Given a size pair
is a manifold of dimension
is an arbitrary real continuous function defined on it, the
-th size functor,[1] with
{\displaystyle Fun(\mathrm {Rord} ,\mathrm {Ab} )\ }
{\displaystyle \mathrm {Rord} \ }
is the category of ordered real numbers, and
{\displaystyle \mathrm {Ab} \ }
is the category of Abelian groups, defined in the following way.
{\displaystyle j_{xy}\ }
equal to the inclusion from
{\displaystyle k_{xy}\ }
equal to the morphism in
, In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes.
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
{\displaystyle H_{0}(j_{xy}):H_{0}(M_{x})\rightarrow H_{0}(M_{y})}
The concept of size functor is strictly related to the concept of persistent homology group,[2] studied in persistent homology.
It is worth to point out that the
-th persistent homology group coincides with the image of the homomorphism
{\displaystyle F_{i}(k_{xy})=H_{i}(j_{xy}):H_{i}(M_{x})\rightarrow H_{i}(M_{y})}