Categorical Foundations of Persistent Homology: Bridging Classical Topology and Topological Data Analysis with Applications
Bibliographic record
Abstract
This paper introduces a novel categorical framework that unifies classical algebraic topology with modern topological data analysis through the lens of category theory. We develop the theory of persistence categories as a natural generalization of persistence modules, establishing functorial relationships between classical topological invariants and their persistent counterparts. Our approach reveals deep connections between sheaf cohomology, spectral sequences, and multi-parameter persistence, providing a rigorous mathematical foundation for understanding the stability and structure of topological features in data. We prove that persistent homology can be viewed as a particular instance of a more general categorical construction that encompasses both classical and computational topology. Furthermore, we establish new stability theorems for categorical persistence and demonstrate how classical results in algebraic topology can be lifted to the persistent setting through appropriate functorial constructions. We present practical applications in data science, computational biology, and machine learning, demonstrating the effectiveness of our theoretical framework through concrete implementations and computational experiments.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.010 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".