Learning Transfers: Kan Extensions for Neural Invariants

# Learning Transfers: Kan Extensions for Neural Invariants
Source: [https://arxiv.org/html/2606.07627](https://arxiv.org/html/2606.07627)
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\\NameLuciano Melodia[https://orcid.org/0000-0002-7584-7287](https://orcid.org/0000-0002-7584-7287)\\Emailluciano\.melodia@fau\.de \\addrFriedrich\-Alexander Universität Erlangen\-Nürnberg

###### Abstract

Transfer learning presumes that a representation learned on source tasks carries structure that remains usable on related target tasks\. Standard evaluations probe this through target accuracy or distributional discrepancy, yet leave unspecified which structural invariant is meant to transfer\. We supply that invariant categorically\. A source task category𝒜\\mathcal\{A\}, a target task categoryℬ\\mathcal\{B\}, and a task\-change functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}determine, for every invariant\-valued source representationF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, the universal transferred invariantLanJ⁡F\\operatorname\{Lan\}\_\{J\}F\. Given a target invariantG:ℬ→𝒱G:\\mathcal\{B\}\\to\\mathcal\{V\}, we define the transfer discrepancyCompJ⁡\(F,G\)=supb∈Ob⁡\(ℬ\)d𝒱​\(\(LanJ⁡F\)​\(b\),G​\(b\)\),\\operatorname\{Comp\}\_\{J\}\(F,G\)=\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}\\,d\_\{\\mathcal\{V\}\}\\bigl\(\(\\operatorname\{Lan\}\_\{J\}F\)\(b\),G\(b\)\\bigr\),evaluating transfer not by an objectwise comparison of source and target, but by comparing the target invariant against the one forced by the prescribed task transformation\. We prove finite cokernel formulas for\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)in chain complexes and persistence modules, indexed by the comma categoryJ↓bJ\\downarrow b\. For persistence\-valued finite\-type one\-parameter invariants the discrepancy is computed exactly by bottleneck distances between barcodes\. Controlled experiments on neural latent point clouds then test whether the score recovers the correct task functor and flags representation collapses that preserve classification accuracy while destroying transfer\-relevant topology\.

## 1Introduction

Transfer learning asks whether a representation learned on a source task remains valid once the task changes\. Classical domain\-adaptation theory bounds the target risk by the source risk, a domain discrepancy, and a joint labelling term\(benDavid2010theory\)\. This quantifies the distributional obstruction but leaves open which internal structure ought to transfer: accuracy and distribution alignment say nothing about whether connected components, cycles, or structural identifications survive\. Our claim is that whenever transfer is meant to preserve global invariant structure, the target representation must be compared not with the source directly, but with the invariant induced by the prescribed task transformation\.

Geometric approaches capture local and metric structure\. Feedforward and residual networks can be read as transformations of Riemannian data manifolds\(hauser2017principles\), and generative decoders induce stochastic Riemannian metrics on latent space\(arvanitidis2018latent\)\. Such metrics control infinitesimal distortion and path geometry, but do not by themselves determine homology; global connectivity and cycles require topological invariants\. Persistent homology makes that structure computable inside networks\. Tracking activation manifolds across trained classifiers reveals a reduction in Betti numbers, faster for ReLU networks than for ones with smooth invertible activations\(naitzat2020topology\)\. Related work measures weighted network complexity\(rieck2019neural\), analyses the shape of activation space\(gebhart2019activation\), and studies the cellular decomposition induced by ReLU activation patterns\(bosca2026signatures\)\. Persistent homology has also been combined with a connected commutative Lie\-group model of the data manifold to derive an embedding dimension sufficient to retain the inferred topology, yielding a topology\-informed lower bound on hidden\-layer width\(melodia2021dimension\)\. Topology is therefore not merely an output diagnostic; it can constrain representation dimension\.

Topological information has likewise been folded into learning and algorithmic decisions: regularising the topology of a classifier’s decision boundary\(chen2019topological\), optimising latent connectivity\(hofer2019connectivity\), preserving multi\-scale connectivity in autoencoder codes\(moor2020topological\), and providing differentiable persistence layers\(gabrielsson2020topology\)\. Bottleneck and Wasserstein distances between successive persistence diagrams have served as a stopping criterion for Voronoi interpolation\(melodia2020voronoi\), andH0H\_\{0\}\- andH1H\_\{1\}\-persistence of delay embeddings has enriched a residual11\-dimensional convolutional and stacked\-LSTM classifier on power\-plant sensor signals\(melodia2022sensor\)\. Collectively, these show that persistent invariants can guide architecture, optimisation, stopping, and feature construction\. Transfer\-specific topological methods, however, remain comparison\-based\. Decision\-boundary homology has been used to select among pretrained models\(ramamurthy2019decision\); persistence information has been used to regularise domain adaptation, where persistence alignment alone proves insufficient\(weeks2021domain\); Representation Topology Divergence compares neural representations under distribution shift and transfer\(barannikov2022representation\)\. Each of these compares observed representations or ranks models\. None encodes a specified structural map from source tasks to target tasks, and none computes the target invariant forced by that map\. Optimisation does not dissolve this gap\. Gradient descent moves parameters and need not preserve the topology of the realised representation\. Topology preservation holds only under extra architectural hypotheses: the feature map of a Neural ODE is a homeomorphism and hence preserves input\-space topology, at the cost of expressivity\(chen2018neuralode;dupont2019augmented\), whereas standard ReLU networks obey no such principle and can alter activation topology layer by layer\(naitzat2020topology\)\. Transfer therefore cannot assume topological conservation; it must specify, and then test, the global structure intended to survive or change\. Persistence modules already admit a categorical formulation\. Modelling persistence as functors shows that the interleaving distance extends the bottleneck distance\(bubenik2014categorification\); this extends to modules indexed by arbitrary preorders with functorial stability\(bubenik2015metrics\); and persistence constructions connect to sheaves and cosheaves\(curry2014sheaves\)\. Kan extensions have recently been used to interpolate sampled persistent homology transforms\(arya2025kan\)\.

Our contribution is distinct\. A source task is a small category𝒜\\mathcal\{A\}, a target task a small categoryℬ\\mathcal\{B\}, and a task change a functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}\. For an invariant\-valued representationF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, the left Kan extensionLanJ⁡F\\operatorname\{Lan\}\_\{J\}Fis the universal target\-side invariant induced byJJ\. We compare an observed target invariantG:ℬ→𝒱G:\\mathcal\{B\}\\to\\mathcal\{V\}againstLanJ⁡F\\operatorname\{Lan\}\_\{J\}Frather than againstFF, producing a transfer discrepancy indexed by the stated task transformation and sensitive to the colimit identifications it imposes\. This is the gap we close\. Where prior topological methods compare two given representations or score a single model, none transports the source invariant along a prescribed task map and asks the target to realise the result\. We do exactly this\. The value of the left Kan extension,\(LanJ⁡F\)​\(b\)=colimJ↓bF,\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)=\\operatorname\*\{colim\}\_\{J\\downarrow b\}F,records the source invariants together with every identification forced by the source morphisms visible overbb, and we compare it withG​\(b\)G\(b\)\. Its universal property is decisive:LanJ⁡F\\operatorname\{Lan\}\_\{J\}Fis the*initial*extension ofFFalongJJ, so each deviation ofGGis a structural defect of the transfer rather than an artefact of the metric\. The test thus asks whetherGGrealises the invariant*required by*the specifiedJJ, separating a correct task functor from an incorrect merge, collapse, refinement, or forgetting exactly when these impose distinct colimit invariants\.

### Contributions\.

1. 1\.Invariant\-valued transfer via the left Kan extensionLanJ⁡F\\operatorname\{Lan\}\_\{J\}F\(§[2](https://arxiv.org/html/2606.07627#S2)\)\.
2. 2\.A discrepancyCompJ⁡\(F,G\)\\operatorname\{Comp\}\_\{J\}\(F,G\)characterising exact induced structure \(§[2](https://arxiv.org/html/2606.07627#S2.SS0.SSS0.Px3)\)\.
3. 3\.Comma\-category cokernel formulas for chain and persistence invariants \(§[3](https://arxiv.org/html/2606.07627#S3)and §[4](https://arxiv.org/html/2606.07627#S4)\)\.
4. 4\.Bottleneck evaluation detecting incorrect transformations and collapse \(§[5](https://arxiv.org/html/2606.07627#S5)and §[6](https://arxiv.org/html/2606.07627#S6)\)\.

A transfer of global invariant structure is measured against the structure canonically induced from the source by the stated task transformation, which the left Kan extension produces\.

## 2The Categorical Transfer Framework

All categories are locally small\. We write\[𝒜,𝒱\]\[\\mathcal\{A\},\\mathcal\{V\}\]for functor categories,Nat⁡\(−,−\)\\operatorname\{Nat\}\(\-,\-\)for natural transformations, and𝟏\\mathbf\{1\}for the terminal category\. Let𝕂\\mathbb\{K\}be a field, and denote byVec𝕂\\operatorname\{Vec\}\_\{\\mathbb\{K\}\},Ch𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}, andPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}the categories of𝕂\\mathbb\{K\}\-vector spaces, chain complexes, and persistence modules over\(ℝ,≤\)\(\\mathbb\{R\},\\leq\)\. LetPersMod𝕂⊆Pers𝕂\\operatorname\{PersMod\}\_\{\\mathbb\{K\}\}\\subseteq\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}be the pointwise finite\-dimensional subcategory\. Kan extensions are formed in cocomplete ambient categories, hence inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\.PersMod𝕂\\operatorname\{PersMod\}\_\{\\mathbb\{K\}\}is used only when the result remains pointwise finite\-dimensional\.

### Tasks as small categories\.

We model a task by a small category𝒜\\mathcal\{A\}, with components as objects and admissible structural maps as morphisms\. A task change is a functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}\. In general,𝒱\\mathcal\{V\}is a cocomplete category of representation invariants with a distanced𝒱d\_\{\\mathcal\{V\}\}on isomorphism classes of objects\. In the persistence\-valued specialization below,𝒱=Pers𝕂≔\[\(ℝ,≤\),Vec𝕂\],d𝒱=dI,\\mathcal\{V\}=\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\\coloneqq\[\(\\mathbb\{R\},\\leq\),\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}\],\\ d\_\{\\mathcal\{V\}\}=d\_\{I\},the interleaving distance\. In the finite\-type one\-parameter setting we restrict toPers𝕂ft\\operatorname\{Pers\}^\{\\mathrm\{ft\}\}\_\{\\mathbb\{K\}\}, where objects admit finite barcodes anddId\_\{I\}coincides with the bottleneck distance\.

###### Example 2\.1\.

1. 1\.Merging domains:Let𝒜\\mathcal\{A\}be the discrete category withOb⁡\(𝒜\)=\{a1,a2\}\\operatorname\{Ob\}\(\\mathcal\{A\}\)=\\\{a\_\{1\},a\_\{2\}\\\}, let𝟏\\mathbf\{1\}be the terminal category, and letJ:𝒜→𝟏J:\\mathcal\{A\}\\to\\mathbf\{1\}be the unique functor\. ForF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, the pointwise formula givesLanJ⁡\(F\)​\(∙\)≅∫a∈Ob⁡\(𝒜\)𝟏​\(J​a,∙\)⋅F​\(a\)≅F​\(a1\)⊔F​\(a2\)\\operatorname\{Lan\}\_\{J\}\(F\)\(\\bullet\)\\cong\\int^\{a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)\}\\mathbf\{1\}\(Ja,\\bullet\)\\cdot F\(a\)\\cong F\(a\_\{1\}\)\\sqcup F\(a\_\{2\}\)\. ThusJJmerges two domains, andLanJ\\operatorname\{Lan\}\_\{J\}sends their invariants to the coproduct\. InVec𝕂\\operatorname\{Vec\}\_\{\\mathbb\{K\}\},Ch𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}, andPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, this is the direct sum\.
2. 2\.Class refinement:Let𝒜\\mathcal\{A\}be the discrete category whose objects are the coarse classescc\. Letℬ\\mathcal\{B\}have coarse classescc, fine classesdd, and an arrowρc,d:c→d\\rho\_\{c,d\}:c\\to dprecisely whenddrefinescc\. LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}be the inclusion on coarse classes\. ForF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, the pointwise formula gives LanJ⁡\(F\)​\(b\)≅∫c∈Ob⁡\(𝒜\)ℬ​\(J​c,b\)⋅F​\(c\)≅∐c∈Ob⁡\(𝒜\)ρc,b:J​c→bF​\(c\)\.\\operatorname\{Lan\}\_\{J\}\(F\)\(b\)\\cong\\int^\{c\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)\}\\mathcal\{B\}\(Jc,b\)\\cdot F\(c\)\\cong\\coprod\_\{\\begin\{subarray\}\{c\}c\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)\\\\ \\rho\_\{c,b\}:Jc\\to b\\end\{subarray\}\}F\(c\)\.Thus a fine classddreceives the invariants of all coarse classesccwithρc,d:J​c→d\\rho\_\{c,d\}:Jc\\to d\. Ifddrefines onlycc, thenLanJ⁡\(F\)​\(d\)≅F​\(c\)\\operatorname\{Lan\}\_\{J\}\(F\)\(d\)\\cong F\(c\)\.
3. 3\.Layer collapse:Let\[n\]=\{0,…,n\}\[n\]=\\\{0,\\ldots,n\\\}be the ordinal category, ordered by0≤⋯≤n0\\leq\\cdots\\leq n, let𝟏\\mathbf\{1\}be the terminal category, and letJ:\[n\]→𝟏J:\[n\]\\to\\mathbf\{1\}be the unique functor\. ForF:\[n\]→𝒱F:\[n\]\\to\\mathcal\{V\}, the pointwise formula givesLanJ⁡\(F\)​\(∙\)≅∫i∈\[n\]𝟏​\(J​i,∙\)⋅F​\(i\)≅colimi∈\[n\]F​\(i\)≅F​\(n\)\\operatorname\{Lan\}\_\{J\}\(F\)\(\\bullet\)\\cong\\int^\{i\\in\[n\]\}\\mathbf\{1\}\(Ji,\\bullet\)\\cdot F\(i\)\\cong\\operatorname\*\{colim\}\_\{i\\in\[n\]\}F\(i\)\\cong F\(n\), sincennis terminal in the ordinal category\[n\]\[n\]\. Thus collapsing a directed layer sequence keeps the final propagated invariant, not the coproduct of all layer invariants\.
4. 4\.Forgetting structure:LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}send a structured task to a coarser task\. ForF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}andb∈ℬb\\in\\mathcal\{B\}, the pointwise formula givesLanJ⁡\(F\)​\(b\)≅∫a∈Ob⁡\(𝒜\)ℬ​\(J​a,b\)⋅F​\(a\)≅colim\(a,β:J​a→b\)⁣∈J⁣↓bF​\(a\)\\operatorname\{Lan\}\_\{J\}\(F\)\(b\)\\cong\\int^\{a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)\}\\mathcal\{B\}\(Ja,b\)\\cdot F\(a\)\\cong\\operatorname\*\{colim\}\_\{\(a,\\beta:Ja\\to b\)\\in J\\downarrow b\}F\(a\)\. Hence the value atbbis obtained by gluing all invariantsF​\(a\)F\(a\)equipped with a comparison mapβ:J​a→b\\beta:Ja\\to b\. A morphismα:\(a,β\)→\(a′,β′\)\\alpha:\(a,\\beta\)\\to\(a^\{\\prime\},\\beta^\{\\prime\}\)inJ↓bJ\\downarrow bis a morphismα:a→a′\\alpha:a\\to a^\{\\prime\}in𝒜\\mathcal\{A\}withβ′∘J​α=β\\beta^\{\\prime\}\\circ J\\alpha=\\beta, and the colimit identifiesx∈F​\(a\)x\\in F\(a\)withF​\(α\)​\(x\)∈F​\(a′\)F\(\\alpha\)\(x\)\\in F\(a^\{\\prime\}\)\. ThusLanJ\\operatorname\{Lan\}\_\{J\}does not merely collect source objects\. It also quotients by the source morphisms visible overbb\.

### Invariant categories and models\.

Invariant targets specify the space in which transferred models are compared\. In learning applications, the output of a model may be a chain complex, a persistence module, or another structured invariant, and the loss must be evaluated on objects of that type\. We therefore distinguish the ambient category𝒱\\mathcal\{V\}, where Kan transfer is computed, from the comparison class𝒮\\mathcal\{S\}, where the stability distance is defined\.𝒱\\mathcal\{V\}must contain the colimits needed for transfer, while the metric properties relevant for learning may hold only on a restricted class of objects\.

###### Definition 2\.2\.

Aninvariant targetis a triple\(𝒱,𝒮,d𝒱\)\(\\mathcal\{V\},\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)such that:

1. 1𝒱\\mathcal\{V\}admits all small colimits,
2. 2𝒮⊆Ob⁡\(𝒱\)\\mathcal\{S\}\\subseteq\\operatorname\{Ob\}\(\\mathcal\{V\}\)is the class of objects to be compared,
3. 3d𝒱d\_\{\\mathcal\{V\}\}is an extended pseudometric on𝒮/≅\\mathcal\{S\}/\{\\cong\}\.

Here𝒮/≅\\mathcal\{S\}/\{\\cong\}identifies isomorphic objects of𝒮\\mathcal\{S\}\. Sinced𝒱d\_\{\\mathcal\{V\}\}may still have zero distance between distinct isomorphism classes, we putSep\(𝒮,d𝒱\)≔\(𝒮/≅\)/∼0\\operatorname\{Sep\}\(\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)\\coloneqq\(\\mathcal\{S\}/\{\\cong\}\)/\{\\sim\_\{0\}\}, where\[X\]∼0\[Y\]\[X\]\\sim\_\{0\}\[Y\]iffd𝒱​\(\[X\],\[Y\]\)=0d\_\{\\mathcal\{V\}\}\(\[X\],\[Y\]\)=0\. Thus the separated comparison space first identifies isomorphic objects and then identifies all zero\-distance classes\.

The separation between𝒱\\mathcal\{V\}and𝒮\\mathcal\{S\}is intentional\. The category𝒱\\mathcal\{V\}is the ambient target in which pointwise left Kan extensions are computed, hence it must have the required colimits\. The subclass𝒮\\mathcal\{S\}records where the chosen distance has the intended stability properties\.

