In mathematics, particularly in algebraic topology, the **n-skeleton** of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace X_{n} that is the union of the simplices of X (resp. cells of X) of dimensions *m* ≤ *n*. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step.

These subspaces increase with n. The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when X has infinite dimension, in the sense that the X_{n} do not become constant as *n* → ∞.

In geometry, a *k*-skeleton of *n*-polytope P (functionally represented as skel_{k}(*P*)) consists of all *i*-polytope elements of dimension up to *k*.^{ [1] }

For example:

- skel
_{0}(cube) = 8 vertices - skel
_{1}(cube) = 8 vertices, 12 edges - skel
_{2}(cube) = 8 vertices, 12 edges, 6 square faces

The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set can be described by a collection of sets , together with face and degeneracy maps between them satisfying a number of equations. The idea of the *n*-skeleton is to first discard the sets with and then to complete the collection of the with to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees .

More precisely, the restriction functor

has a left adjoint, denoted .^{ [2] } (The notations are comparable with the one of image functors for sheaves.) The *n*-skeleton of some simplicial set is defined as

Moreover, has a *right* adjoint . The *n*-coskeleton is defined as

For example, the 0-skeleton of *K* is the constant simplicial set defined by . The 0-coskeleton is given by the Cech nerve

(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)

The above constructions work for more general categories (instead of sets) as well, provided that the category has fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.^{ [3] }

In the mathematical field of algebraic topology, the **fundamental group** of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups. The fundamental group of a topological space is denoted by .

In geometry, a **simplex** is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, a **simplicial complex** is a set composed of points, line segments, triangles, and their *n*-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

In mathematics, a **building** is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.

In the mathematical disciplines of topology and geometry, an **orbifold** is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.

A **CW complex** is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The *C* stands for "closure-finite", and the *W* for "weak" topology. A CW complex can be defined inductively.

In algebraic topology, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In geometry, the **barycentric subdivision** is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.

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 subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles, their edges, and their vertices.

In mathematics, a **simplicial set** is an object composed of *simplices* in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber.

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 a given dimension in the complex. This generalizes the number of connected components.

In mathematics, the **homotopy category** is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In category theory, a discipline within mathematics, the **nerve***N*(*C*) of a 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, called the **classifying space of the category***C*. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory.

In mathematics, **Kan complexes** and **Kan fibrations** are 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 fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.

In mathematics, a **Δ-set***S*, often called a **semi-simplicial set**, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

In mathematics, a **topos** is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The **Grothendieck topoi** find applications in algebraic geometry; the more general **elementary topoi** are used in logic.

In mathematics, more specifically category theory, a **quasi-category** is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.

In algebraic topology, the *n*^{th} symmetric product of a topological space consists of the unordered *n*-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor.

In mathematics, especially in algebraic topology, the **homotopy limit and colimit**^{pg 52} are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram

- ↑ Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)
- ↑ Goerss, P. G.; Jardine, J. F. (1999),
*Simplicial Homotopy Theory*, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1 , section IV.3.2 - ↑ Artin, Michael; Mazur, Barry (1969),
*Etale homotopy*, Lecture Notes in Mathematics, No. 100, Berlin, New York: Springer-Verlag

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