+++ title = “The Playground” section = “features” +++

The Playground

The Playground is where you work with your own spectral sequences. You can import module definitions, explore the resulting Ext charts, and annotate them with differentials, labels, and notes.

Importing modules

There are several ways to add a module to the Playground:

• Build a new module using the Module Builder
• Import a JSON file — either a module definition (which htpy will resolve on the fly) or a pre-computed spectral sequence in v1 or v2 format
• Attach a cell to an existing module to construct a cofiber (see Cell Complexes and Cofibers)
• Compute from a template — select a spectrum from the compute page and specify the desired range

Working with differentials

See Working with Differentials for the full guide. In summary:

• Add differentials — specify $d_r$ by clicking source and target
• Mark $d_r = 0$ — record that a class survives page $r$
• Mark $d_r = {?}$ — flag as unresolved (blocks deduction)
• Delete differentials — removes the differential and all propagated consequences, replaying from scratch

The deduction engine

See The Deduction Engine for the full guide. The “Deduce” button runs the automated reasoning engine; “Deep Deduce” adds elimination by contradiction.

Vanishing lines

You can specify vanishing lines — linear constraints above which all classes vanish. Format: $a \cdot s > b \cdot n + c$ with optional stem bounds. These are used by the deduction engine and are automatically propagated through cofiber sequences.

Labels and annotations

• Custom labels on generators at any bidegree — overrides the auto-inferred name from product propagation
• Named combinations — named linear combinations of basis elements (useful for identifying specific classes in higher-dimensional groups)
• Detects — record which stable homotopy element each generator detects (e.g., $h_0$ detects 2, $h_1$ detects $\eta$)
• Annotations — notes with tags (question, todo, proof, reference) attached to classes, differentials, or structlines

All user data is saved automatically to ~/.htpy/ and persists across sessions.

Recomputing

You can recompute a module at a different range (larger stem or filtration) from its module definition. This re-resolves the module and regenerates the Ext data. You can choose whether to keep or discard existing user differentials.

Computation options

From the compute pages, you can specify: