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...

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...

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...

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

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:...

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

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...

functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on...

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...

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

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...

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...

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

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

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

either the category of sets or the category of simplicial sets to the category of manifolds. A simplicial manifold is a simplicial complex for which the...

A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of...

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

theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object...

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...

if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to...

In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often...

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...

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...

Denis-Charles Cisinksi. It focuses on the theories of model categories and simplicial sets to give access to modern homotopy theory from the perspective of higher...

U, n ≥ 0 {\displaystyle n\geq 0} ) by a single point. This gives a simplicial set which captures some information related to X and the étale topology...

fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more.: 9–11  An abstract simplicial complex (ASC) is family of sets that...

definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations...