Homotopy Pipeline

The homotopy pipeline extends the psi pipeline with three additional stages (psi_7 through psi_9) that compute the full homotopy type of the constraint nerve.

Extended Pipeline Stages

The HomotopyResolver runs stages psi_7 through psi_9 on a CechNerve:

The resolver connects via homotopyTarget (the input CechNerve) and homotopyResult (the output HomotopyGroup observables).

Prerequisite: Kan Promotion

Before entering psi_7, the CechNerve (already a SimplicialComplex from psi_1) must be promoted to a KanComplex. Identity HT_1 certifies that the finite constraint nerve always satisfies the Kan extension condition. The HornFiller witnesses provide constructive proof.

Worked Example

Continuing the 3-constraint example from the psi pipeline guide (constraints C_1, C_2, C_3 on R_4 producing a path nerve C_1 — C_2 — C_3):

psi_7 — Postnikov tower. The nerve has dim = 1 (edges only), so we compute tau_{<=0} and tau_{<=1}:

• tau_{<=0}: the set of path components (one component, pi_0 = Z)
• tau_{<=1}: the full 1-type (pi_1 = 0 since the nerve is a tree)

psi_8 — Homotopy groups. Extract:

• pi_0 = Z (one connected component) — confirms beta_0 = 1 from psi_4
• pi_1 = 0 (no loops) — confirms the nerve is simply connected
• pi_k = 0 for k >= 2 (dim = 1 bound, identity HT_4)

psi_9 — K-invariants. The single k-invariant kappa_1 classifying the extension from tau_{<=0} to tau_{<=1} is trivial (since pi_1 = 0). The homotopy type is a product — confirming the nerve has no higher structure beyond connected components.

Pipeline Composition

Identity HP_1 states that psi_7 composed with psi_1 equals Kan promotion of the nerve. Identity HP_2 ensures that psi_8 restricted to the k-truncation agrees with psi_3 restricted to the k-skeleton. Identity HP_3 connects psi_9 to the QLS_4 spectral sequence convergence.

The overall complexity of the HomotopyResolver is O(n^{d+1}) where n is the number of constraints and d is the nerve dimension (HP_4).