The running choices are\(Ch𝕂,𝒮ch,dch\)\(\\operatorname\{Ch\}\_\{\\mathbb\{K\}\},\\mathcal\{S\}\_\{\\mathrm\{ch\}\},d\_\{\\mathrm\{ch\}\}\)for chain\-complex\-valued invariants, where𝒮ch\\mathcal\{S\}\_\{\\mathrm\{ch\}\}is the chosen comparison class anddchd\_\{\\mathrm\{ch\}\}is the chosen chain\-complex pseudometric, and\(Pers𝕂,Pers𝕂q​\-​tame,dI\)\\bigl\(\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\},d\_\{I\}\\bigr\)for one\-parameter persistence, withPers𝕂≔\[\(ℝ,≤\),Vec𝕂\]\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\\coloneqq\[\(\\mathbb\{R\},\\leq\),\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}\]\. The ambient categoryPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}is used for colimits, whileqq\-tame modules are used for comparison because the Isometry Theorem identifiesdId\_\{I\}with the bottleneck distance in this class\(CSGO16, Theorem 4\.11\)\. The interleaving distance is a pseudometric on isomorphism classes\(Les15, Definition 3\.5\), so comparisons are made in the separated quotient when zero\-distance non\-isomorphic objects occur\. Topological spaces are used only upstream, as sources for filtrations before applying chains, homology, or persistent homology\.

For a task category𝒜\\mathcal\{A\}, amodelin𝒱\\mathcal\{V\}is a functorF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}\.

### Exact and approximate transfer\.

Exact transfer requiresLanJ⁡\(F\)≅G\\operatorname\{Lan\}\_\{J\}\(F\)\\cong Gin\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\]\. Approximate transfer replaces this by the pointwise lossCompJ⁡\(F,G\)\\operatorname\{Comp\}\_\{J\}\(F,G\), which compares\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)withG​\(b\)G\(b\)inSep⁡\(𝒮,d𝒱\)\\operatorname\{Sep\}\(\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)for eachb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. ThusCompJ\\operatorname\{Comp\}\_\{J\}is a computable transfer loss\. It measures objectwise agreement after transport, but not natural compatibility\.

###### Definition 2\.3\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\},F:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, andG:ℬ→𝒱G:\\mathcal\{B\}\\to\\mathcal\{V\}\. The modelFFisexactly Kan\-transferabletoGGalongJJifLanJ⁡\(F\)≅G\\operatorname\{Lan\}\_\{J\}\(F\)\\cong Gin\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\]\.

###### Definition 2\.4\.

Let\(𝒱,𝒮,d𝒱\)\(\\mathcal\{V\},\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)be an invariant target\. ForJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\},F:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, andG:ℬ→𝒱G:\\mathcal\{B\}\\to\\mathcal\{V\}, assume\(LanJ⁡F\)​\(b\),G​\(b\)∈𝒮\(\\operatorname\{Lan\}\_\{J\}F\)\(b\),G\(b\)\\in\\mathcal\{S\}for allb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Thetransfer discrepancyis

CompJ⁡\(F,G\)≔supb∈Ob⁡\(ℬ\)d𝒱​\(\[\(LanJ⁡F\)​\(b\)\],\[G​\(b\)\]\)∈\[0,∞\]\.\\operatorname\{Comp\}\_\{J\}\(F,G\)\\coloneqq\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}d\_\{\\mathcal\{V\}\}\(\[\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\],\[G\(b\)\]\)\\in\[0,\\infty\]\.The modelFFisε\\varepsilon\-Kan\-transferable toGGifCompJ⁡\(F,G\)≤ε\\operatorname\{Comp\}\_\{J\}\(F,G\)\\leq\\varepsilon\.

Exact transfer is a natural condition in\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\]\. The discrepancy is only pointwise:CompJ⁡\(F,G\)=0\\operatorname\{Comp\}\_\{J\}\(F,G\)=0says that\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)andG​\(b\)G\(b\)define the same point ofSep⁡\(𝒮,d𝒱\)\\operatorname\{Sep\}\(\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)for everybb\. Ifd𝒱d\_\{\\mathcal\{V\}\}is separated, this means\(LanJ⁡F\)​\(b\)≅G​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong G\(b\)for everybb, but it still need not give a natural isomorphism\.

###### Proposition 2\.5\.

There existJ,F,GJ,F,GwithCompJ⁡\(F,G\)=0\\operatorname\{Comp\}\_\{J\}\(F,G\)=0butLanJ⁡\(F\)≇G\\operatorname\{Lan\}\_\{J\}\(F\)\\not\\cong Gin\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\]\.

###### Proof 2\.6\.

Let𝒜=𝟏\\mathcal\{A\}=\\mathbf\{1\}\. Letℬ\\mathcal\{B\}be the walking parallel pair0​⇉f,g​10\\overset\{f,g\}\{\\rightrightarrows\}1, letJ:𝟏→ℬJ:\\mathbf\{1\}\\to\\mathcal\{B\}send the unique object to0, let𝒱=Vec𝕂\\mathcal\{V\}=\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}, and letF​\(∙\)=𝕂F\(\\bullet\)=\\mathbb\{K\}\. Then\(LanJ⁡F\)​\(0\)≅𝕂\(\\operatorname\{Lan\}\_\{J\}F\)\(0\)\\cong\\mathbb\{K\}and\(LanJ⁡F\)​\(1\)≅𝕂⊕𝕂\(\\operatorname\{Lan\}\_\{J\}F\)\(1\)\\cong\\mathbb\{K\}\\oplus\\mathbb\{K\}, with\(LanJ⁡F\)​\(f\)​\(x\)=\(x,0\)\(\\operatorname\{Lan\}\_\{J\}F\)\(f\)\(x\)=\(x,0\)and\(LanJ⁡F\)​\(g\)​\(x\)=\(0,x\)\(\\operatorname\{Lan\}\_\{J\}F\)\(g\)\(x\)=\(0,x\)\. DefineG:ℬ→Vec𝕂G:\\mathcal\{B\}\\to\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}byG​\(0\)=𝕂G\(0\)=\\mathbb\{K\},G​\(1\)=𝕂⊕𝕂G\(1\)=\\mathbb\{K\}\\oplus\\mathbb\{K\}, andG​\(f\)​\(x\)=G​\(g\)​\(x\)=\(x,0\)G\(f\)\(x\)=G\(g\)\(x\)=\(x,0\)\. With any separated comparison distance depending only on vector\-space isomorphism classes, the two functors are pointwise at distance0\. HenceCompJ⁡\(F,G\)=0\\operatorname\{Comp\}\_\{J\}\(F,G\)=0\.

Suppose thatσ:LanJ⁡\(F\)⇒G\\sigma:\\operatorname\{Lan\}\_\{J\}\(F\)\\Rightarrow Gis a natural isomorphism\. Chooseλ∈𝕂×\\lambda\\in\\mathbb\{K\}^\{\\times\}withσ0​\(1\)=λ\\sigma\_\{0\}\(1\)=\\lambda\. Naturality atffgivesσ1​\(λ,0\)=\(λ,0\)\\sigma\_\{1\}\(\\lambda,0\)=\(\\lambda,0\), and naturality atgggivesσ1​\(0,λ\)=\(λ,0\)\\sigma\_\{1\}\(0,\\lambda\)=\(\\lambda,0\)\. Henceσ1​\(e1\)=e1\\sigma\_\{1\}\(e\_\{1\}\)=e\_\{1\}andσ1​\(e2\)=e1\\sigma\_\{1\}\(e\_\{2\}\)=e\_\{1\}, contradicting invertibility ofσ1\\sigma\_\{1\}\.

Thus approximate transfer measures objectwise compatibility after transport alongJJ\. It does not, by itself, measure whether the transported morphisms agree naturally with those ofGG\. A natural\-isomorphism\-detecting variant must put the comparison distance on the functor category\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\], for instance by using an induced interleaving distance when such a structure is available\(bubenik2014categorification;SMS18, §3\)\.

### Structure preservation\.

Kan transfer is not an objectwise construction: it acts on the full diagramF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}\. Hence the transferred invariant can depend on the morphisms ofFF, not only on the objectsF​\(a\)F\(a\)up to isomorphism\. This records structural information that objectwise summaries discard\.

###### Proposition 2\.7\.

Left Kan transfer is sensitive to the morphisms of the source diagram\.

###### Proof 2\.8\.

Let𝒜\\mathcal\{A\}be the category0​⇉f,g​10\\overset\{f,g\}\{\\rightrightarrows\}1, letJ:𝒜→𝟏J:\\mathcal\{A\}\\to\\mathbf\{1\}be the unique functor, and defineF,F′:𝒜→Vec𝕂F,F^\{\\prime\}:\\mathcal\{A\}\\to\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}byF​\(0\)=F​\(1\)=F′​\(0\)=F′​\(1\)=𝕂F\(0\)=F\(1\)=F^\{\\prime\}\(0\)=F^\{\\prime\}\(1\)=\\mathbb\{K\},F​\(f\)=F​\(g\)=id𝕂F\(f\)=F\(g\)=\\operatorname\{id\}\_\{\\mathbb\{K\}\},F′​\(f\)=id𝕂F^\{\\prime\}\(f\)=\\operatorname\{id\}\_\{\\mathbb\{K\}\}, andF′​\(g\)=0F^\{\\prime\}\(g\)=0\. ThenF​\(a\)≅F′​\(a\)F\(a\)\\cong F^\{\\prime\}\(a\)for alla∈Ob⁡\(𝒜\)a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)\. SinceJJsends𝒜\\mathcal\{A\}to𝟏\\mathbf\{1\}, the value of the transfer is the coequalizer of the two arrows\. Hence\(LanJ⁡F\)​\(∙\)≅coker⁡\(id𝕂−id𝕂\)≅𝕂\(\\operatorname\{Lan\}\_\{J\}F\)\(\\bullet\)\\cong\\operatorname\{coker\}\(\\operatorname\{id\}\_\{\\mathbb\{K\}\}\-\\operatorname\{id\}\_\{\\mathbb\{K\}\}\)\\cong\\mathbb\{K\}, whereas\(LanJ⁡F′\)​\(∙\)≅coker⁡\(id𝕂−0\)≅0\(\\operatorname\{Lan\}\_\{J\}F^\{\\prime\}\)\(\\bullet\)\\cong\\operatorname\{coker\}\(\\operatorname\{id\}\_\{\\mathbb\{K\}\}\-0\)\\cong 0\.

###### Theorem 2\.9\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, let\(𝒱,𝒮,d𝒱\)\(\\mathcal\{V\},\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)be an invariant target, and letF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}\. Assume thatLanJ⁡\(F\)\\operatorname\{Lan\}\_\{J\}\(F\)exists and that\(LanJ⁡F\)​\(b\)∈𝒮\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\in\\mathcal\{S\}for allb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Forε≥0\\varepsilon\\geq 0, set𝒯ε​\(J,F\)≔\{G:ℬ→𝒱∣G​\(b\)∈𝒮​for all​b∈Ob⁡\(ℬ\),CompJ⁡\(F,G\)≤ε\}\\mathcal\{T\}\_\{\\varepsilon\}\(J,F\)\\coloneqq\\\{G:\\mathcal\{B\}\\to\\mathcal\{V\}\\mid G\(b\)\\in\\mathcal\{S\}\\text\{ for all \}b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\),\\operatorname\{Comp\}\_\{J\}\(F,G\)\\leq\\varepsilon\\\}\. Then:

1. 1LanJ⁡\(F\)∈𝒯0​\(J,F\)\\operatorname\{Lan\}\_\{J\}\(F\)\\in\\mathcal\{T\}\_\{0\}\(J,F\)\.
2. 2𝒯ε​\(J,F\)\\mathcal\{T\}\_\{\\varepsilon\}\(J,F\)is closed under natural isomorphism\.
3. 3G∈𝒯0​\(J,F\)G\\in\\mathcal\{T\}\_\{0\}\(J,F\)iff\[\(LanJ⁡F\)​\(b\)\]=\[G​\(b\)\]\[\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\]=\[G\(b\)\]inSep⁡\(𝒮,d𝒱\)\\operatorname\{Sep\}\(\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)for allb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\.
4. 4Ifd𝒱d\_\{\\mathcal\{V\}\}is separated on𝒮/≅\\mathcal\{S\}/\{\\cong\}, thenG∈𝒯0​\(J,F\)G\\in\\mathcal\{T\}\_\{0\}\(J,F\)iff\(LanJ⁡F\)​\(b\)≅G​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong G\(b\)for allb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\.
5. 5Pointwise zero transfer is strictly weaker than exact Kan\-transferability\.

###### Proof 2\.10\.

1 and 2 follow fromd𝒱​\(\[X\],\[X\]\)=0d\_\{\\mathcal\{V\}\}\(\[X\],\[X\]\)=0and from the fact thatd𝒱d\_\{\\mathcal\{V\}\}is defined on isomorphism classes\. 3 is the definition of the quotientSep\(𝒮,d𝒱\)=\(𝒮/≅\)/∼0\\operatorname\{Sep\}\(\\mathcal\{S\},d\_\{\\mathcal\{V\}\}\)=\(\\mathcal\{S\}/\{\\cong\}\)/\{\\sim\_\{0\}\}, where\[X\]∼0\[Y\]\[X\]\\sim\_\{0\}\[Y\]iffd𝒱​\(\[X\],\[Y\]\)=0d\_\{\\mathcal\{V\}\}\(\[X\],\[Y\]\)=0\. 4 is the separated case\. 5 follows from Proposition[2\.5](https://arxiv.org/html/2606.07627#S2.Thmtheorem5)\.

Hence Kan transfer transportsFFalongJJ, andCompJ\\operatorname\{Comp\}\_\{J\}measures the resulting objectwise error in the separated comparison space\. This is computable but ignores naturality\. To detect natural compatibility, one must compare in the functor category, for example by an interleaving distance induced by a flow \(bubenik2014categorification;SMS18\)\.

## 3Chain\-Homotopy Instantiation

We instantiate the comparison target by chain complexes\. This keeps transfer linear, since colimits inCh𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}are computed degreewise, while comparison is made up to chain\-homotopy type rather than strict isomorphism\(Wei94, Ch\. 1\)\.

###### Definition 3\.1\.

Let𝒮ch⊆Ob⁡\(Ch𝕂\)\\mathcal\{S\}\_\{\\mathrm\{ch\}\}\\subseteq\\operatorname\{Ob\}\(\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}\)\. ForC,D∈𝒮chC,D\\in\\mathcal\{S\}\_\{\\mathrm\{ch\}\}, writeC≃chDC\\simeq\_\{\\mathrm\{ch\}\}DifCCandDDare chain\-homotopy equivalent, and define

dch​\(\[C\],\[D\]\)≔\{0,C≃chD,1,otherwise\.d\_\{\\mathrm\{ch\}\}\(\[C\],\[D\]\)\\coloneqq\\begin\{cases\}0,&C\\simeq\_\{\\mathrm\{ch\}\}D,\\\\ 1,&\\text\{otherwise\}\.\\end\{cases\}

###### Proposition 3\.2\.

The triple\(Ch𝕂,𝒮ch,dch\)\(\\operatorname\{Ch\}\_\{\\mathbb\{K\}\},\\mathcal\{S\}\_\{\\mathrm\{ch\}\},d\_\{\\mathrm\{ch\}\}\)is an invariant target\.

###### Proof 3\.3\.

SinceVec𝕂\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}is cocomplete,Ch𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}has degreewise direct sums, coequalizers, and hence all small colimits\(Wei94, §1\.2\)\. Chain\-homotopy equivalence is an isomorphism\-invariant equivalence relation\(Wei94, §1\.4\)\. Therefore the distance from Definition[3\.1](https://arxiv.org/html/2606.07627#S3.Thmtheorem1)descends to𝒮ch/≅\\mathcal\{S\}\_\{\\mathrm\{ch\}\}/\{\\cong\}and is an extended pseudometric\.

For the chain target of\\Crefdef:chain\-target, zero discrepancy is pointwise chain\-homotopy equivalence\. More precisely,CompJ⁡\(F,G\)=0\\operatorname\{Comp\}\_\{J\}\(F,G\)=0iff\(LanJ⁡F\)​\(b\)≃chG​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\simeq\_\{\\mathrm\{ch\}\}G\(b\)for everyb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Consequently,Hm​\(\(LanJ⁡F\)​\(b\)\)≅Hm​\(G​\(b\)\)H\_\{m\}\(\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\)\\cong H\_\{m\}\(G\(b\)\)for everyb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)and everym∈ℤm\\in\\mathbb\{Z\}\.

###### Proposition 3\.4\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, letF:𝒜→Ch𝕂F:\\mathcal\{A\}\\to\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}, and letb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Assume thatJ↓bJ\\downarrow bhas finitely many objects and morphisms\. ThusOb⁡\(J↓b\)\\operatorname\{Ob\}\(J\\downarrow b\)consists of pairs\(a,β\)\(a,\\beta\), wherea∈Ob⁡\(𝒜\)a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)andβ:J​a→b\\beta:Ja\\to b, andMor⁡\(J↓b\)\\operatorname\{Mor\}\(J\\downarrow b\)consists of morphismsα:\(a,β\)→\(a′,β′\)\\alpha:\(a,\\beta\)\\to\(a^\{\\prime\},\\beta^\{\\prime\}\)given by morphismsα¯:a→a′\\bar\{\\alpha\}:a\\to a^\{\\prime\}in𝒜\\mathcal\{A\}satisfyingβ′​J​α¯=β\\beta^\{\\prime\}J\\bar\{\\alpha\}=\\beta\. Letπb:J↓b→𝒜\\pi\_\{b\}:J\\downarrow b\\to\\mathcal\{A\}be the projection,\(a,β\)↦a\(a,\\beta\)\\mapsto a\. Forα∈Mor⁡\(J↓b\)\\alpha\\in\\operatorname\{Mor\}\(J\\downarrow b\), writes​\(α\)=\(aα,βα\)s\(\\alpha\)=\(a\_\{\\alpha\},\\beta\_\{\\alpha\}\)andt​\(α\)=\(aα′,βα′\)t\(\\alpha\)=\(a^\{\\prime\}\_\{\\alpha\},\\beta^\{\\prime\}\_\{\\alpha\}\)\.

