Simplicial Sets & Higher Categories

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Abstractions about the combinatorial encoding of homotopy structure via simplicial sets, including foundational definitions and variants (simplicial set, delta set, opposite simplicial set), operations combining or transforming them (join of simplicial sets, extension of a simplicial set), and their correspondence to categorical and homotopical structures (Dold-Kan correspondence, simplicially enriched category).

7 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.

• 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.
• Dold–Kan correspondence — An equivalence between simplicial abelian groups and nonnegatively graded chain complexes, matching homotopy groups with homology groups and simplicial homotopy with chain homotopy.
• 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.
• Join (simplicial sets) — A monoidal operation combining two simplicial sets so simplices consist of an ordered simplex from the first followed by one from the second, corresponding under realization to topological join.
• Opposite simplicial set — The simplicial set obtained by precomposing with the order-reversing automorphism of the simplex category, extending categorical arrow reversal to higher categorical models.
• Simplicial set — A contravariant functor from the simplex category to sets, equivalently graded simplices equipped with compatible face and degeneracy maps.
• Simplicially enriched category — A category whose hom-objects are simplicial sets and whose composition and identities are simplicial maps, encoding higher homotopies between morphisms.