Size functor
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, where is the category of ordered real numbers, and is the category of Abelian groups, defined in the following way. For, setting ,, equal to the inclusion from into, and equal to the morphism in from to,
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, 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. The main motivation for introducing the size functor originated by the observation that the size function 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. It is worth to point out that the -th persistent homology group coincides with the image of the homomorphism.