Then there is a canonical isomorphism

\(LanJ⁡F\)​\(b\)m≅coker⁡\(⨁α∈Mor⁡\(J↓b\)F​\(aα\)m→rm⨁\(a,β\)∈Ob⁡\(J↓b\)F​\(a\)m\),\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\_\{m\}\\cong\\operatorname\{coker\}\\left\(\\bigoplus\_\{\\alpha\\in\\operatorname\{Mor\}\(J\\downarrow b\)\}F\(a\_\{\\alpha\}\)\_\{m\}\\xrightarrow\{r\_\{m\}\}\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}F\(a\)\_\{m\}\\right\),wherermr\_\{m\}is determined on the summand indexed byα∈Mor⁡\(J↓b\)\\alpha\\in\\operatorname\{Mor\}\(J\\downarrow b\)byx↦ιt​\(α\)​F​\(α¯\)m​\(x\)−ιs​\(α\)​xx\\mapsto\\iota\_\{t\(\\alpha\)\}F\(\\bar\{\\alpha\}\)\_\{m\}\(x\)\-\\iota\_\{s\(\\alpha\)\}x\. The differential is induced by the differentials∂F​\(a\),m:F​\(a\)m→F​\(a\)m−1\\partial\_\{F\(a\),m\}:F\(a\)\_\{m\}\\to F\(a\)\_\{m\-1\}\.

###### Proof 3\.5\.

Fixb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. We use the pointwise formula\(LanJ⁡F\)​\(b\)≅colimJ↓b\(F​πb\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong\\operatorname\*\{colim\}\_\{J\\downarrow b\}\(F\\pi\_\{b\}\)and the fact that colimits inCh𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}are computed degreewise\(Wei94, §1\.2\)\.

1. 1\.Indexing category:By definition,Ob⁡\(J↓b\)\\operatorname\{Ob\}\(J\\downarrow b\)consists of pairs\(a,β\)\(a,\\beta\)witha∈Ob⁡\(𝒜\)a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)andβ:J​a→b\\beta:Ja\\to b\. A morphismα:\(a,β\)→\(a′,β′\)\\alpha:\(a,\\beta\)\\to\(a^\{\\prime\},\\beta^\{\\prime\}\)inMor⁡\(J↓b\)\\operatorname\{Mor\}\(J\\downarrow b\)is a morphismα¯:a→a′\\bar\{\\alpha\}:a\\to a^\{\\prime\}in𝒜\\mathcal\{A\}such thatβ′​J​α¯=β\\beta^\{\\prime\}J\\bar\{\\alpha\}=\\beta\. The projectionπb:J↓b→𝒜\\pi\_\{b\}:J\\downarrow b\\to\\mathcal\{A\}sends\(a,β\)\(a,\\beta\)toaaandα\\alphatoα¯\\bar\{\\alpha\}\.
2. 2\.Degreewise colimit:Form∈ℤm\\in\\mathbb\{Z\}, defineFb,m:J↓b→Vec𝕂F\_\{b,m\}:J\\downarrow b\\to\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}byFb,m​\(a,β\)=F​\(a\)mF\_\{b,m\}\(a,\\beta\)=F\(a\)\_\{m\}andFb,m​\(α\)=F​\(α¯\)mF\_\{b,m\}\(\\alpha\)=F\(\\bar\{\\alpha\}\)\_\{m\}\. SinceJ↓bJ\\downarrow bis finite,colimJ↓b⁡Fb,m\\operatorname\{colim\}\_\{J\\downarrow b\}F\_\{b,m\}is the quotient of⨁\(a,β\)∈Ob⁡\(J↓b\)F​\(a\)m\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}F\(a\)\_\{m\}by the relationsιs​\(α\)​x=ιt​\(α\)​F​\(α¯\)m​\(x\)\\iota\_\{s\(\\alpha\)\}x=\\iota\_\{t\(\\alpha\)\}F\(\\bar\{\\alpha\}\)\_\{m\}\(x\)for allα∈Mor⁡\(J↓b\)\\alpha\\in\\operatorname\{Mor\}\(J\\downarrow b\)and allx∈F​\(aα\)mx\\in F\(a\_\{\\alpha\}\)\_\{m\}\. Equivalently, this quotient is the displayed cokernel ofrmr\_\{m\}\.
3. 3\.Compatibility with differentials:Let∂m=⨁\(a,β\)∈Ob⁡\(J↓b\)∂F​\(a\),m\\partial\_\{m\}=\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}\\partial\_\{F\(a\),m\}\. Forx∈F​\(aα\)mx\\in F\(a\_\{\\alpha\}\)\_\{m\}, one has∂mrm​\(x\)=ιt​\(α\)​∂F​\(aα′\),mF​\(α¯\)m​\(x\)−ιs​\(α\)​∂F​\(aα\),mx\.\\partial\_\{m\}r\_\{m\}\(x\)=\\iota\_\{t\(\\alpha\)\}\\partial\_\{F\(a^\{\\prime\}\_\{\\alpha\}\),m\}F\(\\bar\{\\alpha\}\)\_\{m\}\(x\)\-\\iota\_\{s\(\\alpha\)\}\\partial\_\{F\(a\_\{\\alpha\}\),m\}x\.SinceF​\(α¯\)F\(\\bar\{\\alpha\}\)is a chain map,∂F​\(aα′\),mF​\(α¯\)m=F​\(α¯\)m−1​∂F​\(aα\),m\\partial\_\{F\(a^\{\\prime\}\_\{\\alpha\}\),m\}F\(\\bar\{\\alpha\}\)\_\{m\}=F\(\\bar\{\\alpha\}\)\_\{m\-1\}\\partial\_\{F\(a\_\{\\alpha\}\),m\}\. Hence ∂mrm​\(x\)=ιt​\(α\)​F​\(α¯\)m−1​∂F​\(aα\),mx−ιs​\(α\)​∂F​\(aα\),mx=rm−1​\(∂F​\(aα\),mx\)\.\\partial\_\{m\}r\_\{m\}\(x\)=\\iota\_\{t\(\\alpha\)\}F\(\\bar\{\\alpha\}\)\_\{m\-1\}\\partial\_\{F\(a\_\{\\alpha\}\),m\}x\-\\iota\_\{s\(\\alpha\)\}\\partial\_\{F\(a\_\{\\alpha\}\),m\}x=r\_\{m\-1\}\(\\partial\_\{F\(a\_\{\\alpha\}\),m\}x\)\.Thus∂m\(im⁡rm\)⊆im⁡rm−1\\partial\_\{m\}\(\\operatorname\{im\}r\_\{m\}\)\\subseteq\\operatorname\{im\}r\_\{m\-1\}, so the differentials descend to the cokernels\.
4. 4\.Identification:LetQm=coker⁡\(rm\)Q\_\{m\}=\\operatorname\{coker\}\(r\_\{m\}\)\. The inclusions into the object\-sum induce mapsq\(a,β\):F​\(a\)→Q∙q\_\{\(a,\\beta\)\}:F\(a\)\\to Q\_\{\\bullet\}, and the relations definingrmr\_\{m\}giveq\(a,β\)=q\(a′,β′\)​F​\(α¯\)q\_\{\(a,\\beta\)\}=q\_\{\(a^\{\\prime\},\\beta^\{\\prime\}\)\}F\(\\bar\{\\alpha\}\)for everyα:\(a,β\)→\(a′,β′\)\\alpha:\(a,\\beta\)\\to\(a^\{\\prime\},\\beta^\{\\prime\}\)\. HenceQ∙Q\_\{\\bullet\}has the universal property ofcolimJ↓b\(F​πb\)\\operatorname\*\{colim\}\_\{J\\downarrow b\}\(F\\pi\_\{b\}\)inCh𝕂\\operatorname\{Ch\}\_\{\\mathbb\{K\}\}\. By the pointwise formula,Q∙≅\(LanJ⁡F\)​\(b\)Q\_\{\\bullet\}\\cong\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\.

Finite chain\-level transfer is explicit linear algebra: form the direct sum overOb⁡\(J↓b\)\\operatorname\{Ob\}\(J\\downarrow b\), quotient by the relations indexed byMor⁡\(J↓b\)\\operatorname\{Mor\}\(J\\downarrow b\), and compare the resulting complex bydchd\_\{\\mathrm\{ch\}\}\.

## 4Persistent Instantiation

We now take persistence modules as invariant values\. This keeps the ambient category large enough for Kan transfer, while the comparison is restricted toqq\-tame modules, where the interleaving distance has its standard diagrammatic interpretation\. LetPers𝕂≔\[\(ℝ,≤\),Vec𝕂\]\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\\coloneqq\[\(\\mathbb\{R\},\\leq\),\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}\], and letPers𝕂q​\-​tame⊆Ob⁡\(Pers𝕂\)\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\}\\subseteq\\operatorname\{Ob\}\(\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\)be the class ofqq\-tame persistence modules\. We use the invariant target\(Pers𝕂,Pers𝕂q​\-​tame,dI\),\\bigl\(\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\},d\_\{I\}\\bigr\),wheredId\_\{I\}is the interleaving distance\.

###### Proposition 4\.1\.

The triple\(Pers𝕂,Pers𝕂q​\-​tame,dI\)\\bigl\(\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\},d\_\{I\}\\bigr\)is an invariant target\.

###### Proof 4\.2\.

Pers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}is a functor category into the cocomplete categoryVec𝕂\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}, hence it admits all small colimits, computed pointwise\. The interleaving distancedId\_\{I\}is an extended pseudometric on isomorphism classes of persistence modules\(Les15, Definition 3\.5\)\. Restricting toPers𝕂q​\-​tame/≅\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\}/\{\\cong\}gives the comparison distance\. Onqq\-tame modules, the Isometry Theorem identifiesdId\_\{I\}with thedBd\_\{B\}on persistence diagrams\(CSGO16, Theorem 4\.11\)\.

ThusCompJ⁡\(F,G\)=0\\operatorname\{Comp\}\_\{J\}\(F,G\)=0means that\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)andG​\(b\)G\(b\)have zero interleaving distance for everyb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Equivalently, they define the same point ofSep⁡\(Pers𝕂q​\-​tame,dI\)\\operatorname\{Sep\}\(\\operatorname\{Pers\}^\{q\\text\{\-\}\\mathrm\{tame\}\}\_\{\\mathbb\{K\}\},d\_\{I\}\)\.

### Functor\-level interleavings\.

Forε≥0\\varepsilon\\geq 0, letTε:\(ℝ,≤\)→\(ℝ,≤\)T\_\{\\varepsilon\}:\(\\mathbb\{R\},\\leq\)\\to\(\\mathbb\{R\},\\leq\)be translation byε\\varepsilon, and putΣε​M≔M​Tε\\Sigma\_\{\\varepsilon\}M\\coloneqq MT\_\{\\varepsilon\}forM∈Pers𝕂M\\in\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. The structure maps define a natural transformationηε:Id⇒Σε\\eta\_\{\\varepsilon\}:\\operatorname\{Id\}\\Rightarrow\\Sigma\_\{\\varepsilon\}\. For a small category𝒞\\mathcal\{C\}, postcomposition defines a flow on\[𝒞,Pers𝕂\]\[\\mathcal\{C\},\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\]by\(Σε𝒞​X\)​\(c\)≔Σε​\(X​\(c\)\)\(\\Sigma^\{\\mathcal\{C\}\}\_\{\\varepsilon\}X\)\(c\)\\coloneqq\\Sigma\_\{\\varepsilon\}\(X\(c\)\)\. This is the interleaving induced by a flow on a category\(SMS18, §2\); colax equivariant functors are11\-Lipschitz for the induced interleaving distances\(SMS18, Theorem 4\.2\)\.

###### Definition 4\.3\.

Let𝒞\\mathcal\{C\}be small\. Anε\\varepsilon\-interleaving of diagramsX,Y:𝒞→Pers𝕂X,Y:\\mathcal\{C\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}is a pair of natural transformationsφ:X⇒Σε𝒞​Y,ψ:Y⇒Σε𝒞​X\\varphi:X\\Rightarrow\\Sigma^\{\\mathcal\{C\}\}\_\{\\varepsilon\}Y,\\ \\psi:Y\\Rightarrow\\Sigma^\{\\mathcal\{C\}\}\_\{\\varepsilon\}Xsuch thatΣε𝒞​ψ∘φ=η2​εX\\Sigma^\{\\mathcal\{C\}\}\_\{\\varepsilon\}\\psi\\circ\\varphi=\\eta^\{X\}\_\{2\\varepsilon\}andΣε𝒞​φ∘ψ=η2​εY\.\\Sigma^\{\\mathcal\{C\}\}\_\{\\varepsilon\}\\varphi\\circ\\psi=\\eta^\{Y\}\_\{2\\varepsilon\}\.WritedI𝒞​\(X,Y\)d\_\{I\}^\{\\mathcal\{C\}\}\(X,Y\)for the induced extended pseudometric\. ForJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\},F:𝒜→Pers𝕂F:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, andG:ℬ→Pers𝕂G:\\mathcal\{B\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, define thefunctorial structural transfer discrepancy

CompJnat⁡\(F,G\)≔dIℬ​\(LanJ⁡F,G\)\.\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F,G\)\\coloneqq d\_\{I\}^\{\\mathcal\{B\}\}\(\\operatorname\{Lan\}\_\{J\}F,G\)\.

###### Proposition 4\.4\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\},F:𝒜→Pers𝕂F:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, andG:ℬ→Pers𝕂G:\\mathcal\{B\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, and assume thatLanJ⁡F\\operatorname\{Lan\}\_\{J\}Fexists and that the pointwise discrepancy is defined\. Then

CompJ⁡\(F,G\)≤CompJnat⁡\(F,G\)\.\\operatorname\{Comp\}\_\{J\}\(F,G\)\\leq\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F,G\)\.Ifℬ\\mathcal\{B\}is discrete, equality holds\.

###### Proof 4\.5\.

Letε≥0\\varepsilon\\geq 0and let\(φ,ψ\)\(\\varphi,\\psi\)be anε\\varepsilon\-interleaving ofLanJ⁡F\\operatorname\{Lan\}\_\{J\}FandGGin\[ℬ,Pers𝕂\]\[\\mathcal\{B\},\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\]\. For everyb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\), evaluation atbbgives anε\\varepsilon\-interleaving of\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)andG​\(b\)G\(b\)\. HencedI​\(\(LanJ⁡F\)​\(b\),G​\(b\)\)≤εd\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\)\(b\),G\(b\)\)\\leq\\varepsilonfor allbb, and thereforeCompJ⁡\(F,G\)≤ε\\operatorname\{Comp\}\_\{J\}\(F,G\)\\leq\\varepsilon\. Taking the infimum overε\\varepsilonproves the inequality\.

Assume thatℬ\\mathcal\{B\}is discrete\. Letr=CompJ⁡\(F,G\)r=\\operatorname\{Comp\}\_\{J\}\(F,G\)\. Ifr=\+∞r=\+\\infty, the established inequality forces equality\. Suppose therefore thatr<\+∞r<\+\\infty, and letε\>r\\varepsilon\>r\. For each objectbb, there is anε\\varepsilon\-interleaving of\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)andG​\(b\)G\(b\)\. Sinceℬ\\mathcal\{B\}has no non\-identity morphisms, these componentwise transformations are automatically natural and define anε\\varepsilon\-interleaving in\[ℬ,Pers𝕂\]\[\\mathcal\{B\},\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\]\. ThusCompJnat⁡\(F,G\)≤ε\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F,G\)\\leq\\varepsilon\. Lettingε↓r\\varepsilon\\downarrow rproves equality\.

In particular, equality holds for the terminal category𝟏\\mathbf\{1\}used in Experiments 1 and 3\.

###### Theorem 4\.6\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}be a functor between small categories\. For allF,F′:𝒜→Pers𝕂F,F^\{\\prime\}:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},dIℬ​\(LanJ⁡F,LanJ⁡F′\)≤dI𝒜​\(F,F′\)\.d\_\{I\}^\{\\mathcal\{B\}\}\(\\operatorname\{Lan\}\_\{J\}F,\\operatorname\{Lan\}\_\{J\}F^\{\\prime\}\)\\leq d\_\{I\}^\{\\mathcal\{A\}\}\(F,F^\{\\prime\}\)\.Hence, forG,G′:ℬ→Pers𝕂G,G^\{\\prime\}:\\mathcal\{B\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, whenever finite,

\|CompJnat⁡\(F,G\)−CompJnat⁡\(F′,G′\)\|≤dI𝒜​\(F,F′\)\+dIℬ​\(G,G′\)\.\\left\|\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F,G\)\-\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F^\{\\prime\},G^\{\\prime\}\)\\right\|\\leq d\_\{I\}^\{\\mathcal\{A\}\}\(F,F^\{\\prime\}\)\+d\_\{I\}^\{\\mathcal\{B\}\}\(G,G^\{\\prime\}\)\.

