A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex.[1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this is formalized by the simplicial approximation theorem.
A simplicial isomorphism is a bijective simplicial map such that both it and its inverse are simplicial.
^Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.
A simplicialmap (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 algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of...
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by...
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...
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 simplicialmap). In each case, the function may be real-valued, or it...
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...
functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on...
each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicialmap φi of Li' onto the link Li...
simplicial setspg 1.3 there is a model category structure where the fibrations are precisely the Kan fibrations, cofibrations are all injective maps,...
{\partial _{4}}}C_{4}{\xleftarrow {\partial _{5}}}\cdots } A simplicialmap is a map between simplicial complexes with the property that the images of the vertices...
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...
together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the...
: 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...
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...
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...
category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used...
geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary representation model; it...
part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore...
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether...
algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the...
\sigma |_{[v_{0},\ldots ,v_{q}]}} indicates the restriction of the simplicialmap σ {\displaystyle \sigma } to its face spanned by the vectors of the...
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...
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 inclusions...
by other models like simplicial and cell complexes. Combinatorial Complexes : Generalize and bridge the gaps between simplicial complexes, cell complexes...
a functor, taking K to XK, from the category of finite simplicial complexes and simplicialmaps and a natural weak homotopy equivalence from |K| to XK...
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