Given a
size pair where
is a
manifold of dimension
and
is an arbitrary real
continuous function defined
on it, the
-th size functor,
with
, denoted by
, is the
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
in
, where
is the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of ordered real numbers, and
is the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
Abelian groups
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, defined in the following way. For
, setting
,
,
equal to the inclusion from
into
, and
equal to the
morphism in
from
to
,
* for each
,
*
In other words, the size functor studies the
process of the birth and death of homology classes as the lower level set changes.
When
is smooth and compact and
is a
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
, the functor
can be
described by oriented trees, called
− trees.
The concept of size functor was introduced as an extension to
homology theory and
category theory of the idea of
size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain Connected component (topology), connected components of a topological space. They a ...
. The main motivation for introducing the size functor originated by the observation that the
size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain Connected component (topology), connected components of a topological space. They a ...
can be seen as the rank
of the image of
.
The concept of size functor is strictly related to the concept of
persistent homology group,
studied in
persistent homology :''See homology for an introduction to the notation.''
Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and a ...
. It is worth to point out that the
-th persistent homology group coincides with the image of the
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
.
See also
*
Size theory In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance ...
*
Size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain Connected component (topology), connected components of a topological space. They a ...
*
Size homotopy group
*
Size pair
References
{{DEFAULTSORT:Size Functor
Algebraic topology
Category theory