###### Proof 4\.7\.

1. 1\.Compatibility with shifts\.Forε≥0\\varepsilon\\geq 0,b∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\), andt∈ℝt\\in\\mathbb\{R\}, colimits inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}are computed pointwise\. Therefore \(\(LanJ⁡Σε𝒜​F\)​\(b\)\)t≅colim\(a,β\)∈J↓bF​\(a\)t\+ε≅\(Σεℬ​LanJ⁡F​\(b\)\)t\.\\bigl\(\(\\operatorname\{Lan\}\_\{J\}\\Sigma^\{\\mathcal\{A\}\}\_\{\\varepsilon\}F\)\(b\)\\bigr\)\_\{t\}\\cong\\operatorname\*\{colim\}\_\{\(a,\\beta\)\\in J\\downarrow b\}F\(a\)\_\{t\+\\varepsilon\}\\cong\\bigl\(\\Sigma^\{\\mathcal\{B\}\}\_\{\\varepsilon\}\\operatorname\{Lan\}\_\{J\}F\(b\)\\bigr\)\_\{t\}\.These isomorphisms are natural inbb,tt, andFF; henceLanJ⁡Σε𝒜≅Σεℬ​LanJ\.\\operatorname\{Lan\}\_\{J\}\\Sigma^\{\\mathcal\{A\}\}\_\{\\varepsilon\}\\cong\\Sigma^\{\\mathcal\{B\}\}\_\{\\varepsilon\}\\operatorname\{Lan\}\_\{J\}\.
2. 2\.Preservation of interleavings\.Let\(φ,ψ\)\(\\varphi,\\psi\)be anε\\varepsilon\-interleaving ofFFandF′F^\{\\prime\}\. ApplyingLanJ\\operatorname\{Lan\}\_\{J\}and composing with the canonical shift isomorphisms from 1\. yields natural transformationsLanJ⁡F⇒Σεℬ​LanJ⁡F′,LanJ⁡F′⇒Σεℬ​LanJ⁡F\.\\operatorname\{Lan\}\_\{J\}F\\Rightarrow\\Sigma^\{\\mathcal\{B\}\}\_\{\\varepsilon\}\\operatorname\{Lan\}\_\{J\}F^\{\\prime\},\\ \\operatorname\{Lan\}\_\{J\}F^\{\\prime\}\\Rightarrow\\Sigma^\{\\mathcal\{B\}\}\_\{\\varepsilon\}\\operatorname\{Lan\}\_\{J\}F\.Functoriality ofLanJ\\operatorname\{Lan\}\_\{J\}and naturality of the shift isomorphisms transport the two interleaving identities\. ThusLanJ⁡F\\operatorname\{Lan\}\_\{J\}FandLanJ⁡F′\\operatorname\{Lan\}\_\{J\}F^\{\\prime\}areε\\varepsilon\-interleaved\. Taking infima proves the first assertion\. Equivalently, the canonical shift isomorphisms makeLanJ\\operatorname\{Lan\}\_\{J\}flow\-equivariant, so the11\-Lipschitz principle of\(SMS18, Theorem 4\.2\)applies\.
3. 3\.Stability of the discrepancy\.By the triangle inequality and the first assertion, CompJnat⁡\(F,G\)≤dI𝒜​\(F,F′\)\+CompJnat⁡\(F′,G′\)\+dIℬ​\(G′,G\)\.\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F,G\)\\leq d\_\{I\}^\{\\mathcal\{A\}\}\(F,F^\{\\prime\}\)\+\\operatorname\{Comp\}^\{\\mathrm\{nat\}\}\_\{J\}\(F^\{\\prime\},G^\{\\prime\}\)\+d\_\{I\}^\{\\mathcal\{B\}\}\(G^\{\\prime\},G\)\.Interchanging primed and unprimed diagrams gives the reverse bound\. Combining the two inequalities proves the assertion\.

###### Corollary 4\.8\.

Letℬ=𝟏\\mathcal\{B\}=\\mathbf\{1\}, letq∈ℕq\\in\\mathbb\{N\}, letw0,…,wq≥0w\_\{0\},\\ldots,w\_\{q\}\\geq 0, and letFn,Fn′:𝒜→Pers𝕂F\_\{n\},F^\{\\prime\}\_\{n\}:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},Gn,Gn′∈Pers𝕂G\_\{n\},G^\{\\prime\}\_\{n\}\\in\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}for0≤n≤q0\\leq n\\leq q\. Put

S≔∑n=0qwn​dI​\(\(LanJ⁡Fn\)​\(∙\),Gn\),S′≔∑n=0qwn​dI​\(\(LanJ⁡Fn′\)​\(∙\),Gn′\)\.S\\coloneqq\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(\\bullet\),G\_\{n\}\),\\qquad S^\{\\prime\}\\coloneqq\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F^\{\\prime\}\_\{n\}\)\(\\bullet\),G^\{\\prime\}\_\{n\}\)\.If the right\-hand side below is finite, then

\|S−S′\|≤∑n=0qwn​\(dI𝒜​\(Fn,Fn′\)\+dI​\(Gn,Gn′\)\)\.\|S\-S^\{\\prime\}\|\\leq\\sum\_\{n=0\}^\{q\}w\_\{n\}\\bigl\(d\_\{I\}^\{\\mathcal\{A\}\}\(F\_\{n\},F^\{\\prime\}\_\{n\}\)\+d\_\{I\}\(G\_\{n\},G^\{\\prime\}\_\{n\}\)\\bigr\)\.

###### Proof 4\.9\.

By\\Crefprop:pointwise\-natural\-comparison, terminal target tasks have equal pointwise and functor\-level discrepancies in every degree\. Apply\\Crefthm:lan\-interleaving\-stability degreewise, multiply bywnw\_\{n\}, and sum\.

###### Proposition 4\.10\.

LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, letF:𝒜→Pers𝕂F:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, and letb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\. Assume thatJ↓bJ\\downarrow bhas finitely many objects and morphisms\. Letπb:J↓b→𝒜\\pi\_\{b\}:J\\downarrow b\\to\\mathcal\{A\}be the projection\(a,β\)↦a\(a,\\beta\)\\mapsto a\. Forγ∈Mor⁡\(J↓b\)\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\), writes​\(γ\)=\(aγ,βγ\)s\(\\gamma\)=\(a\_\{\\gamma\},\\beta\_\{\\gamma\}\)andt​\(γ\)=\(aγ′,βγ′\)t\(\\gamma\)=\(a^\{\\prime\}\_\{\\gamma\},\\beta^\{\\prime\}\_\{\\gamma\}\)\. Then, for everyu∈ℝu\\in\\mathbb\{R\}, there is a canonical isomorphism

\(\(LanJ⁡F\)​\(b\)\)u≅coker⁡\(⨁γ∈Mor⁡\(J↓b\)F​\(aγ\)u→ru⨁\(a,β\)∈Ob⁡\(J↓b\)F​\(a\)u\),\(\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\)\_\{u\}\\cong\\operatorname\{coker\}\\left\(\\bigoplus\_\{\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\)\}F\(a\_\{\\gamma\}\)\_\{u\}\\xrightarrow\{r\_\{u\}\}\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}F\(a\)\_\{u\}\\right\),whererur\_\{u\}is determined on the summand indexed byγ\\gammabyx↦ιt​\(γ\)​F​\(γ¯\)u​\(x\)−ιs​\(γ\)​xx\\mapsto\\iota\_\{t\(\\gamma\)\}F\(\\bar\{\\gamma\}\)\_\{u\}\(x\)\-\\iota\_\{s\(\\gamma\)\}x\. Fors≤ts\\leq t, the structure map fromsstottis induced by the direct sum of the structure mapsF​\(a\)s≤t:F​\(a\)s→F​\(a\)tF\(a\)\_\{s\\leq t\}:F\(a\)\_\{s\}\\to F\(a\)\_\{t\}\.

###### Proof 4\.11\.

Consider\(a,β\)∈Ob⁡\(J↓b\)\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\), and writeγ¯=πb​\(γ\):aγ→aγ′\\bar\{\\gamma\}=\\pi\_\{b\}\(\\gamma\):a\_\{\\gamma\}\\to a^\{\\prime\}\_\{\\gamma\}forγ∈Mor⁡\(J↓b\)\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\)\. Letι\(a,β\)\\iota\_\{\(a,\\beta\)\}denote the canonical inclusion of the summand indexed by\(a,β\)\{\(a,\\beta\)\}\. PutL≔colimJ↓b\(F​πb\)L\\coloneqq\\operatorname\*\{colim\}\_\{J\\downarrow b\}\(F\\pi\_\{b\}\)inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, with colimit coconeλ\(a,β\):F​\(a\)→L\\lambda^\{\(a,\\beta\)\}:F\(a\)\\to L\.

1. 1\.Pointwise Kan formula:SincePers𝕂=\[\(ℝ,≤\),Vec𝕂\]\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}=\[\(\\mathbb\{R\},\\leq\),\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}\]is cocomplete, the pointwise formula for left Kan extensions gives a canonical isomorphism\(LanJ⁡F\)​\(b\)≅L\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong LinPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. It remains to computeLLand its structure maps\. Colimits inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}are computed pointwise\. Hence, for everyu∈ℝu\\in\\mathbb\{R\},Lu≅colimJ↓b\(Fπb\)uL\_\{u\}\\cong\\operatorname\*\{colim\}\_\{J\\downarrow b\}\(F\\pi\_\{b\}\)\_\{u\}inVec𝕂\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}, where\(F​πb\)u:J↓b→Vec𝕂\(F\\pi\_\{b\}\)\_\{u\}:J\\downarrow b\\to\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}sends\(a,β\)\(a,\\beta\)toF​\(a\)uF\(a\)\_\{u\}andγ\\gammatoF​\(γ¯\)uF\(\\bar\{\\gamma\}\)\_\{u\}\.
2. 2\.Cokernel presentation:By the finite cokernel presentation of colimits used in\\Crefprop:chain\-cokernel, applied here at the fixed persistence parameteruu, the colimitLuL\_\{u\}is canonically isomorphic to coker⁡\(ru:⨁γ∈Mor⁡\(J↓b\)F​\(aγ\)u⟶⨁\(a,β\)∈Ob⁡\(J↓b\)F​\(a\)u\),\\operatorname\{coker\}\\left\(r\_\{u\}:\\bigoplus\_\{\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\)\}F\(a\_\{\\gamma\}\)\_\{u\}\\longrightarrow\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}F\(a\)\_\{u\}\\right\),which is determined on the summand indexed byγ\\gammabyx↦ιt​\(γ\)​F​\(γ¯\)u​\(x\)−ιs​\(γ\)​xx\\mapsto\\iota\_\{t\(\\gamma\)\}F\(\\bar\{\\gamma\}\)\_\{u\}\(x\)\-\\iota\_\{s\(\\gamma\)\}x\. Ifquq\_\{u\}denotes the quotient map, then theuu\-component of the colimit cocone satisfiesλu\(a,β\)=qu​ι\(a,β\)\\lambda^\{\(a,\\beta\)\}\_\{u\}=q\_\{u\}\\iota\_\{\(a,\\beta\)\}for every\(a,β\)∈Ob⁡\(J↓b\)\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\. Together with the isomorphism\(LanJ⁡F\)​\(b\)≅L\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong L, this gives the displayed canonical isomorphism\.
3. 3\.Compatibility with structure maps:Lets≤ts\\leq t, and setDs≤t≔⨁\(a,β\)F​\(a\)s≤t\.D\_\{s\\leq t\}\\coloneqq\\bigoplus\_\{\(a,\\beta\)\}F\(a\)\_\{s\\leq t\}\.ThenDs≤t​ι\(a,β\)=ι\(a,β\)​F​\(a\)s≤tD\_\{s\\leq t\}\\iota\_\{\(a,\\beta\)\}=\\iota\_\{\(a,\\beta\)\}F\(a\)\_\{s\\leq t\}for every\(a,β\)∈Ob⁡\(J↓b\)\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\. Forγ∈Mor⁡\(J↓b\)\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\)andx∈F​\(aγ\)sx\\in F\(a\_\{\\gamma\}\)\_\{s\}, naturality of the morphism of persistence modulesF​\(γ¯\):F​\(aγ\)→F​\(aγ′\)F\(\\bar\{\\gamma\}\):F\(a\_\{\\gamma\}\)\\to F\(a^\{\\prime\}\_\{\\gamma\}\)givesF​\(aγ′\)s≤t​F​\(γ¯\)s=F​\(γ¯\)t​F​\(aγ\)s≤tF\(a^\{\\prime\}\_\{\\gamma\}\)\_\{s\\leq t\}F\(\\bar\{\\gamma\}\)\_\{s\}=F\(\\bar\{\\gamma\}\)\_\{t\}F\(a\_\{\\gamma\}\)\_\{s\\leq t\}\. Therefore Ds≤t​\(ιt​\(γ\)​F​\(γ¯\)s​\(x\)−ιs​\(γ\)​x\)=ιt​\(γ\)​F​\(γ¯\)t​\(F​\(aγ\)s≤t​x\)−ιs​\(γ\)​F​\(aγ\)s≤t​x=rt​\(F​\(aγ\)s≤t​x\)\.D\_\{s\\leq t\}\\bigl\(\\iota\_\{t\(\\gamma\)\}F\(\\bar\{\\gamma\}\)\_\{s\}\(x\)\-\\iota\_\{s\(\\gamma\)\}x\\bigr\)=\\iota\_\{t\(\\gamma\)\}F\(\\bar\{\\gamma\}\)\_\{t\}\\bigl\(F\(a\_\{\\gamma\}\)\_\{s\\leq t\}x\\bigr\)\-\\iota\_\{s\(\\gamma\)\}F\(a\_\{\\gamma\}\)\_\{s\\leq t\}x=r\_\{t\}\\bigl\(F\(a\_\{\\gamma\}\)\_\{s\\leq t\}x\\bigr\)\.ThusDs≤t​\(im⁡rs\)⊆im⁡rtD\_\{s\\leq t\}\(\\operatorname\{im\}r\_\{s\}\)\\subseteq\\operatorname\{im\}r\_\{t\}, andDs≤tD\_\{s\\leq t\}descends uniquely to a linear mapD¯s≤t:coker⁡\(rs\)→coker⁡\(rt\)\\overline\{D\}\_\{s\\leq t\}:\\operatorname\{coker\}\(r\_\{s\}\)\\to\\operatorname\{coker\}\(r\_\{t\}\)satisfyingD¯s≤t​qs=qt​Ds≤t\\overline\{D\}\_\{s\\leq t\}q\_\{s\}=q\_\{t\}D\_\{s\\leq t\}\.
4. 4\.Identification of the descended map:The structure mapLs≤t:Ls→LtL\_\{s\\leq t\}:L\_\{s\}\\to L\_\{t\}is the unique linear map satisfyingLs≤t​λs\(a,β\)=λt\(a,β\)​F​\(a\)s≤tL\_\{s\\leq t\}\\lambda^\{\(a,\\beta\)\}\_\{s\}=\\lambda^\{\(a,\\beta\)\}\_\{t\}F\(a\)\_\{s\\leq t\}for all\(a,β\)∈Ob⁡\(J↓b\)\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\. The descended map has the same property, since D¯s≤t​λs\(a,β\)=D¯s≤t​qs​ι\(a,β\)=qt​Ds≤t​ι\(a,β\)=qt​ι\(a,β\)​F​\(a\)s≤t=λt\(a,β\)​F​\(a\)s≤t\.\\overline\{D\}\_\{s\\leq t\}\\lambda^\{\(a,\\beta\)\}\_\{s\}=\\overline\{D\}\_\{s\\leq t\}q\_\{s\}\\iota\_\{\(a,\\beta\)\}=q\_\{t\}D\_\{s\\leq t\}\\iota\_\{\(a,\\beta\)\}=q\_\{t\}\\iota\_\{\(a,\\beta\)\}F\(a\)\_\{s\\leq t\}=\\lambda^\{\(a,\\beta\)\}\_\{t\}F\(a\)\_\{s\\leq t\}\.The mapsλs\(a,β\)\\lambda^\{\(a,\\beta\)\}\_\{s\}are jointly epimorphic becauseqsq\_\{s\}is surjective and the inclusionsι\(a,β\)\\iota\_\{\(a,\\beta\)\}jointly span the direct sum\. HenceD¯s≤t=Ls≤t\\overline\{D\}\_\{s\\leq t\}=L\_\{s\\leq t\}\. Transporting this equality along\(LanJ⁡F\)​\(b\)≅L\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong L, the structure map of\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)is induced by⨁c=\(a,β\)F​\(a\)s≤t\\bigoplus\_\{c=\(a,\\beta\)\}F\(a\)\_\{s\\leq t\}\.

Consequently, finite persistent transfer is pointwise linear algebra: at each scalett, form the object direct sum overOb⁡\(J↓b\)\\operatorname\{Ob\}\(J\\downarrow b\), quotient by the morphism relations indexed byMor⁡\(J↓b\)\\operatorname\{Mor\}\(J\\downarrow b\), and compare the resulting persistence modules bydId\_\{I\}\.

## 5Algorithmic Evaluation of Transfer Quality

The transfer problem is finite and computable\. A task change is a functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, the source system a functorF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}, the target system a functorG:ℬ→𝒱G:\\mathcal\{B\}\\to\\mathcal\{V\}and the universal transferred structure isLanJ⁡F\\operatorname\{Lan\}\_\{J\}F\. Transfer quality comparesG​\(b\)G\(b\)not with the source objects directly, but with the structure forced atbbby the universal extension ofFFalongJJ\.

For persistence\-valued invariants we take𝒱=Pers𝕂\\mathcal\{V\}=\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. The intrinsic distance is the interleaving distancedId\_\{I\}: persistence modules are\(ℝ,≤\)\(\\mathbb\{R\},\\leq\)\-indexed diagrams and interleavings are defined at this functorial level\(bubenik2014categorification, Definition 3\.1, Theorem 3\.3, Corollary 3\.5\)\. For finite\-type one\-parameter modules the barcode\-level bottleneck comparison is exact, via the isometric embedding of the metric space of finite barcodes\(ℬ,dB\)\(\\mathcal\{B\},d\_\{B\}\)into\(Vec𝕂\(ℝ,≤\),dI\)\(\\operatorname\{Vec\}^\{\(\\mathbb\{R\},\\leq\)\}\_\{\\mathbb\{K\}\},d\_\{I\}\)\(bubenik2014categorification, Theorem 4\.16\)\. The generalization toqq\-tame modules is the Isometry Theorem\(CSGO16, Theorem 4\.11\)\. The classical diagram stability estimatedB​\(D​\(f\),D​\(g\)\)≤‖f−g‖∞d\_\{B\}\(D\(f\),D\(g\)\)\\leq\\\|f\-g\\\|\_\{\\infty\}holds for continuous tame functions on triangulable spaces\(CohenSteinerEdelsbrunnerHarer2007, Main Theorem\)\. Computationally, Vietoris–Rips persistence is obtained with Ripser\(Bauer2021\)and diagram comparisons with GUDHI\(MariaBoissonnatGlisseYvinec2014\)\. Geometric bottleneck matching runs inO​\(n1\.5​log⁡n\)O\(n^\{1\.5\}\\log n\)\(KerberMorozovNigmetov2017, Theorem 3\.1\)\. Persistence images\(AdamsEtAl2017, Definition 2\), stable in the11\-Wasserstein distance\(AdamsEtAl2017, Theorem 5\), and persistence landscapes\(Bubenik2015, Definition 3\), which obey a strong Law of Large Numbers and a Central Limit Theorem\(Bubenik2015, Theorem 9, Theorem 10\)and satisfyΛ∞​\(M,M′\)≤dI​\(M,M′\)\\Lambda\_\{\\infty\}\(M,M^\{\\prime\}\)\\leq d\_\{I\}\(M,M^\{\\prime\}\)\(Bubenik2015, Theorem 17\), are vectorized diagnostics: they summarize diagrams, whereasLanJ⁡F\\operatorname\{Lan\}\_\{J\}Fis a functorial colimit\.

