# Size functor

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{{context|date=January 2018}}
Given a [size pair](/source/size_pair)  <math>(M,f)\ </math> where <math>M\ </math> is a [manifold](/source/manifold) of dimension
<math>n\ </math> and <math>f\ </math> is an arbitrary real [continuous function](/source/continuous_function)  defined
on it, the <math>i</math>-th '''size functor''',<ref name="CaFePo01">{{cite journal
| last1=Cagliari | first1=Francesca
| last2=Ferri | first2=Massimo
| last3=Pozzi | first3=Paola
| title=Size functions from a categorical viewpoint
| journal=Acta Applicandae Mathematicae
| volume=67
| issue=3
| pages=225–235
| date=2001
| doi=10.1023/A:1011923819754 | doi-access=free}}</ref> with <math>i=0,\ldots,n\ </math>, denoted by <math>F_i\ </math>, is the [functor](/source/functor) in <math>Fun(\mathrm{Rord},\mathrm{Ab})\ </math>, where  <math>\mathrm{Rord}\ </math> is the [category](/source/category_(mathematics)) of  ordered real numbers, and <math>\mathrm{Ab}\ </math> is the [category](/source/category_(mathematics)) of [Abelian groups](/source/Abelian_groups), defined in the following way. For <math>x\le y\ </math>, setting  <math>M_x=\{p\in M:f(p)\le x\}\ </math>, <math>M_y=\{p\in M:f(p)\le y\}\ </math>, <math>j_{xy}\ </math> equal to the inclusion from <math>M_x\ </math> into <math>M_y\ </math>, and <math> k_{xy}\ </math> equal to the [morphism](/source/morphism) in <math>\mathrm{Rord}\ </math> from <math>x\ </math> to <math>y\ </math>,

* for each <math>x\in\R\ </math>, <math>F_i(x)=H_i(M_x);\ </math>
* <math>F_i(k_{xy})=H_i(j_{xy}).\ </math>

In other words, the size functor  studies the
process of the birth and death of homology classes as the lower level set changes.
When <math>M\ </math> is smooth and compact and <math>f\ </math> is a [Morse function](/source/Morse_function), the functor <math>F_0\ </math> can be
described by oriented trees, called <math>H_0\ </math> − trees.

The concept of size functor was introduced as an extension to [homology theory](/source/homology_theory) and [category theory](/source/category_theory) of the idea of [size function](/source/size_function).  The main motivation for introducing the size functor originated  by the observation that the [size function](/source/size_function) <math>\ell_{(M,f)}(x, y)\ </math> can be seen as the rank
of the image of <math>H_0(j_{xy}) : H_0(M_x) \rightarrow H_0(M_y)</math>.

The concept of size functor is strictly related to the concept of [persistent homology group](/source/persistent_homology_group),<ref name="EdLeZo02">{{cite journal
| last1=Edelsbrunner | first1=Herbert | authorlink1=Herbert Edelsbrunner
| last2=Letscher | first2=David
| last3=Zomorodian | first3=Afra
| title=Topological Persistence and Simplification
| journal=[Discrete & Computational Geometry](/source/Discrete_%26_Computational_Geometry)
| volume=28
| issue=4
| pages=511–533
| date=2002
| doi=10.1007/s00454-002-2885-2 | doi-access=free}}</ref> studied in [persistent homology](/source/persistent_homology). It is worth to point out that the <math>i\ </math>-th persistent homology group coincides with the image of the [homomorphism](/source/homomorphism) <math>F_i(k_{xy})=H_i(j_{xy}): H_i(M_x) \rightarrow H_i(M_y)</math>.

==See also==
* [Size theory](/source/Size_theory)
* [Size function](/source/Size_function)
* [Size homotopy group](/source/Size_homotopy_group)
* [Size pair](/source/Size_pair)

==References==
{{reflist}}

{{DEFAULTSORT:Size Functor}}
Category:Algebraic topology
Category:Category theory

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Adapted from the Wikipedia article [Size functor](https://en.wikipedia.org/wiki/Size_functor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Size_functor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.