Size functor explained

(M,f) where M is a manifold of dimensionn and f is an arbitrary real continuous function definedon it, the i -th size functor,^[1] with i =0 ,\ldots,n , denoted by F _{i} , 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\le y , setting M _{x=\{p\in} M:f(p)\le x\} , M _{y=\{p\in} M:f(p)\le y\} , j _{xy} equal to the inclusion from M _{x} into M _{y} , and k _{xy} equal to the morphism in Rord from x to y ,

• for each

x\in\R , F _{i(x)=H} _{i(M} _{x);} F _{i(k} _{xy} ) = H _{i(j} _{xy} ).

In other words, the size functor studies theprocess 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 F _{0} can bedescribed by oriented trees, called H _{0} − trees.\ell _{(M,f)} (x, y) can be seen as the rankof the image of H _{0(j} _{xy} ) : H _{0(M} _{x)} → 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

i -th persistent homology group coincides with the image of the homomorphism F _{i(k} _{xy} ) = H _{i(j} _{xy} ): H _{i(M} _{x)} → H _{i(M} _{y)} .

See also

• Size theory
• Size function
• Size homotopy group
• Size pair

Notes and References

1. Cagliari . Francesca. Ferri . Massimo. Pozzi . Paola. Size functions from a categorical viewpoint. Acta Applicandae Mathematicae. 67. 3. 225—235. 2001. 10.1023/A:1011923819754 . free.
2. Edelsbrunner . Herbert . Herbert Edelsbrunner. Letscher . David. Zomorodian . Afra. Topological Persistence and Simplification. Discrete & Computational Geometry. 28. 4. 511—533. 2002. 10.1007/s00454-002-2885-2 . free.