Homotopy Nerve

Definition

The homotopy nerve is the full homotopy-theoretic refinement of the constraint nerve. While the basic Homology pipeline extracts chain-level invariants (Betti numbers), the homotopy nerve promotes the nerve to a KanComplex carrying the complete homotopy type — including higher homotopy groups, Postnikov truncations, and k-invariants.

A KanComplex is a simplicial set satisfying the Kan extension condition: every horn (incomplete simplex boundary) can be filled. The HornFiller witnesses certify this condition at each dimension and position, with hornDimension and hornPosition recording the filled horn.

Postnikov Tower

The PostnikovTruncation tau_{<=k} is the k-th stage of the Postnikov tower: a KanComplex whose homotopy groups pi_j vanish for j > k. The tower

tau_{<=0} <- tau_{<=1} <- tau_{<=2} <- ...

successively approximates the full homotopy type. Each stage is linked to its source via truncationSource and carries a truncationLevel.

The KInvariant kappa_k classifies the extension from the (k-1)-truncation to the k-truncation. When kInvariantTrivial is true, the truncation splits as a product — the homotopy type decomposes at that level.

Homotopy Observables

The homotopy nerve yields three families of observables in the observable/ namespace:

• HomotopyGroup — the k-th homotopy group pi_k(N(C), v) based at vertex v. For k=0 this is the set of path components; for k=1 the fundamental group; for k>=2 these are abelian groups detecting higher obstructions. Properties: homotopyDimension, homotopyRank, homotopyBasepoint.

• HigherMonodromy — the image of pi_k into the automorphism group of the site. Generalises the Monodromy pi_1 -> D_{2^n} homomorphism to higher groups. Property: higherMonodromyDimension.

• WhiteheadProduct — the bracket [alpha, beta] detecting non-trivial interactions between homotopy groups that cohomology alone misses. Property: whiteheadTrivial.

Bridge to Spectral Sequences

The postnikovTruncation property on SpectralSequencePage links each page to the corresponding Postnikov truncation level, connecting the spectral sequence machinery of Quantum Spectral Sequence to the homotopy-nerve tower.

Key Identities