Manifold & Simplicial Constructions

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Abstractions about differential structures, manifold doubling, simplicial extensions, toric manifolds, Haefliger structures, and geometric recognition criteria.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Conway criterion — A sufficient boundary-symmetry test guaranteeing that a topological-disk prototile can tile the plane by translations and half-turns.
• Delta set — A semi-simplicial object consisting of sets of n-simplices with face maps satisfying simplicial identities but no required degeneracy maps, providing flexible combinatorial models for gluing and homology.
• Diamond Operation — Combine two simplicial sets by gluing the endpoint faces of their product cylinder to the respective factors, producing a simplicial set over the 1-simplex that is categorically equivalent to their join.
• Differential Structure — A maximal compatible atlas that determines which coordinate descriptions on a topological manifold count as differentiable.
• Double (manifold) — The boundaryless manifold formed by gluing two copies of a manifold with boundary point-for-point along their entire common boundary.
• Extension (simplicial set) — The Ex endofunctor on simplicial sets, right adjoint to subdivision, that replaces a simplicial set by maps from subdivided simplices and iteratively improves horn-filling behavior.
• Haefliger structure — Encode generalized codimension-q foliation data by local maps to transverse Euclidean space whose transition germs form a cocycle, permitting pullback and classification beyond regular foliations.
• Toric manifold — A smooth compact even-dimensional manifold with an effective locally standard action of a half-dimensional torus and a simple convex polytope as orbit space.