Size functor

Given a size pair formula_1 where formula_2 is a manifold of dimension
formula_3 and formula_4 is an arbitrary real continuous function defined
on it, the formula_5-th size functor, with formula_6, denoted by formula_7, is the functor in formula_8, where formula_9 is the category of ordered real numbers, and formula_10 is the category of Abelian groups, defined in the following way. For formula_11, setting formula_12, formula_13, formula_14 equal to the inclusion from formula_15 into formula_16, and formula_17 equal to the morphism in formula_9 from formula_19 to formula_20,
In other words, the size functor studies the
process of the birth and death of homology classes as the lower level set changes.
When formula_2 is smooth and compact and formula_4 is a Morse function, the functor formula_26 can be
described by oriented trees, called formula_27 − 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 formula_28 can be seen as the rank
of the image of formula_29.
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 formula_30-th persistent homology group coincides with the image of the homomorphism formula_31.