Topology Family Coverage Matrix

This page is the honest coverage dashboard for the objective taxonomy. It separates documentation, prototypes, stable APIs, and benchmark evidence.

Family	Current status	Active API	Benchmark status	Next deliverable
Point-set / general topology	Prototype	finite_topology_signature	E2E finite topology check	Tutorial on continuity, compactness, connectedness, and finite covers in data pipelines
Metric, uniform, and proximity spaces	Prototype	metric_cover, nerve_graph	E2E cover and nerve checks	Cover-routing gallery with same-budget random/locality baseline
Domain theory / Scott topology	Prototype	scott_fixed_point	E2E monotone fixed-point check	Monotone dataflow and scheduler invariant design
Homotopy theory	Prototype	path_homotopy_signature	E2E winding-number check	Trajectory path-class tutorial for optimization and systems state
Cohomology beyond Betti counts	Prototype	sheaf_consistency_residual	E2E residual check	Obstruction and interaction diagnostics tutorial
Sheaves and cosheaves	Prototype	sheaf_consistency_residual	E2E residual check	Layer/head/shard residual gallery
Mapper / Reeb / contour topology	Prototype	mapper_graph	E2E Mapper graph check	Activation/loss Mapper visual tutorial
Morse / Conley / dynamical topology	Prototype	dynamical_signature	E2E sampled dynamics check	Training-dynamics topology gallery
Stratified and singular spaces	Prototype	activation_strata	E2E activation-strata check	ReLU region and decision-boundary tutorial
Nerve / Cech / cover calculus	Prototype	metric_cover, nerve_graph	E2E cover and nerve checks	Cech-vs-Rips and routing tutorial
Knot, braid, and link topology	Prototype	braid_crossing_signature	E2E braid crossing check	Thread/trajectory entanglement design note
Fiber bundles and characteristic classes	Docs	TensorBundleSpec as runtime descriptor	E2E TensorBundle interop check	Gauge/equivariant ML tutorial
Topological groups, group actions, quotients	Prototype	finite_orbit_signature, equivariance_residual	E2E finite-orbit and equivariance checks	Equivariance contract utilities
Low-dimensional / geometric topology	Prototype	mesh_euler_characteristic	E2E mesh Euler check	Mesh/scene/handle-surgery explanation
Categorical, Grothendieck, and pointless topology	Prototype	finite_topology_signature	E2E finite open-set lattice check	Sites/locales for observability and privacy
Topological vector spaces / function-space topology	Prototype	weak_convergence_residual	E2E finite-probe weak convergence check	Function-space drift tutorial

The public Python API exposes the same matrix:

import topoml

for row in topoml.topology_family_coverage_matrix():
    print(row["id"], row["status"], row["api_surface"])

Promotion rule: a family moves from docs to prototype only when it has a named object, a computable invariant, a clear ML/system decision, a null baseline, and a graph that makes the behavior inspectable.