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Size theory

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated September 10, 2026

In mathematics, size theory studies the properties of topological spaces endowed with \mathbb {R} ^{k}-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs. A survey of size theory can be found in .

01History and applications

The beginning of size theory is rooted in the concept of size function, introduced by Frosini. Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern recognition.

An extension of the concept of size function to algebraic topology was made in the 1999 Frosini and Mulazzani paper where size homotopy groups were introduced, together with the natural pseudodistance for \mathbb {R} ^{k}-valued functions. An extension to homology theory (the size functor) was introduced in 2001. The size homotopy group and the size functor are strictly related to the concept of persistent homology group studied in persistent homology. It is worth to point out that the size function is the rank of the 0-th persistent homology group, while the relation between the persistent homology group and the size homotopy group is analogous to the one existing between homology groups and homotopy groups.

In size theory, size functions and size homotopy groups are seen as tools to compute lower bounds for the natural pseudodistance. Actually, the following link exists between the values taken by the size functions \ell _{(N,\psi )}({\bar {x}},{\bar {y}}), \ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}}) and the natural pseudodistance d((M,\varphi ),(N,\psi )) between the size pairs (M,\varphi ),\ (N,\psi ) ,

{\text{If }}\ell _{(N,\psi )}({\bar {x}},{\bar {y}})>\ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}}){\text{ then }}d((M,\varphi ),(N,\psi ))\geq \min\{{\tilde {x}}-{\bar {x}},{\bar {y}}-{\tilde {y}}\}.

An analogous result holds for size homotopy group.

The attempt to generalize size theory and the concept of natural pseudodistance to norms that are different from the supremum norm has led to the study of other reparameterization invariant norms.

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Sources and credits

This article is adapted from the Wikipedia article “Size theory”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x < y {\displaystyle x<y} to the natural numbers, counting certain connected components of a topological space.

Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs {\displaystyle (M,\varphi :M\to \mathbb {R} )\ } , ( N , ψ : N → R ) {\displaystyle (N,\psi :N\to \mathbb {R} )\ } is the value inf h ‖ φ − ψ ∘ h ‖ ∞ {\displaystyle \inf _{h}\|\varphi -\psi \circ h\|_{\infty }\ } , where h {\displaystyle h\ } varies in the set of all homeomorphisms from the manifold M {\displaystyle M\ } to the manifold N {\displaystyle N\ } and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }\ } is the supremum norm. If M {\displaystyle M\ } and N {\displaystyle N\ } are not homeomorphic, then the natural pseudodistance is defined to be ∞ {\displaystyle \infty \ } .

Size functor

Given a size pair {\displaystyle (M,f)\ } where M {\displaystyle M\ } is a manifold of dimension n {\displaystyle n\ } and f {\displaystyle f\ } is an arbitrary real continuous function defined on it, the i {\displaystyle i} -th size functor, with i = 0 , … , n {\displaystyle i=0,\ldots ,n\ } , denoted by F i {\displaystyle F_{i}\ } , is the functor in F u n ( R o r d , A b ) {\displaystyle Fun(\mathrm {Rord} ,\mathrm {Ab} )\ } , where R o r d {\displaystyle \mathrm {Rord} \ } is the category of ordered real numbers, and A b {\displaystyle \mathrm {Ab} \ } is the category of Abelian groups, defined in the following way. For x ≤ y {\displaystyle x\leq y\ } , setting M x = { p ∈ M : f ( p ) ≤ x } {\displaystyle M_{x}=\{p\in M:f(p)\leq x\}\ } , M y = { p ∈ M : f ( p ) ≤ y } {\displaystyle M_{y}=\{p\in M:f(p)\leq y\}\ } , j x y {\displaystyle j_{xy}\ } equal to the inclusion from M x {\displaystyle M_{x}\ } into M y {\displaystyle M_{y}\ } , and k x y {\displaystyle k_{xy}\ } equal to the morphism in R o r d {\displaystyle \mathrm {Rord} \ } from x {\displaystyle x\ } to y {\displaystyle y\ } , for each x ∈ R {\displaystyle x\in \mathbb {R} \ } , F i ( x ) = H i ( M x ) ; {\displaystyle F_{i}(x)=H_{i}(M_{x});\ } F i ( k x y ) = H i ( j x y ) .