Size functor

Given a size pair ${\displaystyle (M,f)\ }$ where ${\displaystyle M\ }$ is a manifold of dimension ${\displaystyle n\ }$ and ${\displaystyle f\ }$ is an arbitrary real continuous function defined on it, the ${\displaystyle i}$-th size functor,¹ with ${\displaystyle i=0,\ldots ,n\ }$, denoted by ${\displaystyle F_{i}\ }$, is the functor in ${\displaystyle Fun(\mathrm {Rord} ,\mathrm {Ab} )\ }$, where ${\displaystyle \mathrm {Rord} \ }$ is the category of ordered real numbers, and ${\displaystyle \mathrm {Ab} \ }$ is the category of Abelian groups, defined in the following way. For ${\displaystyle x\leq y\ }$, setting ${\displaystyle M_{x}=\{p\in M:f(p)\leq x\}\ }$, ${\displaystyle M_{y}=\{p\in M:f(p)\leq y\}\ }$, ${\displaystyle j_{xy}\ }$ equal to the inclusion from ${\displaystyle M_{x}\ }$ into ${\displaystyle M_{y}\ }$, and ${\displaystyle k_{xy}\ }$ equal to the morphism in ${\displaystyle \mathrm {Rord} \ }$ from ${\displaystyle x\ }$ to ${\displaystyle y\ }$,

• for each ${\displaystyle x\in \mathbb {R} \ }$, ${\displaystyle F_{i}(x)=H_{i}(M_{x});\ }$
• ${\displaystyle F_{i}(k_{xy})=H_{i}(j_{xy}).\ }$

In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes. When ${\displaystyle M\ }$ is smooth and compact and ${\displaystyle f\ }$ is a Morse function, the functor ${\displaystyle F_{0}\ }$ can be described by oriented trees, called ${\displaystyle H_{0}\ }$ − 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 ${\displaystyle \ell _{(M,f)}(x,y)\ }$ 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 ${\displaystyle i\ }$-th persistent homology group coincides with the image of the homomorphism ${\displaystyle F_{i}(k_{xy})=H_{i}(j_{xy}):H_{i}(M_{x})\rightarrow H_{i}(M_{y})}$.