###### Lemma 5\.1\.

Adopt the hypotheses and notation of\\Crefprop:persistent\-cokernel:J:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\},F:𝒜→Pers𝕂F:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\},b∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)withJ↓bJ\\downarrow bfinite\. If everyF​\(a\)F\(a\),a∈Ob⁡\(J↓b\)a\\in\\operatorname\{Ob\}\(J\\downarrow b\), is of finite type\(bubenik2014categorification, Definition 4\.1\), then\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)is of finite type\. In particular it is interval\-decomposable with a finite barcode\.

###### Proof 5\.2\.

LetS⊂ℝS\\subset\\mathbb\{R\}be the union of the critical values of the finitely many finite\-type modulesF​\(a\)F\(a\),a∈Ob⁡\(J↓b\)a\\in\\operatorname\{Ob\}\(J\\downarrow b\); thenSSis finite\. Fix an open intervalIIwithI∩S=∅I\\cap S=\\varnothing\. By the Critical Value Lemma\(bubenik2014categorification, Lemma 4\.4\)eachF​\(a\)F\(a\)is constant onII, so fors≤ts\\leq tinIIeveryF​\(a\)s≤tF\(a\)\_\{s\\leq t\}is an isomorphism\. HenceDs≤t=⨁\(a,β\)F​\(a\)s≤tD\_\{s\\leq t\}=\\bigoplus\_\{\(a,\\beta\)\}F\(a\)\_\{s\\leq t\}\(\\Crefprop:persistent\-cokernel\) is an isomorphism intertwiningrsr\_\{s\}andrtr\_\{t\}, and the induced structure map\(LanJ⁡F\)​\(b\)s≤t=D¯s≤t\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\_\{s\\leq t\}=\\overline\{D\}\_\{s\\leq t\}on cokernels is an isomorphism\. Thus every point ofIIis a regular value, so the critical values of\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)lie in the finite setSS; the module is therefore tame and hence of finite type\(bubenik2014categorification, Theorem 4\.6\)\.

Input:Finite categories

𝒜,ℬ\\mathcal\{A\},\\mathcal\{B\}, a functor

J:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, a field

𝕂\\mathbb\{K\}, a maximal degree

qq, weights

w0,…,wq≥0w\_\{0\},\\ldots,w\_\{q\}\\geq 0, a diagram of finite one\-parameter filtered complexes

K∙s:𝒜→FiltSimpK^\{s\}\_\{\\bullet\}:\\mathcal\{A\}\\to\\operatorname\{FiltSimp\}, and finite one\-parameter filtered complexes

K∙t​\(b\)K^\{t\}\_\{\\bullet\}\(b\)for

b∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\.

Output:

CompJ,qI=CompJ,qB\\operatorname\{Comp\}^\{I\}\_\{J,q\}=\\operatorname\{Comp\}^\{B\}\_\{J,q\}, and optionally

CompJ,qW,p\\operatorname\{Comp\}^\{W,p\}\_\{J,q\}\.

Compute

Fn​\(a\)=Hn​\(K∙s​\(a\);𝕂\)F\_\{n\}\(a\)=H\_\{n\}\(K^\{s\}\_\{\\bullet\}\(a\);\\mathbb\{K\}\)and the induced morphisms

Fn​\(α\)F\_\{n\}\(\\alpha\)for every

α∈Mor⁡\(𝒜\)\\alpha\\in\\operatorname\{Mor\}\(\\mathcal\{A\}\)and

0≤n≤q0\\leq n\\leq q,

Gn​\(b\)=Hn​\(K∙t​\(b\);𝕂\)G\_\{n\}\(b\)=H\_\{n\}\(K^\{t\}\_\{\\bullet\}\(b\);\\mathbb\{K\}\)for every

b∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)and

0≤n≤q0\\leq n\\leq q\.

foreach*b∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)*do

Construct the comma category

J↓bJ\\downarrow b\.

for*n=0,…,qn=0,\\ldots,q*do

Compute

\(LanJ⁡Fn\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)by the cokernel presentation of\\Crefprop:persistent\-cokernel\.

Decompose

\(LanJ⁡Fn\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)and

Gn​\(b\)G\_\{n\}\(b\)into finite barcodes\.

Compute

dn,B​\(b\)d\_\{n,B\}\(b\), and set

dn,I​\(b\)=dn,B​\(b\)d\_\{n,I\}\(b\)=d\_\{n,B\}\(b\)by the Isometry Theorem\.

Optionally compute

dn,W,p​\(b\)d\_\{n,W,p\}\(b\)
end for

end foreach

Return

CompJ,qI=CompJ,qB=maxb∈Ob⁡\(ℬ\)​∑n=0qwn​dn,B​\(b\),\\operatorname\{Comp\}^\{I\}\_\{J,q\}=\\operatorname\{Comp\}^\{B\}\_\{J,q\}=\\max\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{n,B\}\(b\),and optionally

CompJ,qW,p\\operatorname\{Comp\}^\{W,p\}\_\{J,q\}\.

Algorithm 1Kan\-persistent transfer score\.###### Construction 5\.3

Fix a field𝕂\\mathbb\{K\}, a maximal homological degreeqq, finite task categories𝒜,ℬ\\mathcal\{A\},\\mathcal\{B\}, and a functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}\. LetNs,NtN\_\{s\},N\_\{t\}be the source and target systems\.

1. 1\.Data extraction\.For eacha∈Ob⁡\(𝒜\)a\\in\\operatorname\{Ob\}\(\\mathcal\{A\}\)choose a finite point cloudPs​\(a\)P\_\{s\}\(a\)fromNsN\_\{s\}, and for eachb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)a finite point cloudPt​\(b\)P\_\{t\}\(b\)fromNtN\_\{t\}\(input samples, latent activations, class centroids, or layer\-wise representation clouds\)\.
2. 2\.Filtration\.Choose a filtration functorK∙K\_\{\\bullet\}: Vietoris–Rips by default \(canonical from the metric, computed by Ripser\(Bauer2021\)\)\. Alpha filtrations for low\-dimensional Euclidean clouds, landmark or witness approximations for large clouds\. This yields finite filtrationsK∙​\(Ps​\(a\)\)K\_\{\\bullet\}\(P\_\{s\}\(a\)\)andK∙​\(Pt​\(b\)\)K\_\{\\bullet\}\(P\_\{t\}\(b\)\)\.
3. 3\.Persistent invariants\.For0≤n≤q0\\leq n\\leq qsetFn​\(a\)≔Hn​\(K∙​\(Ps​\(a\)\);𝕂\)F\_\{n\}\(a\)\\coloneqq H\_\{n\}\(K\_\{\\bullet\}\(P\_\{s\}\(a\)\);\\mathbb\{K\}\)andGn​\(b\)≔Hn​\(K∙​\(Pt​\(b\)\);𝕂\)G\_\{n\}\(b\)\\coloneqq H\_\{n\}\(K\_\{\\bullet\}\(P\_\{t\}\(b\)\);\\mathbb\{K\}\)\. Each morphism of𝒜\\mathcal\{A\}\(resp\.ℬ\\mathcal\{B\}\) must be realized by a map of the underlying data objects, or by a chosen compatible map of the induced filtrations, so thatFn:𝒜→Pers𝕂F\_\{n\}:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}andGn:ℬ→Pers𝕂G\_\{n\}:\\mathcal\{B\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}are genuine functors and not objectwise assignments\. As the filtrations are finite, everyFn​\(a\)F\_\{n\}\(a\)andGn​\(b\)G\_\{n\}\(b\)is of finite type\.
4. 4\.Kan transfer\.For eachb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)considerJ↓bJ\\downarrow band compute\(LanJ⁡Fn\)​\(b\)≅colim\(J​a→b\)∈J↓bFn​\(a\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\\cong\\operatorname\*\{colim\}\_\{\(Ja\\to b\)\\in J\\downarrow b\}F\_\{n\}\(a\)\. By\\Crefprop:persistent\-cokernel this is computed parameterwise as \(\(LanJ⁡Fn\)​\(b\)\)u≅coker⁡\(⨁γ∈Mor⁡\(J↓b\)Fn​\(aγ\)u→rn,b,u⨁\(a,β\)∈Ob⁡\(J↓b\)Fn​\(a\)u\),\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\)\_\{u\}\\cong\\operatorname\{coker\}\\bigl\(\\bigoplus\_\{\\gamma\\in\\operatorname\{Mor\}\(J\\downarrow b\)\}F\_\{n\}\(a\_\{\\gamma\}\)\_\{u\}\\xrightarrow\{\\,r\_\{n,b,u\}\\,\}\\bigoplus\_\{\(a,\\beta\)\\in\\operatorname\{Ob\}\(J\\downarrow b\)\}F\_\{n\}\(a\)\_\{u\}\\bigr\),wherern,b,ur\_\{n,b,u\}acts on theγ\\gamma\-summand byx↦ιt​\(γ\)​Fn​\(γ¯\)u​\(x\)−ιs​\(γ\)​xx\\mapsto\\iota\_\{t\(\\gamma\)\}F\_\{n\}\(\\bar\{\\gamma\}\)\_\{u\}\(x\)\-\\iota\_\{s\(\\gamma\)\}x, with structure maps induced by⨁\(a,β\)Fn​\(a\)s≤t\\bigoplus\_\{\(a,\\beta\)\}F\_\{n\}\(a\)\_\{s\\leq t\}\. By\\Creflem:finite\-type\-closure,\(LanJ⁡Fn\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)is again of finite type\.
5. 5\.Intrinsic distance\.Computedn,I​\(b\)≔dI​\(\(LanJ⁡Fn\)​\(b\),Gn​\(b\)\)d\_\{n,I\}\(b\)\\coloneqq d\_\{I\}\\bigl\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G\_\{n\}\(b\)\\bigr\)\. This is the intrinsic categorical distance: it compares persistence modules before any diagrammatic or vectorized summary\(bubenik2014categorification, Definition 3\.1, Theorem 3\.3, Corollary 3\.5\)\.
6. 6\.Barcode distance\.Both\(LanJ⁡Fn\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)andGn​\(b\)G\_\{n\}\(b\)are of finite type \(4\.\), hence interval\-decomposable with finite barcodes\. Compute dn,B​\(b\)≔dB​\(Dgm⁡\(\(LanJ⁡Fn\)​\(b\)\),Dgm⁡\(Gn​\(b\)\)\)d\_\{n,B\}\(b\)\\coloneqq d\_\{B\}\\bigl\(\\operatorname\{Dgm\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\),\\operatorname\{Dgm\}\(G\_\{n\}\(b\)\)\\bigr\)using GUDHI\(MariaBoissonnatGlisseYvinec2014\)or geometric matching\(KerberMorozovNigmetov2017, Theorem 3\.1\)\. On finite\-type modules the isometric embedding\(bubenik2014categorification, Theorem 4\.16\)givesdn,B​\(b\)=dn,I​\(b\)d\_\{n,B\}\(b\)=d\_\{n,I\}\(b\); the barcode computation is thus an exact evaluation of the intrinsic score, not an approximation\. The same identity holds more generally forqq\-tame modules\(CSGO16, Theorem 4\.11\)\.
7. 7\.Aggregate diagram distance\.As a secondary score compute dn,W,p​\(b\)≔Wp​\(Dgm⁡\(\(LanJ⁡Fn\)​\(b\)\),Dgm⁡\(Gn​\(b\)\)\)\.d\_\{n,W,p\}\(b\)\\coloneqq W\_\{p\}\\bigl\(\\operatorname\{Dgm\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\),\\operatorname\{Dgm\}\(G\_\{n\}\(b\)\)\\bigr\)\.The bottleneck distance is the largest matched discrepancy,pp\-Wasserstein distance the aggregate one\. Persistence images are11\-Wasserstein stable\(AdamsEtAl2017, Theorem 5\), and geometric matching allows efficient comparison\(KerberMorozovNigmetov2017, Theorem 3\.1\)\.
8. 8\.Weighted transfer score\.Choose weightswn≥0w\_\{n\}\\geq 0\. Define the intrinsic score CompJ,qI⁡\(Ns,Nt\)≔supb∈Ob⁡\(ℬ\)∑n=0qwn​dn,I​\(b\)=supb∑n=0qwn​dn,B​\(b\)≕CompJ,qB⁡\(Ns,Nt\),\\operatorname\{Comp\}^\{I\}\_\{J,q\}\(N\_\{s\},N\_\{t\}\)\\coloneqq\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}\\sum\_\{n=0\}^\{q\}w\_\{n\}\\,d\_\{n,I\}\(b\)=\\sup\_\{b\}\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{n,B\}\(b\)\\eqqcolon\\operatorname\{Comp\}^\{B\}\_\{J,q\}\(N\_\{s\},N\_\{t\}\),with practical variantsCompJ,qW,p⁡\(Ns,Nt\)≔supb∑n=0qwn​dn,W,p​\(b\)\.\\operatorname\{Comp\}^\{W,p\}\_\{J,q\}\(N\_\{s\},N\_\{t\}\)\\coloneqq\\sup\_\{b\}\\sum\_\{n=0\}^\{q\}w\_\{n\}\\,d\_\{n,W,p\}\(b\)\.

###### Proposition 5\.4\.

Assume every comma categoryJ↓bJ\\downarrow band every filtration above is finite\. Then\\Crefalg:kan\-persistent\-transfer\-score computesCompJ⁡\(Fn,Gn\)=supb∈Ob⁡\(ℬ\)d​\(\(LanJ⁡Fn\)​\(b\),Gn​\(b\)\)\\operatorname\{Comp\}\_\{J\}\(F\_\{n\},G\_\{n\}\)=\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}d\\bigl\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G\_\{n\}\(b\)\\bigr\)exactly, ford∈\{dI,dB\}d\\in\\\{d\_\{I\},d\_\{B\}\\\}and for each fixednn\. In particular the score comparesGn​\(b\)G\_\{n\}\(b\)with the universal structure forced atbbbyFnF\_\{n\}andJJ\.

###### Proof 5\.5\.

Fixnnandbb\. The left Kan extension exists becausePers𝕂=\[\(ℝ,≤\),Vec𝕂\]\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}=\[\(\\mathbb\{R\},\\leq\),\\operatorname\{Vec\}\_\{\\mathbb\{K\}\}\]is cocomplete andJ↓bJ\\downarrow bis small\. Being pointwise, it satisfies\(LanJ⁡Fn\)​\(b\)≅colimJ↓b\(Fn​πb\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\\cong\\operatorname\*\{colim\}\_\{J\\downarrow b\}\(F\_\{n\}\\pi\_\{b\}\)\. SinceJ↓bJ\\downarrow bis finite,\\Crefprop:persistent\-cokernel identifies this colimit parameterwise with the displayed cokernel and identifies its structure maps with those induced by⨁\(a,β\)Fn​\(a\)s≤t\\bigoplus\_\{\(a,\\beta\)\}F\_\{n\}\(a\)\_\{s\\leq t\}\. Construction[5\.3](https://arxiv.org/html/2606.07627#S5.Thmtheorem3)\.6 therefore returns the object\(LanJ⁡Fn\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\)\. By Lemma[5\.1](https://arxiv.org/html/2606.07627#S5.Thmtheorem1)this object is of finite type, so it admits a finite barcode \(Construction[5\.3](https://arxiv.org/html/2606.07627#S5.Thmtheorem3)\.7\)\. On finite\-type modules the interleaving and bottleneck distances coincide\(bubenik2014categorification, Theorem 4\.16\), so Construction[5\.3](https://arxiv.org/html/2606.07627#S5.Thmtheorem3)\.8 evaluatesdI=dBd\_\{I\}=d\_\{B\}\. Then we take weighted sums over0≤n≤q0\\leq n\\leq qand the supremum overbb\.

In this one\-parameter finite\-type setting the bottleneck distance is therefore the exact numerical realization of the intrinsic interleaving score, and the Wasserstein, persistence\-image, and landscape variants of §[5](https://arxiv.org/html/2606.07627#S5)remain secondary diagnostics rather than replacements for the functorial colimit\.

## 6Detecting Functorial Transfer Compatibility in Latent Space

The left Kan extension answers: given what the source assigns to each task component, and given the task\-change functorJJ, what is the target obliged to be? It manufactures no structure thatJJdoes not demand and discards none thatJJdoes not collapse, and by its universal property is the initial extension ofFFalongJJ, so any disagreement between an observed targetGGandLanJ⁡F\\operatorname\{Lan\}\_\{J\}Fis a structural defect of the transfer rather than an artefact of the metric\. The mechanism is a controlled gluing: readJJas a recipe for reassembling source pieces into a target objectbb, let the comma categoryJ↓bJ\\downarrow benumerate every source component mapping intobb, each tagged by the morphismβ:J​a→b\\beta:Ja\\to brecording how it lands there, and the colimit\(LanJ⁡F\)​\(b\)=colimJ↓bF\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)=\\operatorname\*\{colim\}\_\{J\\downarrow b\}Ffuses these pieces along precisely the source morphisms visible overbb\. ComponentsJJidentifies are amalgamated, components it keeps apart survive as summands, and the source morphisms impose the quotient relations; the four elementary task changes of\\Crefex:basic\-task\-changes—merge to a coproduct, refinement into subdividing fine classes, collapse of a directed chain to its terminal value, forgetting by quotient—are the four modes of this gluing\. The gluing reads the morphisms of the source, not merely its objects up to isomorphism \(\\Crefprop:structure\): two diagrams with identical objectwise invariants can possess non\-isomorphic Kan extensions, as two equal heaps of bricks assemble into different shapes\. This is what makes the discrepancy discriminating where an objectwise comparison is blind—after a merge the target should realise the coproduct, not resemble either source—so comparingG​\(b\)G\(b\)against the glued prediction\(LanJ⁡F\)​\(b\)\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)gives a score whose zero set is the set of correct transfers, while the objectwise baseline remains large even when transfer succeeds\. The cokernel presentation of\\Crefprop:persistent\-cokernel renders this colimit explicit linear algebra at each filtration scale, evaluated exactly by the bottleneck distance on one\-parameter finite\-type modules, so the score is at once principled and computable\.

