantitative algebras and a presenting metric term monad. The construction links syntax, semantics, and monadic structure.

  • Syntax and free algebras: Terms over variables are generated by operation symbols in nesting-depth grades, with variables at the base and operations applied to earlier grades.The recursive extension g♯ evaluates variables through g and operations through the algebra structure.
  • Syntax and free algebras: The term construction TΩ is a functor, and substitution is its Kleisli extension under the monad (TΩ, η, µ).The multiplication µ flattens terms of terms.
  • Theories and deduction: Quantitative equations and inferences are generated syntactically, while deducibility is the least relation closed under reflexivity, symmetry, metric rules, nonexpansiveness, substitution, cut, and assumptions.Closure under arbitrary intersections constructs the theory induced by a set of basic inferences.
  • Metric semantics: A quantitative algebra is an extended metric algebra whose operations are nonexpansive, and assignments interpret terms while testing quantitative equations.Models of a theory form a full subcategory of quantitative algebras.
  • Free models and monads: The induced distance dU on terms is an extended pseudometric, whose zero-distance quotient lies in Mod(U), with deduction sound and complete for those models.The forgetful functor has a left adjoint FreeU and is monadic.
  • Free models and monads: The metric term monad TU = GU FreeU presents the category of models, making Mod(U) equivalent to the category of TU-algebras over Met.The paper notes that the monadicity statement transfers from metric equational theories to the discrete-arity setting used here.

4 The Length Nerve and Its Two Homologies

The length nerve yields both persistent magnitude homology through sublevel filtrations and magnitude homology through associated graded pieces. Their chain-level construction also identifies the relevant differentials, functoriality, and degree-zero Vietoris–Rips behavior.

  • Persistent magnitude homology: The length nerve is filtered by simplex-length sublevel sets, whose normalized chain complexes form an inclusion-indexed persistence module.Nonexpansive maps preserve this filtration and induce morphisms of persistence modules.
  • Degree zero: In degree zero, persistent magnitude homology is the persistent homology of the Vietoris–Rips filtration.For finite spaces over a field, its barcode is the single-linkage dendrogram.
  • Magnitude homology: The associated graded at length ℓ is generated by nondegenerate tuples of exact length ℓ, producing the magnitude complex.Its differential deletes an interior point precisely when the corresponding triangle inequality is an equality; outer faces vanish.
  • Magnitude homology: Magnitude homology detects equality in the triangle inequality but not the amount of inequality slack.When no nontrivial equality occurs, the differential vanishes and the homology is free on nondegenerate tuples of that length.
  • Examples: Every finite-distance pair yields a degree-one cycle born no later than its distance and dead no later than twice that distance.For spaces with all distances equal to e, these classes are born at e and die by 2e.

5 The Comparison Theorem

The persistence module and magnitude homology are two views of one length filtration. A long exact sequence relates their changes, allowing magnitude homology to locate possible barcode endpoints.

  • Comparison: Magnitude homology is the associated graded of the length-filtered persistence module.The filtration records sublevel-set homology, while the graded pieces record exact-length contributions.
  • Comparison: A natural long exact sequence connects persistence groups before and at length ℓ with magnitude homology at ℓ.Its maps include the persistence transition, the quotient to the associated graded, and a connecting homomorphism.
  • Barcode critical values: If magnitude homology vanishes in degrees n and n+1 at ℓ, the persistence transition into ℓ is an isomorphism.Thus no degree-n barcode endpoint can occur there under the stated finite-space hypothesis.
  • Barcode critical values: For finite spaces over a field, every birth and death in degree n occurs at a length where magnitude homology is nonzero in degree n or n+1.Magnitude homology therefore bounds the candidate critical lengths, while the persistence module records interval survival.

6 Persistence Modules, Barcodes and Stability

Persistence modules decompose into interval bars, and their interleaving distance quantifies stability. Under pointwise distance perturbations, degree n changes are bounded by (n + 1)δ, with the factor shown to be necessary.

