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A simplicial set X is a contravariant functor. X : Δ → Set. where Set is the category of sets. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the opposite category Δ op →f Set.) Given a simplicial set X, we often write X n instead of X([n]).
A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th
A simplicial set is called a Kan complex if the map from {}, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, { ∗ } {\displaystyle \{*\}} is the terminal object and so a Kan complex is exactly the same as a fibrant object .
Formally, a Δ-set is a sequence of sets {} = together with maps : + for each and =,, …, +, that satisfy + = + whenever <.Often, the superscript of is omitted for brevity.. This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the are the associated face maps, each mapping the -th face of a simplex in + to a simplex in .
Simplicial sets may also be regarded as functors Δ op → Set, where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i ( P ) with objects the elements of P and with a unique morphism from p to q whenever p ≤ q in P .
Simplicial approximation theorem; Simplicial complex; Simplicial complex recognition problem; Simplicial group; Simplicial homotopy; Simplicial manifold; Simplicial map; Simplicial presheaf; Subdivision (simplicial complex) Symmetric spectrum
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A set-family Δ is called an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y ⊆ X, the set Y also belongs to Δ. The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊆ X , so the definition of an abstract simplicial complex can be restated ...