Fixed points of functors
Result description
This is a survey on fixed points of endofunctors, including initial algebras and terminal coalgebras. We also consider the rational fixed point, a canonical domain of behavior for finitely presentable systems. In addition to the basic existence theorems for fixed points, several new results are presented. For example, the Smyth-Plotkin theorem that locally continuous endofunctors of DCPO have terminal coalgebras is derived from a new result stating that every locally monotone endofunctor with a fixed point has a terminal coalgebra. We introduce bounded endofunctors on abstract categories and prove that they have terminal coalgebras. We study well-founded coalgebras and prove that for set functors, the largest well-founded coalgebra of every fixed point is the initial algebra. Another new result concerns mixed fixed points: initial algebras and terminal coalgebras of a parametrized accessible functor always form accessible functors. (C) 2017 Elsevier Inc. All rights reserved.
Keywords
CATEGORY-THEORETIC SOLUTIONTERMINAL COALGEBRASITERATIVE ALGEBRASDOMAIN EQUATIONSINFINITE-TREESMETRIC-SPACESSET FUNCTORSEMANTICSSCHEMESSYSTEMS
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Fixed points of functors
Original language description
This is a survey on fixed points of endofunctors, including initial algebras and terminal coalgebras. We also consider the rational fixed point, a canonical domain of behavior for finitely presentable systems. In addition to the basic existence theorems for fixed points, several new results are presented. For example, the Smyth-Plotkin theorem that locally continuous endofunctors of DCPO have terminal coalgebras is derived from a new result stating that every locally monotone endofunctor with a fixed point has a terminal coalgebra. We introduce bounded endofunctors on abstract categories and prove that they have terminal coalgebras. We study well-founded coalgebras and prove that for set functors, the largest well-founded coalgebra of every fixed point is the initial algebra. Another new result concerns mixed fixed points: initial algebras and terminal coalgebras of a parametrized accessible functor always form accessible functors. (C) 2017 Elsevier Inc. All rights reserved.
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Logical and Algebraic Methods in Programming
ISSN
2352-2208
e-ISSN
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Volume of the periodical
95
Issue of the periodical within the volume
FEB
Country of publishing house
US - UNITED STATES
Number of pages
41
Pages from-to
41-81
UT code for WoS article
000425079300003
EID of the result in the Scopus database
2-s2.0-85043482597
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2018