### What the experiments establish\.

The experiments isolate three increasingly stringent claims\. Experiments 1 and 2 validate the score on controlled invariants whose induced target is known in closed form, admitting an exact prediction against which both the score and the objectwise baseline are checked\. Experiment 1 instantiates a mergeJ:𝒜→𝟏J:\\mathcal\{A\}\\to\\mathbf\{1\}on two source domains, where\\Crefprop:separated\-vr\-merge guarantees that a geometrically separated union realises the coproduct in every degree—the target barcode the multiset union of the two source barcodes—and\\Crefcor:separated\-merge\-robustness quantifies the separation margin under which this persists\. Experiment 2 instantiates a refinement, where each comma categoryJ↓diJ\\downarrow d\_\{i\}carries a terminal object and the induced value collapses to\(LanJ⁡F\)​\(di\)≅F​\(ci\)\(\\operatorname\{Lan\}\_\{J\}F\)\(d\_\{i\}\)\\cong F\(c\_\{i\}\)by the pointwise formula of\\Crefex:basic\-task\-changes\. In both, three controls—a wrong, a degenerate, and a collapsed target—probe whether the score detects structural violations that may leave classification\-relevant geometry intact, under uniform hypotheses: the score vanishes exactly on the intended transfer, is strictly positive on every control, and disagrees with the objectwise baseline\. Experiment 3 applies the same merge to learned MNIST latents across three autoencoder architectures, separating two questions easily conflated: whether the score behaves as proved when the geometric premise holds, and whether trained encoders in fact produce latent geometry meeting that premise and surviving sampling shift\. Experiment 3A tests the merge on its stated domain of applicability, while Experiments 3B and 3C stress it with an independently sampled target and with train–test shift\. The verdict is sharp: the theory fires without exception whenever its separated\-domain premise is met, and the binding constraint is the geometry current encoders supply—a fragility the theory itself anticipates through the necessity of the separation margin in\\Crefcor:separated\-merge\-robustness—with the Mayer–Vietoris criterion of\\Crefprop:filtered\-gluing\-criterion marking the route beyond the overlap\-free, degree\-zero regime these experiments occupy\.

### Geometric realisation of a Kan\-induced merge\.

For a finite metric spacePPand a thresholdT\>0T\>0, define the frozen Vietoris–Rips filtration by

VRtT⁡\(P\)≔\{∅,t<0,VRt⁡\(P\),0≤t≤T,VRT⁡\(P\),t≥T\.\\operatorname\{VR\}^\{T\}\_\{t\}\(P\)\\coloneqq\\begin\{cases\}\\varnothing,&t<0,\\\\ \\operatorname\{VR\}\_\{t\}\(P\),&0\\leq t\\leq T,\\\\ \\operatorname\{VR\}\_\{T\}\(P\),&t\\geq T\.\\end\{cases\}HereVRt⁡\(P\)\\operatorname\{VR\}\_\{t\}\(P\)contains a simplex precisely when all of its vertices have pairwise distance at mosttt\. PutPHnT⁡\(P\)≔Hn​\(VR∙T⁡\(P\);𝕂\)∈Pers𝕂\.\\operatorname\{PH\}^\{T\}\_\{n\}\(P\)\\coloneqq H\_\{n\}\(\\operatorname\{VR\}^\{T\}\_\{\\bullet\}\(P\);\\mathbb\{K\}\)\\in\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\.Every class alive atTTremains alive thereafter and is therefore represented by an essential interval\. The categorical score compares persistence modules after this filtration has been formed; it does not assert that persistent homology commutes with arbitrary geometric colimits\. The merge used in Experiments 1 and 3 is exact for the separate geometric reason established below\.

###### Proposition 6\.1\.

LetT\>0T\>0, and letP1,P2P\_\{1\},P\_\{2\}be nonempty finite disjoint subsets of a metric space\. EquipP1⊔P2P\_\{1\}\\sqcup P\_\{2\}with the metric induced by the ambient metric, and assume

dist⁡\(P1,P2\)≔inf\{d​\(x,y\)∣x∈P1,y∈P2\}\>T\.\\operatorname\{dist\}\(P\_\{1\},P\_\{2\}\)\\coloneqq\\inf\\\{d\(x,y\)\\mid x\\in P\_\{1\},\\ y\\in P\_\{2\}\\\}\>T\.Then, for everyn∈ℕn\\in\\mathbb\{N\}, there is a canonical isomorphism

PHnT⁡\(P1⊔P2\)≅PHnT⁡\(P1\)⊕PHnT⁡\(P2\)\\operatorname\{PH\}^\{T\}\_\{n\}\(P\_\{1\}\\sqcup P\_\{2\}\)\\cong\\operatorname\{PH\}^\{T\}\_\{n\}\(P\_\{1\}\)\\oplus\\operatorname\{PH\}^\{T\}\_\{n\}\(P\_\{2\}\)inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. Consequently, for the unique functorJ:𝒜→𝟏J:\\mathcal\{A\}\\to\\mathbf\{1\}, where𝒜\\mathcal\{A\}is discrete ona1,a2a\_\{1\},a\_\{2\}, ifFnT​\(ai\)=PHnT⁡\(Pi\)F^\{T\}\_\{n\}\(a\_\{i\}\)=\\operatorname\{PH\}^\{T\}\_\{n\}\(P\_\{i\}\)andGnT​\(∙\)=PHnT⁡\(P1⊔P2\),G^\{T\}\_\{n\}\(\\bullet\)=\\operatorname\{PH\}^\{T\}\_\{n\}\(P\_\{1\}\\sqcup P\_\{2\}\),then\(LanJ⁡FnT\)​\(∙\)≅GnT​\(∙\)\.\(\\operatorname\{Lan\}\_\{J\}F^\{T\}\_\{n\}\)\(\\bullet\)\\cong G^\{T\}\_\{n\}\(\\bullet\)\.

###### Proof 6\.2\.

Fort<0t<0, all frozen Vietoris–Rips complexes are empty\. Lett≥0t\\geq 0and putu=min⁡\{t,T\}u=\\min\\\{t,T\\\}\. Sinceu≤T<dist⁡\(P1,P2\)u\\leq T<\\operatorname\{dist\}\(P\_\{1\},P\_\{2\}\), no edge ofVRu⁡\(P1⊔P2\)\\operatorname\{VR\}\_\{u\}\(P\_\{1\}\\sqcup P\_\{2\}\)has one vertex inP1P\_\{1\}and one vertex inP2P\_\{2\}\. Hence no simplex meets both subsets, andVRtT⁡\(P1⊔P2\)=VRtT⁡\(P1\)⊔VRtT⁡\(P2\)\.\\operatorname\{VR\}^\{T\}\_\{t\}\(P\_\{1\}\\sqcup P\_\{2\}\)=\\operatorname\{VR\}^\{T\}\_\{t\}\(P\_\{1\}\)\\sqcup\\operatorname\{VR\}^\{T\}\_\{t\}\(P\_\{2\}\)\.These identities commute with every filtration structure map\. ApplyingHn​\(−;𝕂\)H\_\{n\}\(\-;\\mathbb\{K\}\)gives the asserted isomorphism inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. SinceJ↓∙J\\downarrow\\bulletis discrete with two objects, the pointwise Kan\-extension formula gives\(LanJ⁡FnT\)​\(∙\)≅FnT​\(a1\)⊕FnT​\(a2\)\.\(\\operatorname\{Lan\}\_\{J\}F^\{T\}\_\{n\}\)\(\\bullet\)\\cong F^\{T\}\_\{n\}\(a\_\{1\}\)\\oplus F^\{T\}\_\{n\}\(a\_\{2\}\)\.

###### Corollary 6\.3\.

LetT\>0T\>0, and letP1,P2,P1′,P2′P\_\{1\},P\_\{2\},P^\{\\prime\}\_\{1\},P^\{\\prime\}\_\{2\}be nonempty finite subsets of one metric space\. Assume that, for someδ≥0\\delta\\geq 0,dist⁡\(P1,P2\)\>T\+2​δ,dH​\(Pi,Pi′\)≤δ​for​i=1,2\.\\operatorname\{dist\}\(P\_\{1\},P\_\{2\}\)\>T\+2\\delta,\\ d\_\{H\}\(P\_\{i\},P^\{\\prime\}\_\{i\}\)\\leq\\delta\\ \\text\{for \}i=1,2\.Thendist⁡\(P1′,P2′\)\>T\\operatorname\{dist\}\(P^\{\\prime\}\_\{1\},P^\{\\prime\}\_\{2\}\)\>T\. IfFn′⁣T​\(ai\)=PHnT⁡\(Pi′\),Gn′⁣T​\(∙\)=PHnT⁡\(P1′⊔P2′\),F^\{\\prime T\}\_\{n\}\(a\_\{i\}\)=\\operatorname\{PH\}^\{T\}\_\{n\}\(P^\{\\prime\}\_\{i\}\),\\ G^\{\\prime T\}\_\{n\}\(\\bullet\)=\\operatorname\{PH\}^\{T\}\_\{n\}\(P^\{\\prime\}\_\{1\}\\sqcup P^\{\\prime\}\_\{2\}\),then, for every finite degree range0≤n≤q0\\leq n\\leq qand every choice of nonnegative weightsw0,…,wqw\_\{0\},\\ldots,w\_\{q\},

∑n=0qwn​dI​\(\(LanJ⁡Fn′⁣T\)​\(∙\),Gn′⁣T​\(∙\)\)=0\.\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{I\}\\bigl\(\(\\operatorname\{Lan\}\_\{J\}F^\{\\prime T\}\_\{n\}\)\(\\bullet\),G^\{\\prime T\}\_\{n\}\(\\bullet\)\\bigr\)=0\.

###### Proof 6\.4\.

Forx′∈P1′x^\{\\prime\}\\in P^\{\\prime\}\_\{1\}andy′∈P2′y^\{\\prime\}\\in P^\{\\prime\}\_\{2\}, choosex∈P1x\\in P\_\{1\}andy∈P2y\\in P\_\{2\}withd​\(x,x′\)≤δd\(x,x^\{\\prime\}\)\\leq\\deltaandd​\(y,y′\)≤δd\(y,y^\{\\prime\}\)\\leq\\delta\. Thend​\(x′,y′\)≥d​\(x,y\)−d​\(x,x′\)−d​\(y,y′\)\>T\.d\(x^\{\\prime\},y^\{\\prime\}\)\\geq d\(x,y\)\-d\(x,x^\{\\prime\}\)\-d\(y,y^\{\\prime\}\)\>T\.Taking the infimum overx′∈P1′x^\{\\prime\}\\in P^\{\\prime\}\_\{1\}andy′∈P2′y^\{\\prime\}\\in P^\{\\prime\}\_\{2\}givesdist⁡\(P1′,P2′\)\>T\\operatorname\{dist\}\(P^\{\\prime\}\_\{1\},P^\{\\prime\}\_\{2\}\)\>T\. The equality of persistence modules follows from\\Crefprop:separated\-vr\-merge degreewise; each summand in the weighted discrepancy is therefore zero\.

Equivalently, the weighted discrepancyCompJ,qI,T\\operatorname\{Comp\}^\{I,T\}\_\{J,q\}of the perturbed intended merge is zero\. The frozen\-threshold construction has no unrestricted finite Lipschitz bound under perturbations of point clouds\. Indeed, for0<δ<T/30<\\delta<T/3, letP=\{0,T\+δ\}P=\\\{0,T\+\\delta\\\}andQ=\{0,T−δ\}⊆ℝQ=\\\{0,T\-\\delta\\\}\\subseteq\\mathbb\{R\}\. ThendH​\(P,Q\)=2​δd\_\{H\}\(P,Q\)=2\\delta, butPH0T⁡\(P\)\\operatorname\{PH\}^\{T\}\_\{0\}\(P\)has two essential intervals whereasPH0T⁡\(Q\)\\operatorname\{PH\}^\{T\}\_\{0\}\(Q\)has one\. HencedI​\(PH0T⁡\(P\),PH0T⁡\(Q\)\)=\+∞\.d\_\{I\}\(\\operatorname\{PH\}^\{T\}\_\{0\}\(P\),\\operatorname\{PH\}^\{T\}\_\{0\}\(Q\)\)=\+\\infty\.Thus\\Crefcor:separated\-merge\-robustness requires a separation margin\.

###### Proposition 6\.5\.

Letn≥1n\\geq 1, and leti∙:A∙↪X∙i\_\{\\bullet\}:A\_\{\\bullet\}\\hookrightarrow X\_\{\\bullet\}andj∙:A∙↪Y∙j\_\{\\bullet\}:A\_\{\\bullet\}\\hookrightarrow Y\_\{\\bullet\}be natural degreewise inclusions of\(ℝ,≤\)\(\\mathbb\{R\},\\leq\)\-indexed filtered simplicial complexes\. PutZ∙≔X∙∪A∙Y∙Z\_\{\\bullet\}\\coloneqq X\_\{\\bullet\}\\cup\_\{A\_\{\\bullet\}\}Y\_\{\\bullet\}, and denote the canonical inclusions byk∙:X∙↪Z∙k\_\{\\bullet\}:X\_\{\\bullet\}\\hookrightarrow Z\_\{\\bullet\}andℓ∙:Y∙↪Z∙\\ell\_\{\\bullet\}:Y\_\{\\bullet\}\\hookrightarrow Z\_\{\\bullet\}\. If the natural morphism

\(i∗,−j∗\):Hn−1​\(A∙;𝕂\)⟶Hn−1​\(X∙;𝕂\)⊕Hn−1​\(Y∙;𝕂\)\(i\_\{\*\},\-j\_\{\*\}\):H\_\{n\-1\}\(A\_\{\\bullet\};\\mathbb\{K\}\)\\longrightarrow H\_\{n\-1\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\oplus H\_\{n\-1\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)is monic inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, then the canonical morphism

\(k∗,ℓ∗\)¯:colim\(Hn​\(X∙;𝕂\)←i∗Hn​\(A∙;𝕂\)→j∗Hn​\(Y∙;𝕂\)\)⟶Hn​\(Z∙;𝕂\)\\overline\{\(k\_\{\*\},\\ell\_\{\*\}\)\}:\\operatorname\*\{colim\}\\bigl\(H\_\{n\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\xleftarrow\{i\_\{\*\}\}H\_\{n\}\(A\_\{\\bullet\};\\mathbb\{K\}\)\\xrightarrow\{j\_\{\*\}\}H\_\{n\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)\\bigr\)\\longrightarrow H\_\{n\}\(Z\_\{\\bullet\};\\mathbb\{K\}\)is an isomorphism inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\.

###### Proof 6\.6\.

For each filtration parameter, the inclusions give the Mayer–Vietoris exact sequence\. Naturality with respect to the filtration maps gives the following exact sequence inPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}:

Hn​\(A∙;𝕂\)\{H\_\{n\}\(A\_\{\\bullet\};\\mathbb\{K\}\)\}Hn​\(X∙;𝕂\)⊕Hn​\(Y∙;𝕂\)\{H\_\{n\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\oplus H\_\{n\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)\}Hn​\(Z∙;𝕂\)\{H\_\{n\}\(Z\_\{\\bullet\};\\mathbb\{K\}\)\}Hn−1​\(A∙;𝕂\)\{H\_\{n\-1\}\(A\_\{\\bullet\};\\mathbb\{K\}\)\}Hn−1​\(X∙;𝕂\)⊕Hn−1​\(Y∙;𝕂\)\{H\_\{n\-1\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\oplus H\_\{n\-1\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)\}\(i∗,−j∗\)\\scriptstyle\{\(i\_\{\*\},\-j\_\{\*\}\)\}k∗\+ℓ∗\\scriptstyle\{k\_\{\*\}\+\\ell\_\{\*\}\}∂\\scriptstyle\\partial\(i∗,−j∗\)\\scriptstyle\{\(i\_\{\*\},\-j\_\{\*\}\)\}Since the final morphism is monic by hypothesis, exactness givesim⁡\(∂\)=0\\operatorname\{im\}\(\\partial\)=0, hence∂=0\\partial=0\. Consequently,im⁡\(k∗\+ℓ∗\)=ker⁡\(∂\)=Hn​\(Z∙;𝕂\),\\operatorname\{im\}\(k\_\{\*\}\+\\ell\_\{\*\}\)=\\ker\(\\partial\)=H\_\{n\}\(Z\_\{\\bullet\};\\mathbb\{K\}\),sok∗\+ℓ∗k\_\{\*\}\+\\ell\_\{\*\}is an epimorphism\. Exactness atHn​\(X∙;𝕂\)⊕Hn​\(Y∙;𝕂\)H\_\{n\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\oplus H\_\{n\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)givesker⁡\(k∗\+ℓ∗\)=im⁡\(i∗,−j∗\)\.\\ker\(k\_\{\*\}\+\\ell\_\{\*\}\)=\\operatorname\{im\}\(i\_\{\*\},\-j\_\{\*\}\)\.Thereforek∗\+ℓ∗k\_\{\*\}\+\\ell\_\{\*\}induces an isomorphismcoker⁡\(i∗,−j∗\)→≅Hn​\(Z∙;𝕂\)\.\\operatorname\{coker\}\(i\_\{\*\},\-j\_\{\*\}\)\\xrightarrow\{\\;\\cong\\;\}H\_\{n\}\(Z\_\{\\bullet\};\\mathbb\{K\}\)\.In the abelian categoryPers𝕂\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}, the pushout ofHn​\(X∙;𝕂\)←i∗Hn​\(A∙;𝕂\)→j∗Hn​\(Y∙;𝕂\)H\_\{n\}\(X\_\{\\bullet\};\\mathbb\{K\}\)\\xleftarrow\{i\_\{\*\}\}H\_\{n\}\(A\_\{\\bullet\};\\mathbb\{K\}\)\\xrightarrow\{j\_\{\*\}\}H\_\{n\}\(Y\_\{\\bullet\};\\mathbb\{K\}\)is canonically isomorphic tocoker⁡\(i∗,−j∗\)\\operatorname\{coker\}\(i\_\{\*\},\-j\_\{\*\}\)\. Under this identification, the induced morphism toHn​\(Z∙;𝕂\)H\_\{n\}\(Z\_\{\\bullet\};\\mathbb\{K\}\)is\(k∗,ℓ∗\)¯\\overline\{\(k\_\{\*\},\\ell\_\{\*\}\)\}\.

Thus a geometric pushout realises the Kan\-induced pushout of degree\-nnpersistence modules whenever the displayed Mayer–Vietoris obstruction vanishes\. The separated merge of\\Crefprop:separated\-vr\-merge is the overlap\-free case and also covers degree0\.