  • Scope: Tameness requires finite dimension at every length, but many theories have infinite total chain rank despite finite-dimensional homology in each computed degree.The examples use finite spaces, where degreewise homology finiteness is immediate.
  • Barcodes: Tame persistence modules over a field decompose into interval modules, with the multiset of intervals determined up to bijection.The intervals constitute the barcode, whose finite endpoints are deaths and whose lower endpoints are births.
  • Stability: The interleaving distance is defined by mutually compatible shifts of two persistence modules and is symmetric with the triangle inequality.Persistence modules thereby form a symmetric V-category, becoming separated only after quotienting zero distance.
  • Stability: (n + 1)δ bounds the interleaving distance between degree-n persistence modules when corresponding metrics differ pointwise by at most δ.The factor counts the distances summed in simplicial degree n+1, the top degree needed to compute Hn.
  • Stability: The degree-n invariant is Lipschitz but not nonexpansive for the supremum metric on distances.A computed perturbation shows barcode movement exceeding δ, so the factor cannot generally be removed.

7 The Composite Construction

Applying the construction to free algebras of quantitative equational theories produces theory-dependent persistence modules and magnitude homology. Inclusions of theories induce monad morphisms and explicit barcode comparisons, while examples illustrate the resulting invariant.

  • Composite construction: The composite construction applies persistent magnitude homology to free algebras generated by metric spaces.The resulting barcode depends only on the theory’s axioms and the generating metric space.
  • Composite construction: The homology of each associated graded piece of the free-algebra filtration is the free algebra’s magnitude homology.This identification is natural in the generating metric space.
  • Theory comparison: An inclusion U ⊆ U′ induces a natural transformation between the presenting metric term monads and a corresponding comparison of persistence modules.The induced components are surjective and nonexpansive; under bijectivity and a δ distance bound, stability applies.
  • Examples: The computations compare a hexagon, a free quantitative semilattice, an axiom-constrained semilattice, and a perturbed hexagon.The displayed examples include distance collapse under an axiom and degree-dependent barcode scattering under perturbation.
  • Theory comparison: The axioms added by U′ shorten free-algebra distances by at most δ, so degree-n barcodes move by at most (n + 1)δ.This makes the barcode a measure of axiomatic strength within the stated comparison setting.

8 Computations

Four examples test the barcode computations across metric spaces and quantitative theories, showing how graded magnitude homology identifies barcode endpoint lengths and how theory axioms alter the resulting invariant.

  • Computation checks: All barcodes are computed over F2 and checked against single-linkage clustering, degree-one generators, and critical lengths.These checks compare PH0, MH1,ℓ, and barcode endpoints with independent structural expectations.
  • Example 8.1: Example 8.1 finds nonzero MHn,ℓ(X) in degrees n ≤ 2 only at lengths {0, 1, 3, 4}.Every barcode endpoint in those degrees is among these lengths or a length carried by degree 3.
  • Example 8.2: Uniform seven-point semilattice spaces have 6n+1 finite bars in degree n, all equal to [n, n + 1), plus one essential degree-zero bar.More generally, the uniform space on m points has (m − 1)n+1 bars.
  • Example 8.3: Adding the axiom ⊢x ∨y =2/5 x reduces the degree-one finite bars from 196 to 146 and yields bottleneck distance 1/2 in degrees 0 and 1.The axiom shortens some distances by δ = 2 − 2/5, and the observed bottleneck distance lies within the (n + 1)δ bound.
  • Example 8.4: Perturbing the hexagon distances by δ = 1/10 produces bottleneck distances 0.082, 0.164, and 0.246 in degrees 0, 1, and 2.The distances grow linearly with degree and exceed δ from degree 1 onward.

9 What the Invariant Sees

The invariant records both length-critical information and persistence, but its graded and filtered forms have distinct limitations concerning slack, stability, and theory dependence.

