mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional...

In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices...

In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn...

join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to construct...

homology of the simplicial set S ∗ X {\displaystyle S_{*}X} . Also, the geometric realization | ⋅ | {\displaystyle |\cdot |} of a simplicial set is a CW complex...

simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets...

combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking...

pro-simplicial set is an inverse system of simplicial sets. A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets...

mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable...

In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation...

complex can be defined as the set of homotopy classes of 1-simplices. The fundamental group of an arbitrary simplicial set X are defined to be the homotopy...

skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set K ∗ {\displaystyle...

In higher category theory in mathematics, the opposite simplicial set (or dual simplicial set) is an operation extending the opposite category (or dual...

spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R-modules...

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if f , g : X →...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

a simplicial set A {\displaystyle A} define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:...

theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition. Rational Hurewicz theorem: Let X be a...

part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of...

especially algebraic topology, a weak equivalence between simplicial sets is a map between simplicial sets that is invertible in some weak sense. Formally, it...

that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the...

bisimplicial set is a simplicial object in the category of simplicial sets, which themselves are simplicial objects in the category of sets. Many concepts...

topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse system of simplicial sets one may...

product. This is the Čech nerve. By taking connected components we get a simplicial set, which we can realise topologically: | S ( π 0 ( C ) ) | {\displaystyle...

the independent sets of the graph. Formally, the independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that...

Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these...

simplicial sets is an operation taking two simplicial sets to construct another simplicial set. It is closely related to the join of simplicial sets and...

small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space...

{V} } as a V {\displaystyle \mathbf {V} } -valued presheaf. A simplicial set is a Set-valued presheaf on the simplex category C = Δ {\displaystyle C=\Delta...

A subdivision (also called refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original...