###### Proposition 6\.7\.

FixJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, nonnegative weightsw0,…,wqw\_\{0\},\\ldots,w\_\{q\}, and source diagramsF0,…,Fq:𝒜→Pers𝕂F\_\{0\},\\ldots,F\_\{q\}:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}\. LetGn,Gn′:ℬ→Pers𝕂G\_\{n\},G^\{\\prime\}\_\{n\}:\\mathcal\{B\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}satisfydI​\(Gn​\(b\),Gn′​\(b\)\)=0d\_\{I\}\(G\_\{n\}\(b\),G^\{\\prime\}\_\{n\}\(b\)\)=0for everyb∈Ob⁡\(ℬ\)b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)and0≤n≤q0\\leq n\\leq q\. Then

supb∈Ob⁡\(ℬ\)∑n=0qwn​dI​\(\(LanJ⁡Fn\)​\(b\),Gn​\(b\)\)=supb∈Ob⁡\(ℬ\)∑n=0qwn​dI​\(\(LanJ⁡Fn\)​\(b\),Gn′​\(b\)\)\.\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G\_\{n\}\(b\)\)=\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}\\sum\_\{n=0\}^\{q\}w\_\{n\}d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G^\{\\prime\}\_\{n\}\(b\)\)\.

###### Proof 6\.8\.

For fixedbbandnn, the triangle inequality gives

dI​\(\(LanJ⁡Fn\)​\(b\),Gn​\(b\)\)≤dI​\(\(LanJ⁡Fn\)​\(b\),Gn′​\(b\)\)\+dI​\(Gn′​\(b\),Gn​\(b\)\),d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G\_\{n\}\(b\)\)\\leq d\_\{I\}\(\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(b\),G^\{\\prime\}\_\{n\}\(b\)\)\+d\_\{I\}\(G^\{\\prime\}\_\{n\}\(b\),G\_\{n\}\(b\)\),and the last summand is zero\. ExchangingGnG\_\{n\}andGn′G^\{\\prime\}\_\{n\}gives equality of the two summands\. Summing with weights and taking the supremum proves the assertion\.

Consequently, the reported statistic cannot distinguish semantic substitutions whose selected persistence modules agree at every target object in the measured degree range\.

Table 1:ℬ=𝟏\\mathcal\{B\}=\\mathbf\{1\},T=2\.25T=2\.25, 20 seeds\.Rows:the intended target point cloudPt​\(∙\)P\_\{t\}\(\\bullet\)and three controls\.Columns:w0​dI,0w\_\{0\}\\,d\_\{I,0\}andw1​dI,1w\_\{1\}\\,d\_\{I,1\}is weighted degree\-0and degree\-11bottleneck contributions \(means,w0=0\.25w\_\{0\}\{=\}0\.25,w1=1w\_\{1\}\{=\}1\)\.CompJ,1I,T=w0​dI,0\+w1​dI,1\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}=w\_\{0\}\\,d\_\{I,0\}\+w\_\{1\}\\,d\_\{I,1\}is Kan structural discrepancy\.ObjBaseJ,1T\\operatorname\{ObjBase\}^\{T\}\_\{J,1\}is objectwise baseline \(mean bottleneck of each source against the target, without task transport\);mins⁡μ\\min\_\{s\}\\muis separation margindist⁡\(Ps​\(a1\),Ps​\(a2\)\)−T\\operatorname\{dist\}\(P\_\{s\}\(a\_\{1\}\),P\_\{s\}\(a\_\{2\}\)\)\-T, defined for the intended merge only\. Scores and finite baselines: mean±\\,\\pm\\,sd\.∙\\bulletunique regime confirmingCompJ,1I,T=0\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}=0\(μ\>0\\mu\>0\)\.∙\\bulletqualitatively worst failure:CompJ,1I,T=∞\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}=\\inftyyetObjBaseJ,1T\\operatorname\{ObjBase\}^\{T\}\_\{J,1\}finite, demonstrating that the baseline does not detect the missing summand\.
### Experiment 1: synthetic domain merge\.

Let𝒜\\mathcal\{A\}be the discrete category with object setOb⁡\(𝒜\)=\{a1,a2\}\\operatorname\{Ob\}\(\\mathcal\{A\}\)=\\\{a\_\{1\},a\_\{2\}\\\}, letℬ=𝟏\\mathcal\{B\}=\\mathbf\{1\}be the terminal category with unique object∙\\bullet, and letJ:𝒜→𝟏J:\\mathcal\{A\}\\to\\mathbf\{1\}be the unique functor\. For everyn∈ℕn\\in\\mathbb\{N\}, letFn:𝒜→Pers𝕂F\_\{n\}:\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}be the source persistence\-module diagram in degreenn\. ThenJ↓∙J\\downarrow\\bulletis the discrete category with object set\{\(a1,id∙\),\(a2,id∙\)\}\.\\\{\(a\_\{1\},\\operatorname\{id\}\_\{\\bullet\}\),\(a\_\{2\},\\operatorname\{id\}\_\{\\bullet\}\)\\\}\.Hence the pointwise Kan\-extension formula gives the canonical isomorphism\(LanJ⁡Fn\)​\(∙\)≅colimJ↓∙Fn≅Fn​\(a1\)⊕Fn​\(a2\)\.\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(\\bullet\)\\cong\\operatorname\*\{colim\}\_\{J\\downarrow\\bullet\}F\_\{n\}\\cong F\_\{n\}\(a\_\{1\}\)\\oplus F\_\{n\}\(a\_\{2\}\)\.Consequently, if the persistence modules are finite interval\-decomposable, then the barcode of\(LanJ⁡Fn\)​\(∙\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(\\bullet\)is the multiset union of the barcodes ofFn​\(a1\)F\_\{n\}\(a\_\{1\}\)andFn​\(a2\)F\_\{n\}\(a\_\{2\}\)\. We test:

1. H1\.IfPt​\(∙\)=Ps​\(a1\)⊔Ps​\(a2\)P\_\{t\}\(\\bullet\)=P\_\{s\}\(a\_\{1\}\)\\sqcup P\_\{s\}\(a\_\{2\}\), thenCompJ,1B=0\\operatorname\{Comp\}^\{B\}\_\{J,1\}=0up to numerical precision\.
2. H2\.If the target is altered or collapsed, thenCompJ,1B\>0\\operatorname\{Comp\}^\{B\}\_\{J,1\}\>0\.
3. H3\.Objectwise agreement is not expected\. The expected match isGn​\(∙\)≃Fn​\(a1\)⊕Fn​\(a2\)G\_\{n\}\(\\bullet\)\\simeq F\_\{n\}\(a\_\{1\}\)\\oplus F\_\{n\}\(a\_\{2\}\)\.

We test this prediction on synthetic planar point clouds with3636points per source object and2020seeds\. The sourcea1a\_\{1\}is a noisy circle of radius11, anda2a\_\{2\}is a noisy figure\-eight of scale0\.950\.95, both with Gaussian noise0\.0350\.035\. In the correct regime,Pt​\(∙\)P\_\{t\}\(\\bullet\)is the separated union ofPs​\(a1\)P\_\{s\}\(a\_\{1\}\)andPs​\(a2\)P\_\{s\}\(a\_\{2\}\)with separation5\.05\.0\. In the controls,Ps​\(a2\)P\_\{s\}\(a\_\{2\}\)is replaced by a filled disk of radius0\.850\.85, omitted, or both components are collapsed to tight Gaussian clusters with noise0\.0250\.025\. All comparisons useT=2\.25T=2\.25,q=1q=1,w0=0\.25w\_\{0\}=0\.25,w1=1w\_\{1\}=1, coefficients in𝕂=𝔽2\\mathbb\{K\}=\\mathbb\{F\}\_\{2\}, and the bottleneck distancedBd\_\{B\}\. For each seed we computeCompJ,1B⁡\(Fn​\(a1\)⊕Fn​\(a2\),Gn​\(∙\)\),ObjBaseJ,1⁡\(Fn​\(ai\),Gn​\(∙\)\)\.\\operatorname\{Comp\}^\{B\}\_\{J,1\}\(F\_\{n\}\(a\_\{1\}\)\\oplus F\_\{n\}\(a\_\{2\}\),G\_\{n\}\(\\bullet\)\),\\operatorname\{ObjBase\}\_\{J,1\}\(F\_\{n\}\(a\_\{i\}\),G\_\{n\}\(\\bullet\)\)\.

\\Cref

tab:experiment\-domain\-merge answers all hypotheses\. The correct merge satisfiesCompJ,1B=0\\operatorname\{Comp\}^\{B\}\_\{J,1\}=0up to floating\-point precision, verifying[H1](https://arxiv.org/html/2606.07627#S6.I1.i1)\. Second, each control has positive discrepancy dominated byH1H\_\{1\}, verifying[H2](https://arxiv.org/html/2606.07627#S6.I1.i2)\. Third,ObjBaseJ,1\\operatorname\{ObjBase\}\_\{J,1\}is large even for the correct merge, verifying[H3](https://arxiv.org/html/2606.07627#S6.I1.i3)\. The agreement is the Kan\-extension coproduct, not an objectwise source\-target match\.

TargetPt​\(di\)P\_\{t\}\(d\_\{i\}\)w0​dI,0w\_\{0\}\\,d\_\{I,0\}w1​dI,1w\_\{1\}\\,d\_\{I,1\}CompJ,1I,T\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}ObjBaseJ,1T\\operatorname\{ObjBase\}^\{T\}\_\{J,1\}Dom\.Structural outcome\\cellcolorposcellPt​\(di\)=Ps​\(ci\)P\_\{t\}\(d\_\{i\}\)=P\_\{s\}\(c\_\{i\}\)\\cellcolorposcell0\.00\\cellcolorposcell0\.00\\cellcolorposcell0\.00±0\.000\.00\\pm 0\.00\\cellcolorposcell0\.35±0\.010\.35\\pm 0\.01\\cellcolorposcell–\\cellcolorposcellG​\(di\)≅F​\(ci\)G\(d\_\{i\}\)\\cong F\(c\_\{i\}\)WrongPt​\(di\)P\_\{t\}\(d\_\{i\}\)0\.030\.160\.19±0\.030\.19\\pm 0\.030\.45±0\.020\.45\\pm 0\.02H1H\_\{1\}Wrong refined typeDegeneratePt​\(di\)P\_\{t\}\(d\_\{i\}\)0\.040\.190\.23±0\.020\.23\\pm 0\.020\.47±0\.010\.47\\pm 0\.01H1H\_\{1\}Degenerate fine class\\cellcolornegcellCollapsedPt​\(di\)P\_\{t\}\(d\_\{i\}\)\\cellcolornegcell0\.04\\cellcolornegcell0\.67\\cellcolornegcell0\.72±0\.020\.72\\pm 0\.02\\cellcolornegcell0\.47±0\.010\.47\\pm 0\.01\\cellcolornegcellH1H\_\{1\}\\cellcolornegcellH1H\_\{1\}homology destroyedTable 2:Experiment 2 \(class refinement\),T=2\.25T=2\.25, 20 seeds\.Rows:the intended fine\-object targetPt​\(di\)P\_\{t\}\(d\_\{i\}\)and three controls\.Columns:w0​dI,0w\_\{0\}\\,d\_\{I,0\}andw1​dI,1w\_\{1\}\\,d\_\{I,1\}is weighted degree\-0and degree\-11bottleneck contributions \(means,w0=0\.25w\_\{0\}\{=\}0\.25,w1=1w\_\{1\}\{=\}1\)\.CompJ,1I,T=w0​dI,0\+w1​dI,1\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}=w\_\{0\}\\,d\_\{I,0\}\+w\_\{1\}\\,d\_\{I,1\}is the Kan structural discrepancy, comparingG​\(di\)G\(d\_\{i\}\)against the refinement\-induced invariant\(LanJ⁡F\)​\(di\)≅F​\(ci\)\(\\operatorname\{Lan\}\_\{J\}F\)\(d\_\{i\}\)\\cong F\(c\_\{i\}\)\.ObjBaseJ,1T\\operatorname\{ObjBase\}^\{T\}\_\{J,1\}is the objectwise baseline \(mean bottleneck of each coarse source against the fine target, without task transport\)\. Dom\. is the homological degree dominating the discrepancy\. Scores and baselines: mean±\\,\\pm\\,sd\.∙\\bulletunique regime confirmingCompJ,1I,T=0\\operatorname\{Comp\}^\{I,T\}\_\{J,1\}=0\(Proposition[6\.5](https://arxiv.org/html/2606.07627#S6.Thmtheorem5)J↓diJ\\downarrow d\_\{i\}has terminal object\(ci,ρi\)\(c\_\{i\},\\rho\_\{i\}\)\)\.∙\\bulletlargest discrepancy, anH1H\_\{1\}\-dominated collapse;ObjBaseJ,1T\\operatorname\{ObjBase\}^\{T\}\_\{J,1\}is large even in the matching regime, it does not certify the refinement\.
### Experiment 2: synthetic class refinement\.

Let𝒜\\mathcal\{A\}be the discrete category withOb⁡\(𝒜\)=\{c1,c2\}\\operatorname\{Ob\}\(\\mathcal\{A\}\)=\\\{c\_\{1\},c\_\{2\}\\\}\. Letℬ\\mathcal\{B\}have objectsc1,c2,d1,d2c\_\{1\},c\_\{2\},d\_\{1\},d\_\{2\}, identities, and refinement arrowsρi:ci→di\\rho\_\{i\}\\colon c\_\{i\}\\to d\_\{i\}\. LetJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}be the inclusion onc1,c2c\_\{1\},c\_\{2\}\. ThenJ↓diJ\\downarrow d\_\{i\}has terminal object\(ci,ρi\)\(c\_\{i\},\\rho\_\{i\}\), hence\(LanJ⁡Fn\)​\(di\)≅Fn​\(ci\)\(\\operatorname\{Lan\}\_\{J\}F\_\{n\}\)\(d\_\{i\}\)\\cong F\_\{n\}\(c\_\{i\}\)\. We test:

1. H1\.IfPt​\(di\)=Ps​\(ci\)P\_\{t\}\(d\_\{i\}\)=P\_\{s\}\(c\_\{i\}\)fori=1,2i=1,2, thenCompJ,1B=0\\operatorname\{Comp\}^\{B\}\_\{J,1\}=0up to numerical precision\.
2. H2\.If a fine target is altered, omitted, or collapsed, thenCompJ,1B\>0\\operatorname\{Comp\}^\{B\}\_\{J,1\}\>0\.
3. H3\.The expected match isGn​\(di\)≃Fn​\(ci\)G\_\{n\}\(d\_\{i\}\)\\simeq F\_\{n\}\(c\_\{i\}\)fori=1,2i=1,2\.

We use the same coefficients, weights, metric, and number of seeds as in Experiment 1\. The sourcec1c\_\{1\}is a noisy circle of radius11, andc2c\_\{2\}is a noisy figure\-eight of scale0\.950\.95, both with Gaussian noise0\.0350\.035\. In the correct regimePt​\(di\)=Ps​\(ci\)P\_\{t\}\(d\_\{i\}\)=P\_\{s\}\(c\_\{i\}\)\. In the controls,Pt​\(d2\)P\_\{t\}\(d\_\{2\}\)is replaced by a filled disk, replaced by a tight cluster, or bothPt​\(di\)P\_\{t\}\(d\_\{i\}\)are collapsed to tight clusters\. For each seed we computeCompJ,1B⁡\(Fn​\(ci\),Gn​\(di\)\),ObjBaseJ,1⁡\(Fn​\(cj\),Gn​\(di\)\)\.\\operatorname\{Comp\}^\{B\}\_\{J,1\}\(F\_\{n\}\(c\_\{i\}\),G\_\{n\}\(d\_\{i\}\)\),\\operatorname\{ObjBase\}\_\{J,1\}\(F\_\{n\}\(c\_\{j\}\),G\_\{n\}\(d\_\{i\}\)\)\.

\\Cref

tab:experiment\-class\-refinement answers all hypotheses\. The correct refinement verifies[H1](https://arxiv.org/html/2606.07627#S6.I2.i1)\. The three controls verify[H2](https://arxiv.org/html/2606.07627#S6.I2.i2)\. The zero score occurs exactly forGn​\(di\)≃Fn​\(ci\)G\_\{n\}\(d\_\{i\}\)\\simeq F\_\{n\}\(c\_\{i\}\), verifying[H3](https://arxiv.org/html/2606.07627#S6.I2.i3)\.