  • What the invariant sees: Magnitude homology is blind to slack in the triangle inequality at a single length, whereas persistence records the resulting interval through bar length.A face survives on the associated graded only when d(x, y) + d(y, z) = d(x, z).
  • Stability: The stability bound (n + 1)δ is degree-dependent: Example 8.4 exceeds 2.45δ in degree 2, so no constant below 2.45 covers every degree.The proposed Vietoris–Rips maximum filtration would be degree-independent but does not have the magnitude complex as its associated graded.
  • Theory dependence: Theory inclusion compares barcodes with an explicit bound, but Example 8.3 shows that shortening a distance by 2 − 2/5 moves the barcode by only 1/2.The gap between the distance change and barcode movement is left open for some classes of theories.
  • Open directions: For finite metric spaces, the length filtration is bounded below, exhaustive when distances are finite, and has finitely many values per degree.These properties support a spectral sequence with magnitude homology as its E1 page, while its later differentials remain unknown.
  • Length nerve: A simplicial n-simplex is an ordered tuple of objects, with faces deleting entries and degeneracies repeating them.The length nerve filters these tuples by the sum of consecutive distances, λx.

B Proofs of Section 2

The proofs establish structural properties of the enrichment base V, including closure, completeness, symmetric monoidality, and the fact that all diagrams commute because V is thin.

  • Thinness: V is thin, so any two parallel morphisms agree and every diagram in V commutes.This discharges coherence conditions for enrichment, monoidal structure, limits, and the closed adjunction.
  • Limits and colimits: Limits and colimits in V are respectively suprema and infima of the diagram’s object values.The empty diagram gives terminal object 0 and initial object ∞.
  • Closed structure: The truncated subtraction operation c ⊖a satisfies a + b ≥c exactly when b ≥c ⊖a.The proof checks the cases c = ∞, c ≤a, and a < c < ∞.
  • Monoidal structure: The unit 0 is terminal, making V semicartesian, and addition supplies its symmetric monoidal structure.Associators, unitors, and the symmetry are identities because their sources and targets coincide.
  • Monoidal structure: The tensor product is addition, while the product of two objects is their maximum, so 1 ⊗1 = 2 but 1 × 1 = 1.The distinction shows that tensor and categorical product are not interchangeable in V.

C Proofs of Section 3

The proofs construct free algebras from syntax, establish the associated monad, and verify freeness, termination, disjoint term representation, and functoriality.

  • Free algebra construction: The coproduct construction separates variables from operation applications, yielding a fixed-point syntax set TΩW.The injections are used to distinguish the two summands and encode terms uniquely.
  • Termination: Finite tuples of terms lie at a common grade determined by the maximum component grade, enabling recursive evaluation.The recursion calls g♯ only at strictly smaller grade and therefore terminates by induction.
  • Unique representation: Pairwise-disjoint syntax summands make term readings unique: equal operation terms have the same operation, arity, and arguments.This injectivity supports the well-defined algebra operations and the freeness proof.
  • Freeness: The recursive extension g♯ is the unique Ω-homomorphism extending a map g: W →TΩW′.Its two defining clauses establish both the unit condition and preservation of operations.
  • Monad structure: The unit η and multiplication µ are natural, and the monad laws follow as identities of natural transformations.The graphical calculus represents η as a dot and µ as a merge of two wires.

D Proofs of Section 4

Section 4 proves that length sublevel sets form a simplicial filtration, while faces and degeneracies do not increase simplex length. In degree zero, persistent homology is described by graph components and magnitude chains are controlled by length-preserving faces.

  • Length filtration: Faces and degeneracies preserve the length sublevel sets because simplicial reindexing cannot increase the sum of consecutive distances.Inner faces decrease length by the triangle-inequality defect, while degeneracies insert zero-length steps.
  • Length filtration: An inner face preserves length exactly when the corresponding triangle inequality is an equality.The length difference is d(x_i−1, x_i) + d(x_i, x_i+1) − d(x_i−1, x_i+1).
  • Functoriality: Nonexpansive maps induce simplicial maps between the filtered nerves and restrict to every length sublevel.Distance inequalities summed over consecutive entries show that the induced map does not increase simplex length.
  • Degree-zero chains: Degree-zero persistent homology is the free abelian group on connected components of the graph whose edges have distance at most ℓ.The graph is the 1-skeleton of the Vietoris–Rips complex at scale ℓ.
  • Degree-zero chains: The degree-zero barcode is the merge-tree barcode: each merge contributes a bar [0, u-height), while each component contributes [0, ∞).Here u(x, y) is the smallest scale at which a chain joins x to y.

E Proofs of Section 5

Section 5 constructs a short exact sequence from the strict and non-strict filtered chain complexes and derives a long exact sequence linking persistent homology with magnitude homology. This sequence identifies barcode endpoints through nonvanishing magnitude groups.

  • Exact sequences: The filtered chain complexes before ℓ, at ℓ, and their quotient form a short exact sequence whose quotient is the magnitude complex.The inclusion is injective, the quotient map is surjective, and its kernel is precisely the strict sublevel complex.
  • Exact sequences: The short exact sequence yields a natural long exact homology sequence connecting PH_n(X)(ℓ−), PH_n(X)(ℓ), and MH_n,ℓ(X).Over a field, exactness is preserved because the magnitude chain groups are free abelian degreewise.
  • Running example: In the running example, the barcode is [0, ∞), [0, 2), and [0, 1), with births at 0 and deaths at 1 and 2.At ℓ = 3 both magnitude groups vanish and the barcode does not change, despite the block boundary.
  • Barcode endpoints: A birth at ℓ makes the transition map τ_n,ℓ fail to be surjective, forcing MH_n,ℓ(X; F) ≠ 0.Thus births occur among lengths detected by degree-n magnitude homology.
  • Barcode endpoints: A death at ℓ makes τ_n,ℓ fail to be injective, forcing MH_n+1,ℓ(X; F) ≠ 0.The next magnitude-homology degree detects finite right endpoints.

F Proofs of Section 6

Section 6 develops interval decompositions and interleaving distance for persistence modules. It proves interval-module uniqueness and shows that interleaving distance satisfies the triangle inequality but is not separated before quotienting.

  • Interval decomposition: Every pointwise finite-dimensional persistence module decomposes as a direct sum of interval modules.The indexing order is countable and dense in the relevant order topology, enabling the interval-decomposition theorem.
  • Interval decomposition: Interval modules are indecomposable because their endomorphism rings are isomorphic to the field and therefore local.The proof identifies every natural endomorphism with multiplication by one scalar across the interval.
  • Interval decomposition: Interval decompositions are unique up to a bijection of summands because interval modules are indecomposable with local endomorphism rings.Isomorphic interval modules have identical support intervals, determined by the dimensions of their components.
  • Interleaving distance: Composing a δ-interleaving with an ε-interleaving produces a (δ + ε)-interleaving.The construction uses whiskering, vertical composition, and the interchange law for natural transformations.
  • Interleaving distance: Interleaving distance is a pseudometric rather than a metric: distinct interval modules F[0,1) and F[0,1] can have distance zero.It becomes a metric only after quotienting by the relation dI = 0.

G Proofs of Section 7

The proofs establish the functorial connection between magnitude homology, persistent homology, and quantitative equational theories, including monad comparisons and the stability bound. They also explain the bound’s sharp dependence on degree and support the computed barcode examples.

  • Magnitude and persistent homology: The length filtration of the enriched nerve yields the persistence module PH_n(T_U A), while its associated graded gives magnitude homology.The construction is functorial in the metric-space input and identifies the resulting groups by equality, not merely isomorphism.
  • Monad comparison: An inclusion U ⊆ U′ makes Mod(U′) a full subcategory of Mod(U), enabling a comparison map q: T_U ⇒ T_U′ between the free-algebra monads.The proof establishes restriction, uniqueness, generation, naturality, and the monad axioms for q.
  • Monad comparison: The comparison map q is surjective and nonexpansive, and applying persistent homology produces a natural transformation between the corresponding persistence modules.The construction uses the functor PH_n on the nonexpansive map q_A.
  • Stability bound: The proof requires q_A to be bijective and assumes d_TU A(t, s) ≤ d_TU′ A(q_A t, q_A s) + δ before constructing the auxiliary metric space Y.These hypotheses let the two metric spaces share an underlying set and satisfy the comparison inequalities needed for stability.
  • Stability bound: (n + 1)δ is the interleaving bound in degree n when each relevant distance changes by at most δ.An n-simplex calculation gives at most n + 1 affected distances, each contributing at most δ; the resulting interleaving passes through an intermediate metric space.
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