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

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed...

In 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...

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of...

the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation...

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by...

linear on the simplices and which are homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement...

affine space, as well as on piecewise linear manifolds and simplicial complexes (see simplicial map). In each case, the function may be real-valued, or it...

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

fixed-point-free simplicial map (i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced...

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...

each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link Li...

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...

right by 0. An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain complex is bounded above if all modules...

{\partial _{4}}}C_{4}{\xleftarrow {\partial _{5}}}\cdots } A simplicial map is a map between simplicial complexes with the property that the images of the vertices...

simplicial setspg 1.3 there is a model category structure where the fibrations are precisely the Kan fibrations, cofibrations are all injective maps,...

: X → Y {\displaystyle f,g:X\to Y} are maps between simplicial sets, a simplicial homotopy from f to g is a map h : X × Δ 1 → Y {\displaystyle h:X\times...

all maps from the n-simplex to X {\displaystyle X} . Then the sequence S n X {\displaystyle S_{n}X} of sets is a simplicial set. Each simplicial set K...

category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used...

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

together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the...

geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary representation model; it...

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

algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the...

pyramid is a 3-simplex.) Simplicial homology can in turn be generalized to singular homology, which allows more general maps of simplices into the topological...

\sigma |_{[v_{0},\ldots ,v_{q}]}} indicates the restriction of the simplicial map σ {\displaystyle \sigma } to its face spanned by the vectors of the...

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

The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether...

to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations. If X is a connected simplicial complex...

the k-th homology group of a simplicial complex depends only on the simplices of dimension at most k+1 (see simplicial homology). Therefore, the above...