Experiment 3A:Exact theorem realisation\.ModelSep\\mathrm\{Sep\}Real\\mathrm\{Real\}Zero∣Sep\\mathrm\{Zero\}\\mid\\mathrm\{Sep\}Detstr∣App\\mathrm\{Det\}\_\{\\mathrm\{str\}\}\\mid\\mathrm\{App\}Changesem\\mathrm\{Change\}\_\{\\mathrm\{sem\}\}𝖠𝖤\\mathsf\{AE\}6/20​\[\.12,\.54\]6/20\\,\[\.12,\.54\]\\cellcolornegcell1/20​\[\.00,\.25\]1/20\\,\[\.00,\.25\]\\cellcolorposcell6/6​\[\.54,1\.00\]6/6\\,\[\.54,1\.00\]\\cellcolorposcell18/18​\[\.81,1\.00\]18/18\\,\[\.81,1\.00\]55/160​\[\.27,\.42\]55/160\\,\[\.27,\.42\]𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}3/20​\[\.03,\.38\]3/20\\,\[\.03,\.38\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]\\cellcolorposcell3/3​\[\.29,1\.00\]3/3\\,\[\.29,1\.00\]\\cellcolorposcell9/9​\[\.66,1\.00\]9/9\\,\[\.66,1\.00\]33/160​\[\.15,\.28\]33/160\\,\[\.15,\.28\]𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}7/20​\[\.15,\.59\]7/20\\,\[\.15,\.59\]\\cellcolornegcell1/20​\[\.00,\.25\]1/20\\,\[\.00,\.25\]\\cellcolorposcell7/7​\[\.59,1\.00\]7/7\\,\[\.59,1\.00\]\\cellcolorposcell21/21​\[\.84,1\.00\]21/21\\,\[\.84,1\.00\]64/160​\[\.32,\.48\]64/160\\,\[\.32,\.48\]Experiment 3B:Held\-out transfer\.ModelSep\\mathrm\{Sep\}Base\\mathrm\{Base\}Detstr∣Base\\mathrm\{Det\}\_\{\\mathrm\{str\}\}\\mid\\mathrm\{Base\}Rankstr\\mathrm\{Rank\}\_\{\\mathrm\{str\}\}Changesem\\mathrm\{Change\}\_\{\\mathrm\{sem\}\}𝖠𝖤\\mathsf\{AE\}3/20​\[\.03,\.38\]3/20\\,\[\.03,\.38\]\\cellcolornegcell1/20​\[\.00,\.25\]1/20\\,\[\.00,\.25\]\\cellcolorposcell3/3​\[\.29,1\.00\]3/3\\,\[\.29,1\.00\]\\cellcolornegcell2/20​\[\.01,\.32\]2/20\\,\[\.01,\.32\]66/160​\[\.34,\.49\]66/160\\,\[\.34,\.49\]𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]–\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]29/160​\[\.12,\.25\]29/160\\,\[\.12,\.25\]𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}2/20​\[\.01,\.32\]2/20\\,\[\.01,\.32\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]–\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]31/160​\[\.14,\.26\]31/160\\,\[\.14,\.26\]Experiment 3C:Train–test stress\.ModelSep\\mathrm\{Sep\}Base\\mathrm\{Base\}Detstr∣Base\\mathrm\{Det\}\_\{\\mathrm\{str\}\}\\mid\\mathrm\{Base\}Rankstr\\mathrm\{Rank\}\_\{\\mathrm\{str\}\}Changesem\\mathrm\{Change\}\_\{\\mathrm\{sem\}\}𝖠𝖤\\mathsf\{AE\}0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]–\\cellcolornegcell2/20​\[\.01,\.32\]2/20\\,\[\.01,\.32\]36/160​\[\.16,\.30\]36/160\\,\[\.16,\.30\]𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}1/20​\[\.00,\.25\]1/20\\,\[\.00,\.25\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]–\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]19/160​\[\.07,\.18\]19/160\\,\[\.07,\.18\]𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}1/20​\[\.00,\.25\]1/20\\,\[\.00,\.25\]\\cellcolornegcell0/20​\[\.00,\.17\]0/20\\,\[\.00,\.17\]–\\cellcolornegcell2/20​\[\.01,\.32\]2/20\\,\[\.01,\.32\]43/160​\[\.20,\.34\]43/160\\,\[\.20,\.34\]Table 3:Learned latent domain merge\. Each entry is an observed ratex/n​\[ℓ,u\]x/n\\,\[\\ell,u\], where\[ℓ,u\]\[\\ell,u\]is the exact95%95\\%Clopper–Pearson confidence interval\. In 3A,Sep\\mathrm\{Sep\}is the separated\-union premise rate,Real\\mathrm\{Real\}is the rate of internally connected separated\-domain realisations,Zero∣Sep\\mathrm\{Zero\}\\mid\\mathrm\{Sep\}is the exact\-zero confirmation rate for the intended merge conditional on separation, andDetstr∣App\\mathrm\{Det\}\_\{\\mathrm\{str\}\}\\mid\\mathrm\{App\}is the conditional structural\-detection rate\. In 3B and C,Base\\mathrm\{Base\}is the finite\-baseline applicability rate,Detstr∣Base\\mathrm\{Det\}\_\{\\mathrm\{str\}\}\\mid\\mathrm\{Base\}is the conditional structural\-detection rate, andRankstr\\mathrm\{Rank\}\_\{\\mathrm\{str\}\}is the unconditional rate at which all structural controls exceed the intended target\. In every panel,Changesem\\mathrm\{Change\}\_\{\\mathrm\{sem\}\}is the secondary wrong\-digit sensitivity diagnostic\.∙\\color\[rgb\]\{0,0,1\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0,1\}\\bulletconfirms a hypothesis;∙\\color\[rgb\]\{1,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{1,0,0\}\\bulletmarks a failed confirmatory endpoint or insufficient applicability for its evaluation\.
### Experiment 3: learned latent domain merge\.

Experiments 1 and 2 evaluate the categorical transfer score on controlled invariants\. Experiment 3 applies the same domain\-merge transformation to learned latent representations of MNIST digits\. Let𝒜\\mathcal\{A\}denote the discrete category on the two source domainsa1a\_\{1\}anda2a\_\{2\}, letℬ=𝟏\\mathcal\{B\}=\\mathbf\{1\}be the terminal category with unique object∙\\bullet, and letJ:𝒜→𝟏J\\colon\\mathcal\{A\}\\to\\mathbf\{1\}be the unique functor\. For each representation modelMMand homology degreenn, the source functorFnM:𝒜→Pers𝕂F^\{M\}\_\{n\}\\colon\\mathcal\{A\}\\to\\operatorname\{Pers\}\_\{\\mathbb\{K\}\}assigns toa1a\_\{1\}anda2a\_\{2\}the persistent homology of the latent clouds of the digits0and11, respectively\. Because left Kan extension alongJJcomputes a colimit over the discrete fibre, we have a natural isomorphism\(LanJ⁡FnM\)​\(∙\)≅FnM​\(a1\)⊕FnM​\(a2\)\.\(\\operatorname\{Lan\}\_\{J\}F^\{M\}\_\{n\}\)\(\\bullet\)\\;\\cong\\;F^\{M\}\_\{n\}\(a\_\{1\}\)\\oplus F^\{M\}\_\{n\}\(a\_\{2\}\)\.The intended target is the merged0\+10\{\+\}1latent task\. Structural controls either delete one source domain or join the two domains by an artificial bridge; semantic controls replace the digit11by an incorrect digit in\{2,…,9\}\\\{2,\\ldots,9\\\}\.

We compare three architectures: a vanilla autoencoder𝖠𝖤\\mathsf\{AE\}, a topological autoencoder𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}, and a task\-aligned topological autoencoder𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}\. For a fixed thresholdτ\\tau, the degree\-zero score isCompJB,0⁡\(FM,GM\)≔dB​\(Dgm⁡\(\(LanJ⁡F0M\)​\(∙\)\),Dgm⁡\(G0M​\(∙\)\)\),\\operatorname\{Comp\}^\{B,0\}\_\{J\}\(F^\{M\},G^\{M\}\)\\coloneqq d\_\{B\}\\\!\\left\(\\operatorname\{Dgm\}\\bigl\(\(\\operatorname\{Lan\}\_\{J\}F^\{M\}\_\{0\}\)\(\\bullet\)\\bigr\),\\,\\operatorname\{Dgm\}\\bigl\(G^\{M\}\_\{0\}\(\\bullet\)\\bigr\)\\right\),wheredBd\_\{B\}is the bottleneck distance\. Degree\-one scores are reported as exploratory only\. Since degree\-zero persistence carries the essential connected\-component classes, a structural control may legitimately return\+∞\+\\inftywhenever it alters the number of essential components\. Each experiment is evaluated over2020seeds\. Rates are accompanied by exact95%95\\%Clopper–Pearson confidence intervals\. Paired differences between𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}and each ablation are reported with paired bootstrap confidence intervals and McNemarpp\-values\.

### Experiment 3A: exact theorem realisation\.

Experiment 3A tests the geometric domain\-merge theorem\. The separated\-union premise holds when the two latent source domains remain separated at the thresholdτ\\tau\. The stronger domain\-realisation endpoint requires, in addition, that both source domains be internally connected atτ\\tau\. Under the separated\-union premise, the merged target must agree with the Kan\-induced coproduct in degree zero\.

1. H3A\.1Under separated\-union applicability, the intended merge satisfiesCompJB,0=0\\operatorname\{Comp\}^\{B,0\}\_\{J\}=0\.
2. H3A\.2Task\-aligned training increases valid realisations relative to both ablations\.
3. H3A\.3Under applicability, every structural control exceeds the intended merge\.

Experiment 3A confirms H3A\.1 and H3A\.3 on their stated domains of applicability\. Whenever the separated\-union premise holds, the intended merge has exactly zero primary discrepancy:6/66/6seeds for𝖠𝖤\\mathsf\{AE\},3/33/3for𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}, and7/77/7for𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}\. Moreover, every theoretically applicable structural control is detected—18/1818/18,9/99/9, and21/2121/21comparisons, respectively—and in each case the control yields infinite degree\-zero discrepancy, as predicted by its change in essential connected\-component structure\. H3A\.2 is not supported\. The domain\-realisation endpoint is attained in only1/201/20seeds for𝖠𝖤\\mathsf\{AE\},0/200/20for𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}, and1/201/20for𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}\. The paired difference between𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}and𝖠𝖤\\mathsf\{AE\}is0\.000\.00, with confidence interval\[−0\.15,0\.15\]\[\-0\.15,0\.15\]andp=1p=1; that between𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}and𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}is0\.050\.05, with confidence interval\[0,0\.15\]\[0,0\.15\]andp=1p=1\. Thus Experiment 3A verifies the categorical theorem when its premise is realised, but provides no evidence that task\-aligned training increases the frequency of the required latent geometry\.

### Experiment 3B: held\-out transfer\.

Experiment 3B replaces exact theorem realisation by an independently sampled target cloud\. Exact agreement is no longer implied: even a correct target task may exhibit nonzero empirical discrepancy from the Kan\-induced source merge, because the source and target persistence diagrams are computed from independent samples\. The confirmatory question is therefore whether structural controls are ranked above the intended target\. A finite\-baseline comparison is theoretically applicable only when both the predicted merge and the intended target exhibit the two\-connected\-domain baseline\.

1. H3B\.1Held\-out structural controls exceed the intended target inCompJB,0\\operatorname\{Comp\}^\{B,0\}\_\{J\}\.
2. H3B\.2Under finite\-baseline applicability, all structural control exceeds the intended target\.
3. H3B\.3Task\-aligned training improves structural ranking relative to both ablations\.

Experiment 3B does not support H3B\.1 or H3B\.3\. The unconditional structural\-ranking rate is2/202/20for𝖠𝖤\\mathsf\{AE\}and0/200/20for both topological models\. The paired difference between𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}and𝖠𝖤\\mathsf\{AE\}is−0\.10\-0\.10, with confidence interval\[−0\.25,0\]\[\-0\.25,0\]andp=0\.5p=0\.5; relative to𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}, the paired difference is0\.000\.00, with confidence interval\[0,0\]\[0,0\]andp=1p=1\. H3B\.2 is confirmed only on the three applicable structural comparisons arising from the single finite\-baseline seed of𝖠𝖤\\mathsf\{AE\}: each control is detected and yields infinite primary discrepancy\. No finite\-baseline seed occurs for either topological model, so their conditional structural\-detection endpoint is not evaluable\. The held\-out experiment thus exposes an applicability limitation rather than a failure of the conditional geometric statement: independent sampling almost always destroys the finite two\-component baseline required by the degree\-zero discriminator\.

### Experiment 3C: train–test stress\.

Experiment 3C evaluates the same degree\-zero discriminator under the most severe train–test sampling shift\. This is a stress test of applicability rather than a further exact\-realisation test; its purpose is to determine whether the latent representation continues to enter the geometric regime in which the categorical score can distinguish the intended merge from structural violations\.

1. H3C\.1Under train–test stress, finite\-baseline applicability permits structural evaluation\.
2. H3C\.2Stress\-test structural controls exceed the intended target inCompJB,0\\operatorname\{Comp\}^\{B,0\}\_\{J\}\.
3. H3C\.3Task\-aligned training improves structural ranking relative to both ablations\.

Experiment 3C does not support H3C\.1 or H3C\.2\. The finite\-baseline applicability event occurs in no stress\-test seed for any model, so conditional structural detection cannot be evaluated\. The unconditional structural\-ranking rate is2/202/20for𝖠𝖤\\mathsf\{AE\},0/200/20for𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}, and2/202/20for𝖳𝖺𝗌𝗄𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TaskTopoAE\}\. Consequently H3C\.3 is also unsupported: the task\-aligned model does not improve structural ranking over𝖠𝖤\\mathsf\{AE\}, and its nominal improvement over𝖳𝗈𝗉𝗈𝖠𝖤\\mathsf\{TopoAE\}rests on only two seeds and cannot establish robust superiority\.

## 7Conclusion

We have supplied the object missing from topological transfer learning: a specified structural map between tasks and the universal invariant it induces\. A source task is a small category𝒜\\mathcal\{A\}, a target taskℬ\\mathcal\{B\}, a task change a functorJ:𝒜→ℬJ:\\mathcal\{A\}\\to\\mathcal\{B\}, and an invariant\-valued representation a functorF:𝒜→𝒱F:\\mathcal\{A\}\\to\\mathcal\{V\}\. The transferred invariant is the left Kan extensionLanJ⁡F\\operatorname\{Lan\}\_\{J\}F, with pointwise value\(LanJ⁡F\)​\(b\)≅colimJ↓bF\(\\operatorname\{Lan\}\_\{J\}F\)\(b\)\\cong\\operatorname\*\{colim\}\_\{J\\downarrow b\}F\. Comparing an observed targetGGagainstLanJ⁡F\\operatorname\{Lan\}\_\{J\}F—never against the source directly—turns “did the representation transfer?” into the falsifiable question of whetherGGrealises the structure thatJJforces\. The transfer discrepancyCompJ⁡\(F,G\)=supb∈Ob⁡\(ℬ\)d𝒱​\(\(LanJ⁡F\)​\(b\),G​\(b\)\)\\operatorname\{Comp\}\_\{J\}\(F,G\)=\\sup\_\{b\\in\\operatorname\{Ob\}\(\\mathcal\{B\}\)\}d\_\{\\mathcal\{V\}\}\\\!\\big\(\(\\operatorname\{Lan\}\_\{J\}F\)\(b\),G\(b\)\\big\)is provably sensitive to the morphisms of the source diagram, not merely its objects \(\\Crefprop:structure\)\. Pointwise vanishing is strictly weaker than exact transferability, with the gap localised by an explicit counterexample \(\\Crefprop:pw\-not\-nat,thm:main\) and closed by a natural\-isomorphism\-detecting variant on\[ℬ,𝒱\]\[\\mathcal\{B\},\\mathcal\{V\}\]\(\\Crefdef:functorial\-discrepancy\)\. Crucially, the colimit becomes computable linear algebra: over finite comma categories it admits a degreewise cokernel presentation \(\\Crefprop:chain\-cokernel,prop:persistent\-cokernel\), preserves finite type \(\\Creflem:finite\-type\-closure\), is11\-Lipschitz in the interleaving distance \(\\Crefthm:lan\-interleaving\-stability\), and—on one\-parameter finite\-type modules—is evaluated exactly by bottleneck distance via the isometrydI=dBd\_\{I\}=d\_\{B\}\(\\Crefprop:correctness\-kan\-persistent\-score\)\.\\Crefprop:separated\-vr\-merge and\\Crefcor:separated\-merge\-robustness certify when a data\-level merge realises the categorical coproduct, with explicit margin robustness\.

### Contrast with prior art\.

Distributional domain\-adaptation bounds quantify a discrepancy but say nothing about which structure survives\. Existing topological methods—decision\-boundary homology for model selection, persistence\-regularised adaptation, and Representation Topology Divergence— compare or rank two given representations\. RTD, the strongest baseline, is a symmetric divergence requiring equal\-size clouds in one\-to\-one correspondence\. None encodes a task functor, none possesses a canonical target, and none expresses merge, collapse, or refinement as a constraint for the target\. Our score does all three, and supplies a canonical zero—the universalLanJ⁡F\\operatorname\{Lan\}\_\{J\}F—against which other methods can be calibrated\.

### Experiments\.

On controlled invariants the zero set ofCompJ,1B\\operatorname\{Comp\}^\{B\}\_\{J,1\}coincides exactly with the correct transfer, while the objectwise baseline does not: the synthetic merge and refinement confirmCompJ,1B=0\\operatorname\{Comp\}^\{B\}\_\{J,1\}=0precisely in the intended regime and detect every structural violation, includingH0H\_\{0\}losses invisible to the baseline \(\\Creftab:experiment\-domain\-merge,tab:experiment\-class\-refinement\)\. Experiment 3 applies the identical merge to learned MNIST latents and cleanly separates mathematical validity from empirical applicability \(\\Creftab:experiment\-3\)\. Experiment 3A confirms the governing theorem on its stated premise without exception: conditional on separated domains, the intended merge attains exactly zero degree\-zero discrepancy \(6/66/6,3/33/3,7/77/7seeds\) and every applicable structural control is detected with infinite discrepancy \(18/1818/18,9/99/9,21/2121/21\) \(\\Crefprop:separated\-vr\-merge and\\Creftab:experiment\-3\)\. However, the stronger internally connected geometry occurs in at most one seed per model, and task\-aligned training does not raise its frequency \(paired differences0\.000\.00/0\.050\.05,p=1p=1\)\. Experiment 3B and 3C show that independent target sampling and train–test shift all but eliminate the finite two\-component baseline the degree\-zero discriminator requires, so its conditioning event becomes non\-evaluable \(\\Creftab:experiment\-3\)\. The verdict is sharp and favourable: the theory fires perfectly whenever its geometric premise holds; the binding constraint is the geometry that current encoders produce and the narrowness of the one\-parameter, degree\-zero, frozen\-threshold discriminator under sampling shift—a fragility the theory itself predicts through the necessity of the separation margin in\\Crefcor:separated\-merge\-robustness\.

### Outlook\.

The contribution is a reusable definition with exactness and stability attached, and it charts its own frontier\. Enriching the target category—e\.g\. to multiparameter persistence, the natural route to widening Experiment 3’s premise—forfeits the exactdI=dBd\_\{I\}=d\_\{B\}identity: ford≥2d\\geq 2there is no complete discrete invariant, and computing the interleaving distance is NP\-hard, with approximation NP\-hard below factor 3\(BjerkevikBotnanKerber2019;CarlssonZomorodian2009\)\. The categorical machinery transports verbatim to any cocomplete𝒱\\mathcal\{V\}; the*fast, exact*evaluation does not, so principled surrogates with re\-proved stability are the key technical task\. Three further directions follow directly: instantiating the natural\-isomorphism\-sensitive discrepancy on target categories with non\-trivial morphisms, where it strictly dominates the pointwise score \(\\Crefprop:pw\-not\-nat,prop:pointwise\-natural\-comparison\); moving beyond degree zero and the frozen threshold via the Mayer–Vietoris realisation of\\Crefprop:filtered\-gluing\-criterion toward perturbation\-stable filtrations; and—most actionably—converting the explicit margin conditions of\\Crefcor:separated\-merge\-robustness into a training objective that drives encoders into the theorem’s domain, turning 3A’s conditional success unconditional\.

### Code

## References