In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
presented as a
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
(resp.
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
) refers to the
subspace that is the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of the
simplices
In geometry, a simplex (plural: simplexes or simplices) 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 dimension. ...
of (resp. cells of ) of dimensions In other words, given an
inductive definition
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include fact ...
of a complex, the is obtained by stopping at the .
These subspaces increase with . The is a
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, and the a
topological graph. The skeletons of a space are used in
obstruction theory In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the ex ...
, to construct
spectral sequences by means of
filtrations, and generally to make
inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as
In geometry
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a of P (functionally represented as skel
''k''(''P'')) consists of all elements of dimension up to ''k''.
For example:
: skel
0(cube) = 8 vertices
: skel
1(cube) = 8 vertices, 12 edges
: skel
2(cube) = 8 vertices, 12 edges, 6 square faces
For simplicial sets
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a
simplicial set
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 a ...
. 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
. (The notations
are comparable with the one of
image functors for sheaves
In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses.
Given a continuous mapping ''f'': ''X'' → ...
.) The ''n''-skeleton of some simplicial set
is defined as
:
Coskeleton
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
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
:
(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 product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...
s. The coskeleton is needed to define the concept of
hypercovering In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every ...
in
homotopical algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.
References
External links
* {{MathWorld , urlname=Skeleton , title=Skeleton
Algebraic topology
General topology