About: Size functor

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Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ordered real numbers, and is the category of Abelian groups, defined in the following way. For , setting , , equal to the inclusion from into , and equal to the morphism in from to , * for each , *

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rdfs:label	Size functor (en)
rdfs:comment	Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ordered real numbers, and is the category of Abelian groups, defined in the following way. For , setting , , equal to the inclusion from into , and equal to the morphism in from to , * for each , * (en)
dcterms:subject	Algebraic topology Category theory
Wikipage page ID	16845636 (xsd:integer)
Wikipage revision ID	1116644088 (xsd:integer)
Link from a Wikipage to another Wikipage	Homology theory Homomorphism Continuous function Size pair Persistent homology Size homotopy group Morphism Functor Abelian groups Algebraic topology Category theory Manifold Category (mathematics) Category theory Size theory Size function Morse function dbr:Persistent_homology_group
sameAs	Size functor Size functor Size functor
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Context
has abstract	Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ordered real numbers, and is the category of Abelian groups, defined in the following way. For , setting , , equal to the inclusion from into , and equal to the morphism in from to , * for each , * In other words, the size functor studies theprocess of the birth and death of homology classes as the lower level set changes.When is smooth and compact and is a Morse function, the functor can bedescribed by oriented trees, called − 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 can be seen as the rankof the image of . The concept of size functor is strictly related to the concept of , studied in persistent homology. It is worth to point out that the -th persistent homology group coincides with the image of the homomorphism . (en)
prov:wasDerivedFrom	wikipedia-en:Size_functor?oldid=1116644088&ns=0
page length (characters) of wiki page	3044 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Size_functor
is Link from a Wikipage to another Wikipage of	Size homotopy group Matching distance Natural pseudodistance Size theory Size function
is foaf:primaryTopic of	wikipedia-en:Size_functor

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