Size functor

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Given a size pair (M,f) where M is a manifold of dimension n and f is an arbitrary real continuous function defined on it, the i-th size functor,[1] with i=0,…,n , denoted by Fi , is the functor in Fun(Rord,Ab) , where Rord is the category of ordered real numbers, and Ab is the category of Abelian groups, defined in the following way. For x≤y , setting Mx={p∈M:f(p)≤x} , My={p∈M:f(p)≤y} , jxy equal to the inclusion from Mx into My , and kxy equal to the morphism in Rord from x to y ,

* for each x∈ℝ , Fi(x)=Hi(Mx);
* Fi(kxy)=Hi(jxy).

In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes. When M is smooth and compact and f is a Morse function, the functor F0 can be described by oriented trees, called H0 − 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 ℓ(M,f)(x,y) can be seen as the rank of the image of H0(jxy):H0(Mx)->H0(My).

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 i -th persistent homology group coincides with the image of the homomorphism Fi(kxy)=Hi(jxy):Hi(Mx)->Hi(My).

## See also

* Size theory
* Size function
* Size homotopy group
* Size pair

## References

1. ↑ Cagliari, Francesca; Ferri, Massimo; Pozzi, Paola (2001). "Size functions from a categorical viewpoint". Acta Applicandae Mathematicae 67 (3): 225—235. doi:10.1023/A:1011923819754.
2. ↑ "Topological Persistence and Simplification". Discrete & Computational Geometry 28 (4): 511—533. 2002. doi:10.1007/s00454-002-2885-2.

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